Algorithms Composition Approach Based on Difference Potentials Method
for Parabolic Problems
Yekaterina Epshteyn1
Abstract. In this work we develop an efficient and flexible Algorithms Composition Approach based on the idea of the
difference potentials method (DPM) for parabolic problems in composite and complex domains. Here, the parabolic equation
serves both as the simplified model, and as the first step towards future development of the proposed framework for more
realistic systems of materials, fluids, or chemicals with different properties in the different domains. Some examples of such
models include the ocean-atmosphere models, chemotaxis models in biology, and blood flow models. Very often, such models
are heterogeneous systems - described by different types of partial differential equations (PDEs) in different domains, and they
have to take into consideration the complex structure of the computational subdomains. The major challenge here is to design
an efficient and flexible numerical method that can capture certain properties of analytical solutions in different domains, while
handling the arbitrary geometries and complex structures of the subdomains. The Algorithms Compositions principle, as well as
the Domain Decomposition idea, is one way to overcome these difficulties while developing very efficient and accurate numerical
schemes for the problems. The Algorithms Composition Approach proposed here can handle the complex geometries of the
domains without the use of unstructured meshes, and can be employed with fast Poisson solvers. Our method combines the
simplicity of the finite difference methods on Cartesian meshes with the flexibility of the Difference Potentials method. The
developed method is very well suited for parallel computations as well, since most of the computations in each domain are
performed independently of the others.
AMS subject classification: 65M06, 65M22, 65M55, 65M70, 35K05, 35K20
Key words: Parabolic equation, heat equation, Calderon’s potentials, Calderon’s boundary equations with
projections, difference potentials methods, finite difference, Cartesian meshes, complex and composite domains, curvilinear boundaries, algorithms composition, and domain decomposition.
1. Introduction
In this work we develop an efficient and flexible Algorithms Composition Approach based on the idea of
the Difference Potentials Method (DPM) for parabolic problems in composite and complex domains.
Here, a parabolic equation (2.1) serves both as the simplified model, and as the first step towards future
development of the proposed scheme for more realistic systems of materials, fluids, or chemicals with different
properties in the different domains (or in the different parts of the domains). Some examples of such models
include the ocean-atmosphere models, chemotaxis models in biology, and blood flow models (see for example
[2, 36, 53, 10, 9, 43, 42]).
Numerical approximations and modeling of many physical, biological, and biomedical problems often deal
with heterogeneous models (described by different types of partial differential equations (PDEs) in different
domains), and/or they have to take into consideration the complex structure of the computational subdomains. The major challenge here is to design an efficient and flexible numerical method that can capture
certain properties of analytical solutions in different domains/subdomains (such as positivity, different regularity/smoothness of the solutions in the domains/subdomains, etc), while handling the arbitrary geometries
and complex structures of the domains. The Algorithms Compositions principle, as well as the Domain
Decomposition idea, is one way to overcome these difficulties while developing very efficient and accurate
numerical schemes for the problems. This methodology can be used within any discretization for PDEs (such
as finite differences, finite volumes, finite elements, or spectral methods). It provides great opportunities
to subdivide problems into subproblems, and to design the most suitable numerical approximation for each
of them independently. After that, one can compose the problems and algorithms together by imposing
some interface conditions. Such schemes can be used for parallel computations as well, since most of the
computations in each subdomain are performed independently of the others (see for example, [41, 51]).
There is an extensive literature that addresses problems in domains with complex geometries and interface
problems. We will briefly discuss below some established finite difference methods for such problems. For
more detailed review on the subject the reader can consult for example [25]. The immersed boundary method
(IB) was originally proposed by Peskin to model blood flow in a human heart (see for example [38, 39]).
One of the essential ideas of the IB method is to employ a discrete delta function to place/spread a singular
source to neighboring mesh points. The IB method is simple and efficient but in most cases is a low-order
(first-order) method. The higher-order (second order) version of the IB method has been recently proposed in
[19]. The IB method has been applied to many problems in computational fluid dynamics and mathematical
biology (see for example [37, 40, 12, 54]), and it has been parallelized [33].
∗ Department
of Mathematics, The University of Utah, Salt Lake City, UT, 84112, epshteyn@math.utah.edu
1
The immersed interface method (IIM) is designed for interface problems and problems defined on irregular
domains [23, 24, 25]. This method is a sharp interface method for PDEs that can have discontinuities in the
coefficients, the solution and its derivatives, and it can handle Dirac singularities in the source terms. The
IIM is based on Cartesian meshes with second order or fourth order (for some problems) accuracy. Standard
finite difference or some standard finite element schemes can be employed as the core discretization in IIM.
The IIM modifies these schemes near or at the interfaces/boundaries through the interface relations so that
second order/or fourth order accuracy can be achieved in the whole domain. However, some of the difficulties
with IIM schemes are the requirement for the explicit knowledge of the jump conditions at the interfaces and
boundaries for the development of IIM (in order to derive the correction terms for the schemes at the mesh
points near the interface), and that the parallelization of IIM is not a trivial task due to global coupling of
the solutions in the subdomains. The IIM is used for many problems such as Stefan problems, incompressible
Stokes and Navier-Stokes flow problems, etc, (see for example [25]). The other sharp interface method is
the ghost fluid method (GFM). This method was originally proposed for the accurate approximation of the
boundary conditions for hyperbolic systems [13], and was later developed in [26] (the convergence was proved
in [27]) for the elliptic interface problems. The GFM is based on a Cartesian grid finite-difference method.
Similar to IIM, the main idea of GFM [26] is to incorporate the jump conditions into the finite difference
scheme. However, the idea of GFM is to decompose the flux jump in dimension by dimension fashion which
may decrease the accuracy of the method in some cases. The main advantages of GFM are its second-order
accuracy, simplicity, and ease of implementation. Some shortcomings of GFM are that fast Poisson solvers
cannot be employed with the method, since the coefficients of the finite difference discretization are modified
at the mesh points near the boundaries; the GFM may produce only a first-order accurate solution for more
general boundary conditions (for example for mixed boundary conditions). The GFM has been applied
to several problems including the simulation of incompressible flame, incompressible multiphase flows, etc,
and it has been applied to complex domain problems with Dirichlet boundary conditions (see for example
[35, 20, 15]). Let us comment that in [15], the ghost values across the interface were defined using dimension
by dimension extrapolation, and the overall second order accuracy was achieved for the complex domain
problems with Dirichlet boundary conditions (instead of explicit information about the jump conditions
across the interface). Also, based on the ideas from [15], a fourth-order scheme has been developed in [14]
for Dirichlet boundary conditions on domains with complex geometry.
Let us now mention the method based on the integral equations approach. In [30], the fast and high-order
methods were developed for solving Laplace’s and biharmonic equations on complex domains with smooth
boundaries in 2D. The idea of the method proposed in [30] is to combine integral equations based on the
single and double layer theory with finite difference method to solve a Laplace’s equation on a complex
domain. In this method the domain is embedded in a computationally simple region (auxiliary domain)
where a fast Poisson solver can be used on a uniform mesh. The right-hand side of the original equation
is modified appropriately in the auxiliary domain using the information about the values of the extended
solution at the mesh points near (inside and outside) the original continuous boundary (at the irregular
mesh points). The approximations to the unknown values of the extended solution at these irregular mesh
points are constructed through Taylor expansion and the solution of the Fredholm integral equation of the
second kind. Finally, the approximation of the solution in the original region is obtained by the use of the
fast Poisson solver in the simple auxiliary domain. Based on these ideas, the method introduced in [30]
avoids the common difficulty with the solution of the integral boundary equations (for example problems
near the original boundary). This approach has been parallelized in [29, 17]. The method can be accelerated
if coupled with a fast multipole method [32]. The possibility of extension to elliptic problems with variable
coefficients is mentioned in [31]. Note that, in general, the methods based on integral equations approaches
are very efficient for homogeneous source terms and for the specific type of boundary conditions (see also
discussion below on Difference Potentials Method and Boundary Element Method).
Similar to the method in [30], the idea of our method here and in [50, 48] is to first introduce a computationally simple auxiliary domains. After that, the original domains/subdomains are embedded into simple
auxiliary domains (and the auxiliary domains are discretized using Cartesian meshes). However, compared to
the integral approach in [30], we construct discrete Boundary Equations with projections (discrete generalized
Calderon’s boundary equations with projections) to obtain the values of the solutions at the points near the
continuous boundaries of the original domains (at the points of the discrete grid boundaries which approximate the continuous boundaries from the inside and outside of the domains). Using the obtained values of
2
the solutions at the discrete grid boundaries, the approximation to the solutions in each domain/subdomain
is constructed through the discrete generalized Green’s formulas. The main complexity of our approach
reduces to the several solutions of simple auxiliary problems on structured Cartesian grids, and similar to
[30] the solutions of these auxiliary problems can be combined with fast Poisson solvers. Like method in [30],
and IIM and GFM, our method (and in general, methods based on Difference Potentials Method [49, 34])
preserve the underlying accuracy of the schemes being used for the space discretization of the continuous
PDEs in each domain/subdomain (here, and in [50, 48] we considered second-order finite difference scheme
for the space approximation). But compared to [30], and to IIM and GFM, our approach is not restricted
by the type of the boundary or interface conditions (as long as the continuous problems are well-posed). Let
us mention that the accuracy of our approach is confirmed by several numerical experiments in the current
paper (see Section 5) and in papers [50, 48]. The reader can consult [49] for the theoretical convergence
study of the methods based on Difference Potentials.
In this work, we will consider the heat equation (2.5) in a composite domain with curvilinear smooth
boundaries in 2D (even though, the proposed framework is general and can be extended in the future to
the arbitrary 1D, 2D and 3D domains). We will further develop, as well as numerically test the Algorithms
Composition Scheme proposed originally in [50, 48] for linear elliptic problems. The Algorithms Composition
Approach developed in this paper is an accurate, simple, and robust scheme of algorithms composition for the
numerical approximations of the boundary value problems in composite and complex domains. The proposed
method can handle complex geometries without the use of unstructured meshes (with the consideration of
only regular Cartesian grids), and can be employed with fast Poisson solvers. Our method combines the
simplicity of the finite difference methods on Cartesian meshes with the flexibility of the Difference Potentials
method [49].
Difference Potentials Method (DPM) can be understood as the discrete version of the method of generalized Calderon’s potentials and Calderon’s boundary equations with projections in the theory of PDEs.
The DPM on its own, or in combination with other numerical methods, is an efficient tool for the numerical solution of the interior and exterior boundary value problems in arbitrary domains (see for example,
[49, 46, 28, 47, 52, 34, 50, 48, 10, 9]). Viktor S. Ryaben’kii originally introduced DPM in his Doctor of
Science thesis (Habilitation thesis) in 1969. The DPM allows one to reduce uniquely solvable and well-posed
boundary value problems into pseudo-differential boundary equations. In some respect, the difference potentials method (DPM) is related in spirit to the boundary element method (BEM). The idea of the BEM
(see for example [1, 8]) is to reduce the boundary value problem to Fredholm-type integral equations with
respect to equivalent boundary sources, and these equations are discretized accordingly. One shortcoming
of the BEM is the full structure of the resulting systems/matrices (as opposed to the sparse nature of the
systems/matrices produced by the finite differences (FD), or finite element methods (FEM)). Recent progress
on fast multipole methods considerably accelerated solutions of such full systems [18]. However, the most
serious drawback of BEM methods is the requirement for the explicit knowledge of the fundamental solution
of given differential operators. This can impose several restrictions on the practical applications of BEM
methods. The essence of the DPM is to transform uniquely solvable and well-posed boundary value problems
into so-called pseudo-differential boundary equations that do not employ a fundamental solution. Hence,
on one hand, the DPM enjoys geometric flexibility of the BEM. On the other hand, it can be applied to a
wider class of problems than the BEM, and it can have several advantages over the BEM. We will make a
few comments about this below:
Main features and advantages of the DPM (see also Section 2 in this paper and [49, 34] for more detailed
discussion):
(i) The original PDEs (without imposed boundary conditions) are reduced to an equivalent generalized
Calderon’s boundary equations with projections. These equations are supplemented by the given boundary
conditions. Compared to BEM, DPM does not employ Fredholm equations of the first or second kind;
(ii) The derived Calderon’s problem can be discretized on structured Cartesian grids. Discrete inverse
operators which are introduced in the DPM for the approximation of the Calderon’s potentials and projections do not contain any singularities or convolutions. These inverse operators can be obtained using
fast numerical calculations as the solution of the simple and computationally efficient auxiliary problems
[49, 34, 50, 48, 10, 9];
(iii) DPM can treat arbitrary smooth boundaries of the domains, and the boundaries do not need to
align/conform with the grid - this does not produce any loss of accuracy. The DPM provides flexibility to
3
handle general boundary conditions in an efficient and universal way, and the method always produces a
well-posed discrete version of the problem (if the original continuous boundary value problem is well-posed)
[49, 34, 3]. Note that well posedness for BEM may not be a trivial task for problems with general boundary
conditions;
(iv) DPM gives flexibility to construct high-order schemes on regular Cartesian grids for problems with
complex geometries [49, 34, 3]. Some of the advantages of numerical methods on Cartesian grids are that
the grid generation for these schemes is trivial, the design of the high-order numerical methods that satisfy
certain stability properties are usually much more straightforward on regular structured grids, and the numerical methods on Cartesian meshes are more robust than those of body-fitted grids;
(v) DPM can approximate both variable coefficients and constant coefficients problems: the main steps in
the construction of the Calderon’s potentials and projections for variable coefficients and constant coefficients
stay essentially the same [49, 34, 50, 48, 10, 9].
The developed Algorithms Composition Framework combines the above advantages of the DPM and offers
novel flexibility to the DPM (for a more detailed discussion, see Sections 3 - 6 in this paper, as well as
[50, 48]):
(i) the Algorithms Composition Framework is well-suited for heterogenous problems and complex interface
problems, as well as for the development of the adaptive schemes and domain decomposition approaches;
(ii) our method provides the flexibility to consider a non-difference approximation of the general boundary
and interface (/or matching) conditions, which automatically takes into account the smoothness of the solution. Such self-tuning is impossible, for example in the difference or finite-element approximations of these
conditions;
(iii) the proposed framework can be further developed for the efficient computation of the solutions in arbitrary domains with arbitrary boundaries and interfaces;
(iv) the numerical schemes, as well as meshes can be chosen totally independently for each subdomain/
domain; the boundaries of the subdomains and interfaces do not need to conform/align with the grids;
(v) the main complexity of the developed algorithm reduces to the several solutions of simple auxiliary
problems on structured Cartesian grids;
(vi) the proposed approach is general and can be developed as a high-order method (both in time and space:
higher than first order in time and higher than second order in space). For example, by considering highorder methods such as high-order finite difference, finite element, or spectral methods for the construction
of the discrete parts of generalized Calderon’s potentials, as well as for the approximations of the particular
solutions to the inhomogeneous equations;
(vii) since the schemes for constructing the solutions in each domain/subdomain are independent of each
other, our approach is very well suited for parallel computations;
(viii) the proposed method is not restricted to 2D and can be applied to variable coefficients problems. The
method can be generalized to the equations not necessarily of the elliptic/parabolic type as well.
The paper is organized as follows. First, in Section 2 we give some preliminaries that will be helpful for
the introduction of the proposed algorithm. Then, we introduce the idea of DPM for the parabolic equation
in the single domain, and we make an overview of the important properties of the DPM for the model under
consideration in Section 2.1. In Section 3, we develop the Algorithms Composition Scheme for the parabolic
model in a composite domain using the flexibility of the DPM. In Section 4, we state the main steps of the
proposed algorithm. Finally, we illustrate the flexibility and performance of the proposed scheme in several
numerical experiments in Section 5. Some concluding remarks are given in Section 6, and some technical
details about the algorithm are provided in Appendix Section 7.
2. Preliminaries
In this section, we will introduce some preliminaries that will be helpful for the discussions in the following
sections. We are concerned in this paper with the parabolic initial and boundary value problem (IVP) in
some bounded domain Ω ⊂ Ω0 ⊂ R2 , and its neighborhood Ω0 over the time interval [0, T ].
∂u
+ Lu = q in Ω × (0, T ),
∂t
l(u) = ψ on ∂Ω × (0, T ),
u|t=0 = u0 in Ω.
4
(2.1)
(2.2)
(2.3)
a
1$
a
" #"
! ï
" #" "
!
10
1
Fig. 2.1: Example (a sketch) of the original domain Ω and auxiliary domain Ω0
Here, q(x1 , x2 , t) is the sufficiently regular source function in the domain Ω0 (function q is originally given
Ω, and extended to the larger domain Ω0 ), and L is the second order linear symmetric elliptic differential
operator
Lu := −
2
X
k,j=1
∂
∂u
(akj (x)
) + a0 u,
∂xk
∂xj
(2.4)
where coefficients a0 and akj are assumed to be sufficiently smooth functions in Ω0 , akj (x) = ajk (x),
(k, j) = 1, 2 for almost every x := (x1 , x2 ) := (x, y) ∈ Ω0 . Moreover, there exists a constant α0 > 0 such
P2
that k,j=1 akj (x)ξk ξj ≥ α0 |ξ|2 for each ξ ∈ R2 , and for almost every x ∈ Ω0 . The classical example of the
above parabolic problem (2.1) is the heat equation
∂u
− ∆u = q.
∂t
(2.5)
To simplify the presentation of the ideas, we will be concerned below with the heat equation (2.5) subject
to (2.2) - (2.3). However, let us mention that the same idea can be extended in a straightforward way to a
general parabolic equation (2.1).
Next, at the given time t, we denote by vΓ the Cauchy data of an arbitrary continuous piecewise smooth
function v(x, y, t) defined on Γ and in some of its neighborhoods:
∂v vΓ := v ,
(2.6)
.
Γ ∂n Γ
∂
Here, ∂n
is the inward (with respect to Ω) normal derivative to Γ, and Γ := ∂Ω is a piecewise smooth
boundary of Ω.
Now, in order to introduce our ideas, let us consider model problem (2.5) in its time-discrete form.
Towards this end, we subdivide time interval [0, T ] into Nt time steps and denote by ti := i∆t, ∆t > 0.
1
Setting m := ∆t
and considering the Backward Euler method, we obtain below the following time discrete
reformulation of the parabolic equation (2.5) in some domain Ω0
Let us introduce
L∆t [ui+1 ] := ∆ui+1 − mui+1 ,
(x, y) ∈ Ω0 ,
(2.7)
where L∆t denotes the linear elliptic operator applied to ui+1 .
We also denote the right-hand side as:
Gu∆t := −q i+1 − mui ,
where q i+1 := q(x, y, ti+1 ).
5
(x, y) ∈ Ω0 ,
(2.8)
Then the semi-discrete formulation of the model (2.5) is stated as follows:
Find some ui+1 := ui+1 (x, y) ≈ u(x, y, ti+1 ) such that
L∆t [ui+1 ] = Gu∆t ,
(x, y) ∈ Ω0 ,
(2.9)
Remark: For each i this is a time-independent elliptic problem.
Let us now introduce the auxiliary problem. For the purposes of the discussion below, we will suppress
for now the explicit dependence on the time level i. We place the original domain Ω in the auxiliary domain
Ω0 ⊂ R2 : Ω̄ ⊂ Ω0 . Next, we will formulate a discrete in time and continuous in space Auxiliary Problem
(AP):
Definition 2.1. For any given sufficiently regular right-hand side G∆t , find F such that,
L∆t [F ] = G∆t ,
F = 0,
(x, y) ∈ Ω0 ,
0
(x, y) ∈ ∂Ω ,
(2.10)
(2.11)
where L∆t is the same linear elliptic operator as in (2.9) and is applied here to F . Since the auxiliary
domain Ω0 can be arbitrary, we can choose it to be a square. The above (AP) Dirichlet problem is uniquely
solvable.
Remark: This is a continuous in space Dirichlet problem for each fixed time level.
Let us now construct a potential with a density vΓ . Define the vector function
vΓ := (φ0 (s), φ1 (s)),
(2.12)
where φ0 (s) and φ1 (s) are two piecewise smooth continuous functions on |Γ| that are s−periodic with a
period of |Γ|, s is the arc length along |Γ|, and |Γ| is the length of the boundary. Here, the arc length is
chosen as a parameter only for definiteness.
Let v(x, y) = vΩ0 be an arbitrary sufficiently smooth function on Ω0 that satisfies condition (2.11) on
0
∂Ω : v|∂Ω0 = 0. Assume that its Cauchy data vΓ is defined as in (2.6) and is equal to the vector function
vΓ in (2.12). Then, we can recall the following definition of the potential PΩΓ vΓ below [48, 49].
Definition 2.2. A Potential uΩ := PΩΓ vΓ defined on Ω with density vΓ is equal on Ω to the solution of
(AP, Def. 2.1), with the right hand-side G∆t defined as follows:
0, (x, y) ∈ Ω,
G∆t :=
(2.13)
L∆t [v], (x, y) ∈ Ω0 \Ω
It can be shown that at each time level the above potential uΩ := PΩΓ vΓ is well-defined: it depends only on
Cauchy data (2.12), but is independent of the choice of a particular function v(x, y) satisfying (2.11) on ∂Ω0
whose Cauchy data coincides with (2.12). For a more detailed discussion on the potentials with projectors,
see [48, 49].
Remark: The potential uΩ := PΩΓ vΓ can be viewed as the modification [45] of the Calderon potential
[4]. However, in comparison to the Calderon potential, the potential PΩΓ vΓ admits a finite-dimensional
constructive approximation by Difference Potentials [49, 10, 9], as will be illustrated below in Section 2.1.
2.1. Scheme Based on the Difference Potentials
We will develop our Algorithms Composition Approach based on the idea of the Difference Potentials
Method (DPM) [49, 48, 50, 10, 9]. Difference Potentials Method (DPM) can be viewed as the method of
building and computing the discrete parts of the modified Calderon’s potentials (see for example, [49, 28,
47, 52, 50, 48, 10, 9] and Introduction Section 1).
We will present in this Section the necessary preliminaries and some overview of the numerical scheme
based on the difference potentials for the single arbitrary domain Ω ∈ R2 . This discussion will be important
for the development of our approach for the composite/complex domain Ω ∈ R2 , which we will present in
Sections 3 - 5.
At this point, let us assume that we consider (2.5) in some domain Ω - an arbitrary bounded domain
in R2 with the boundary ∂Ω. First, let us introduce some preliminary notations and definitions that will
6
Ω0
γ
M
M =M
−
+
Ω
Fig. 2.2: Example (a sketch) of the auxiliary domain Ω0 , original domain Ω; the example of some points
(xj , yk ) in the set γ: the points which are outside Ω are from γex , the points which are inside Ω are from
γin ∈ M
be used in this section. Denote ΠSR vR as the extension operator of function v from set/domain R to the
set/domain S, πS as the restriction operator to the set/domain S, wS := πS w as the restriction of function
w to the set/domain S, and χS as the characteristic function of the set S.
Next, we introduce here the auxiliary fully discrete problem. Let us place the original domain Ω in
the auxiliary domain Ω0 ⊂ R2 . The choice of the domain Ω0 should be convenient for the computations,
so we will choose it to be a square, and we will introduce a Cartesian mesh for Ω0 , with points xj =
j∆x, yk = k∆y (k, j = 0, ±1, ...). Let us assume for simplicity that ∆x = ∆y := h. Let us also define
a five-point stencil Nj,k with its center placed at (xj , yk ) to be the set of the following points: Nj,k :=
{(xj , yk ), (xj±1 , yk ), (xj , yk±1 )}.
In addition, we also introduce point sets M := M + := (xj , yk ) ∈ Ω - the sets ofSall the points (xj , yk ) that
belong to the interior of the original domain Ω. We now define N := N + := { j,k Nj,k |(xj , yk ) ∈ M }- the
set of all points covered by five-point stencils when center point (xj , yk ) of the stencil goes through all the
points of the set M . Note that the points in the set N will be both inside and outside of the original domain
Ω.
Now, let us introduce the grid boundaries γex := γex = N \M is the exterior grid boundary layer for domain
Ω. γin := {(xj , yk )|(xj , yk ) ∈ M : Nj,k 6⊂ M } is the interior grid boundary layer for domain Ω (in other
words, this is the set of all the nodes (xj , yk ) in M for which stencil Nj,k is not the subset of M : Nj,k contains
nodes outside of M , but the center point (xj , yk ) of the stencil Nj,k belongs to M ). Define γ := γex ∪ γin a narrow set of nodes that surrounds the continuous boundary ∂Ω, Figure 2.2.
Next, we construct the auxiliary set M 1 by completing the set N to a rectangle, and adding one extra layer
of grid points to each side of the rectangle, hence N ⊂ M 1 . Also, as before, define N 1 := {∪j,k Nj,k |(xj , yk ) ∈
M 1 }, and finally, let us introduce γ 1 := N 1 \M 1 .
We can now introduce the space approximation of (2.9) and consider the fully discrete version of equation
(2.5). Therefore, the computed quantity will be the point values uj,k (t) ≈ u(xj , yk , t). We denote by uij,k
the computed uj,k (ti ): uij,k :≈ uj,k (ti ) at the discrete time level ti := i∆t, with time step ∆t. Additionally,
we denote by ∆j,k the discrete Laplacian obtained using second order central difference formulas for the x
and y variables, and by
L∆t,h [ui+1 ] := ∆j,k ui+1 − mui+1
j,k ,
(xj , yk ) ∈ Ω0 .
(2.14)
Finally, we denote by g u ,
i+1
u
g u := gj,k
= −qj,k
− muij,k ,
7
(2.15)
i+1
where, as before, qj,k
≡ q(xj , yk , ti+1 ) is the value of the source function q(xj , yk , t) at ti+1 .
Thus, the fully discrete finite-difference based version of the parabolic equation (2.5) is:
Find some ui+1 such that it satisfies
L∆t,h [ui+1 ] := g u ,
(xj , yk ) ∈ Ω0 .
(2.16)
Again, as in the “Preliminaries” in Section 2, we will suppress for now the explicit dependence on the
time level i for the clarity of the discussion. Based on a central finite difference approximation (2.16), we will
now formulate the fully discrete analog of the auxiliary problem (AP), Def. 2.1 in Section 2 - the Discrete
Auxiliary Problem (DAP):
Definition 2.3. For the given grid function g, find the solution of the scheme f such that:
(xj , yk ) ∈ M1 ,
L∆t,h [f ] = g,
1
(xj , yk ) ∈ γ ,
f = 0,
(2.17)
(2.18)
where, as before in (2.14), L∆t,h [f ] ≡ ∆j,k f − mfj,k . We note that the ((DAP), Def. 2.3) is well defined for
any right hand side g - it has a unique solution f defined on the set N 1 .
Also, it should be noted that the solution of ((DAP), Def. 2.3) can be efficiently computed using the Fast
Fourier Transform (FFT) with the appropriate choice of the auxiliary set M 1 .
We now introduce a linear space Vγ of all the grid functions vγ defined on γ, similar to [48, 10, 9, 49].
We will extend by zero the value of vγ to other points of the grid N 1 . Let us now recall that,
Definition 2.4. A Difference Potential [48, 10, 9, 49] with the given density vγ ∈ Vγ is the grid function
u := PN γ vγ defined on the set N , which coincides (on the set N ) with the solution of ((DAP), Def. 2.3)
when the right hand-side is defined as follows:
0, (xj , yk ) ∈ M,
g :=
(2.19)
L∆t,h [vγ ], (xj , yk ) ∈ M 1 \M.
The Difference Potential can be viewed as the discrete analog of the modified potential of Calderon’s type,
or as the discrete analog of the space-continuous potential, Def. 2.2 in Section 2 (similar to Def. 2.4, the
definition of Difference Potential can be extended by considering the set M 1 \M as the “interior” set, and
set M as the “exterior” one, [49]). Here, PN γ denotes the operator that constructs the difference potential
u = PN γ vγ from the given density vγ ∈ Vγ . The operator PN γ is the linear operator of density vγ , and it can
be easily constructed (see for example [10, 9, 49]). Again, for our problem, L∆t,h [vγ ] ≡ ∆j,k vγ − m(vγ )j,k .
As in the space-continuous case, (Def. 2.2, Section 2), the concept of the difference potential is well-defined
at each time level due to the following statement:
Theorem 2.5. The difference potential PN γ vγ depends only on vγ ∈ Vγ , but is independent of the choice of
the function v defined on N 1 (satisfying condition (2.18) on γ 1 : v = 0, (xj , yk ) ∈ γ 1 ), and coinciding with
vγ on γ: v|γ ≡ T rγ v = vγ .
Let us recall the proof for the reader’s convenience (for the general proof and discussion,Ssee [49, 50, 48]).
Proof: Let us define the sets N + := N , Ω− := Ω0 \Ω, M − := (xj , yk ) ∈ Ω− , and N − := { j,k Nj,k |(xj , yk ) ∈
M − }. Recall that, πN + denotes the restriction operator to the set N := N + , ΠN 1 γ vγ is the arbitrary
extension vγ to N 1 , and χM − is the characteristic function of set M − .
Let us notice now that the difference potential, Def. 2.4, can be represented in the following operator
form:
u = PN + γ vγ = πN + L−1
∆t,h [χM − L∆t,h [vγ ]],
where vγ is extended by zero outside of set γ to N 1 \γ. Now, let us represent the arbitrary function z defined
on N 1 , such that: z = ΠN 1 γ vγ , as the sum of the three terms
z = χN 1 \N + z + χN 1 \N − z + vγ .
8
Notice that, for the first term χM − L∆t,h [χN 1 \N + z] = L∆t,h [χN 1 \N + z], and for the second term
χM − L∆t,h [χN 1 \N − z] = 0. From this we obtain that
−1
πN + L−1
∆t,h [χM − L∆t,h [χN 1 \N + z]] = πN + L∆t,h [L∆t,h [χN 1 \N + z]] = 0.
Hence, the only remaining contribution to the potential u is the contribution from the third term :
u ≡ PN + γ vγ = πN + L−1
∆t,h [χM − L∆t,h [vγ ]].
Next, let us recall [10, 9, 49] and define another operator Pγ : Vγ → Vγ as the trace (or restriction) of
the Difference Potentials PN γ vγ on the grid boundary γ: Pγ vγ := T rγ PN γ vγ = PN γ vγ |γ . We now state an
important theorem for the further development of our scheme (see for example [49, 48].) The details of the
general proof can be found in [49]:
Theorem 2.6. At each time level ti+1 , density vγ ∈ Vγ is the trace of some solution
u := uN ∈ N : vγ = T rγ uN ≡ uγ to the homogeneous equation
(xj , yk ) ∈ M
L∆t,h [u] = 0,
(2.20)
if and only if we have
Qγ vγ := vγ − Pγ vγ = 0,
(xj , yk ) ∈ γ.
(2.21)
Moreover, the solution uN defined on the set N can be reconstructed from its boundary value vγ using the
formula u := PN γ vγ .
Remark: It can be shown that Pγ is the projector, hence Vγ = Im Pγ ⊕ Ker Pγ .
Thus, Theorem 2.6 implies that the problem of finding a unique solution to (2.20) subject to the appropriate approximation of the boundary conditions on ∂Ω denoted by l(u) = ψ, in other words the problem:
L∆t,h [u] = 0,
(2.22)
l(u) = ψ,
(2.23)
is equivalent to the problem of finding the unique density function vγ ≡ uγ from the system of the Boundary
Equations:
Qγ vγ = 0,
(2.24)
l(PNγ vγ ) = ψ.
(2.25)
After that, at each time level the solution uN to (2.22)-(2.23) is reconstructed from uN = PN γ vγ .
Remark: Let us note that the equation (2.24) can be viewed as the generalized Poincaré-Steklov interface
equation.
From the above discussion and, in particular, from Theorem 2.6 and from (2.24) - (2.25), the next result
follows:
Proposition 2.7. At each time level ti+1 , the approximation uN to the solution u of the (IVP) problem
(2.5), (2.2) - (2.3) in domain Ω can be obtained as
uN = PN γ vγ + ūN ,
l(v + ū) = ψ,
(2.26)
(2.27)
where ūN is the approximation of any particular solution to the inhomogeneous equation (2.5) in Ω̄, at
the given time level ti+1 . vN := PN γ vγ is the difference potential in Ω̄ at the same time level ti+1 , with
a density vγ that satisfies equation (2.24): Qγ vγ = 0. Equation (2.27) denotes the approximation of the
boundary condition l(u) = ψ on ∂Ω, and the representation of the boundary conditions in the form (2.27)
can be viewed as the consequence of the equality (2.26) and (2.24) (projected on the continuous boundary).
Finally, define ūγ := T rγ ūN .
9
Remark: Let us mention that in (2.26), we can consider any particular solution: Let ūN be some solution
to L∆t,h [ūN ] = g u , (xj , yk ) ∈ M . We extend the solution arbitrarily to N 1 , and we will apply operator L∆t,h
to the extended solution ūN 1 . Hence, we will get the right-hand side of the equation L∆t,h [ūN 1 ] = g, which
coincides on M with g u . For example, we can construct the particular solution ūN as follows:
Definition 2.8. Define, ū to be the solution f of the auxiliary problem (DAP, Def. 2.3) with the right
hand-side defined as
u
g , (xj , yk ) ∈ M,
g :=
(2.28)
g̃ u , (xj , yk ) ∈ M 1 \M
where g u is given in (2.15) and g̃ u is some extension of g u to M 1 \M . Set ūN := πN ū. This point will be
discussed in more details in Section 5.
Remark: Finally, let us comment briefly about the accuracy in space of the approximation uN =
PNγ vγ + ūN in (2.26)-(2.27), to the solution u of (2.5), (2.2) - (2.3). One would expect that the solution
uN will converge to the continuous solution u in the discrete Hölder norm of order q + (with the arbitrary
0 < < 1), with the rate O(hp− ) as h → 0. Here, p is the order of the accuracy of the approximation
of the continuous differential operator L∆t by the discrete operator L∆t,h (we assume that the boundary
condition (2.2) is approximated by (2.27) with the same or a higher order than p). Hence, for the central
finite difference scheme that is discussed here, we will expect the O(h2 ) rate in the maximum norm in space.
For a more detailed and general discussion, and the proof of the accuracy of the (DPM) method, the reader
can consult [49, 34].
Remark: Let us emphasize that one could develop any other numerical approximation in space (such as
high-order finite difference, finite-volume, finite element, or spectral methods) for (2.1) within the presented
framework of the potentials and of the difference potentials. But for our goals in this paper, we will consider
the second order finite-difference approximation for the model equation (2.5).
3. Algorithms Composition Approach based on the Difference Potentials Method
We will now develop, as well as numerically test the Algorithms Composition Approach for a parabolic
model under consideration (2.5), (2.2) - (2.3).
We start the introduction of the scheme and the illustration of the ideas by considering the system (2.5),
(2.2) - (2.3) (or formulation (2.9) in its time-discrete form) in some composite domain Ω. Ω is an arbitrary
bounded domain in R2 with the boundary ∂Ω, and such that it consists of two disjoint subdomains Ω1 and
Ω2 : Ω = Ω1 ∪ Ω2 , Γ = Ω1 ∩ Ω2 , with piecewise smooth boundaries Γ1 := ∂Ω1 ∪ Γ and Γ2 := ∂Ω2 ∪ Γ (see
Figure 3.1). We would like to also emphasize that the two subdomains are considered here only for the
δ
Ω
Ω2
Ω1
Γ
Fig. 3.1: Sketch of domain Ω with boundary ∂Ω, two subdomains Ω1 , Ω2 , and the interface Γ.
simplicity of the presentation. The discussed idea can be extended in a straightforward way to multiple
subdomains or composite domains.
Our goal at each time level ti+1 is to find an approximations to ui+1 in domain Ω:
( i+1
uΩ1 , (x, y) ∈ Ω1 ,
ui+1
:=
(3.1)
Ω
ui+1
Ω2 , (x, y) ∈ Ω2 ,
10
where ui+1
Ωp (here p = 1, 2) are the solutions to (2.9) in each subdomain Ωp :
uΩ
p
L∆t [ui+1
Ωp ] = G∆t ,
(x, y) ∈ Ωp , and (p = 1, 2).
(3.2)
Solution ui+1
is subject to the appropriate boundary conditions on the boundary ∂Ω of the original domain
Ω
i+1
l(ui+1
,
Ω )=ψ
(3.3)
and to the interface conditions on the interface boundary Γ, which we will select to be the following:
i+1
uΩ1 = ui+1
(x, y) ∈ Γ,
i+1
i+1
Ω2 ,
lΓ (uΩ1 , uΩ2 ) :=
(3.4)
i+1
∇ui+1
·
n
=
β∇u
(x, y) ∈ Γ.
Γ
Ω1
Ω2 · nΓ ,
Here, nΓ is a unit outward normal vector to the interface boundary Γ (with respect to Ω1 ). To develop our
idea, we will consider the difference potentials scheme (2.26) - (2.27) from Section 2.1 (which uses Backward
Euler in time and central finite difference space discretization) as the fully discrete approximation of (2.5),
(2.2) - (2.3) in each subdomain Ωp , and we will build an algorithm for the approximation of ui+1
in (3.1)
Ω
based on the algorithms composition idea. As before, we will remove below the explicit dependence on time
for the clarity of the presentation.
First, as we have done in (Section 2.1) for the single domain Ω, we will introduce auxiliary difference
problems for each subdomain Ωp , (p = 1, 2): we will place each of the original subdomains Ωp in the auxiliary
domains Ω0p ⊂ R2 , (p = 1, 2). As before, the choice of each domain Ω0p should be convenient for computations,
and the choice of these auxiliary domains do not need to depend on each other. Again, for each subdomain,
we will proceed in a similar way as we did in (Section 2.1). For each Ω0p we will introduce, for example, a
Cartesian mesh (again the choice of the meshes for the auxiliary problems in each subdomain can be totally
independent. The choice for each subdomain is based on the considerations of the efficiency and simplicity
of the resulting discrete problems.) After that, all the definitions, notations, and properties introduced in
(Section 2.1) extends to each subdomain Ωp in a direct and straightforward way: we will use index p, (p = 1, 2)
to distinguish each subdomain.
The cornerstone of our approach is the following proposition, which is a consequence of Proposition 2.7
in Section 2.1 for a single domain.
Proposition 3.1. At each time level ti+1 , the fully discrete approximation uNp to the solution uΩp , (p = 1, 2)
in (3.2) - (3.4) is obtained as
uNp = PNp γp vγp + ūNp ,
(3.5)
l(vΓ1 + ūΓ1 , vΓ2 + ūΓ2 ) = ψ,
(3.6)
lΓ (vΓ1 + ūΓ1 , vΓ2 + ūΓ2 ) = φ,
(3.7)
where ūNp is the approximation of the particular solution to the inhomogeneous equation (2.5) in each
subdomain Ωp at the given time level ti+1 . PNp γp vγp is the difference potential with a density vγp in each
domain Ωp at the same time level ti+1 . Expressions in (3.6)-(3.7) denote the approximations of the boundary
and interface conditions, respectively, on the continuous boundaries ∂Ω1 , ∂Ω2 , and interface boundary Γ (in
other words on the boundaries Γ1 and Γ2 ). Denote ūγp = T rγp ūN . The construction of (3.6)-(3.7) will be
discussed in more detail in Section 3.2. Density/trace vγp in the equation (3.5) ranges over the solution of
the Boundary Equation on each discrete grid boundary set γp :
Qγp vγp ≡ vγp − Pγp vγp = 0,
(3.8)
see (2.24) in Section 2.1. However, there are multiple solutions (vγ1 , vγ2 ) to (3.8), and as before, the unique
pair of the densities/traces vγ1 ∈ Vγ1 and vγ2 ∈ Vγ2 will satisfy the above Boundary Equation (3.8), as well
as the boundary and interface conditions (3.6) - (3.7).
3.1. System of Boundary Equations: Weak Formulation
At each time level, the unique pair of the densities/traces (vγ1 , vγ2 ) ≡ (T rγ1 vN1 , T rγ2 vN2 ) ∈ Vγ1 × Vγ2 of
the approximation (3.5)-(3.7) is the unique solution to the Boundary Equation (3.8), subject to the boundary
11
and interface conditions (3.6) - (3.7). There are different ways to solve (3.8) subject to (3.6) - (3.7). One
possibility is to directly consider the original formulation (3.8) and employ finite difference approximation
of the boundary and interface conditions (3.6) - (3.7) (see for example [10, 49] for more details on this
approach). However, to avoid the difficulties associated with the finite difference approximations of (3.6)
- (3.7) in arbitrary domains, we will take advantage of the weak formulation of (3.8) and the spectral
approximation of the Cauchy data.
To define a weak formulation, let us first introduce the discrete norm for the space of grid functions
vγp ∈ Vγp . Similar to [48], we will consider the following norm:
hX
X vν ,ν +1 − vν 2 i
X vν +1,ν − vν 2
2
||vγp ||2Vγp := h
|vν |2 + α
+α
1 2
, (p = 1, 2).
1
h
h
(3.9)
Here, the sum is extended over all ν := (ν1 , ν2 ) ∈ γp for the first term in (3.9), over all ν := (ν1 , ν2 ) that,
together with (ν1 + 1, ν2 ) belong to γp in the second term of (3.9), and over all ν := (ν1 , ν2 ) that, together
with (ν1 , ν2 + 1) belong to γp in the third term of (3.9). α is a nonnegative coefficient which will be defined
in the numerical experiments.
Remark: The norm (3.9) is the discrete analog of the continuous norm for the Cauchy data vΓp ∈ VΓp :
||vΓp ||2VΓp
:=
Z
Γp
h
dv 0 (s) 2
i
p |vp0 (s)|2 + α + |vp1 (s)|2 ds
ds
(3.10)
Let us introduce ṽΓp ∈ ṼΓp as a finite dimensional approximation of the continuous Cauchy data vΓp ∈ VΓp :
ṽΓp :≈ vΓp , and ṼΓp as the finite dimensional subspace of VΓp . The details on the construction of the
approximation ṽΓp will be presented in Section 3.2. Finally, let us also introduce
Definition 3.2. Let Πγp Γp be the operator that, to every vΓp ≡ (vp0 (s), vp1 (s)) from the space of continuous
Cauchy data VΓp , assigns vγp from the space of the discrete densities Vγp , (p = 1, 2):
vγp = Πγp Γp vΓp ,
(3.11)
Moreover, we have from Def. 3.2 that for ṽΓp ∈ ṼΓp , we can define ṽγp ∈ Vγp to be:
ṽγp := Πγp Γp ṽΓp
(3.12)
The exact form of the operator Πγp Γp will be given in (3.22) - (3.23) in Section 3.2 as well.
Therefore, the weak formulation of the Boundary Equations (3.8) subject to (3.6)-(3.7) is formulated as
follows:
Definition 3.3.
At each time level, find (ṽΓ1 , ṽΓ2 ) ≡ (ũΓ1 , ũΓ2 ) ∈ ṼΓ1 × ṼΓ2 that minimizes the functional
˜ Γ1 , ṽΓ2 + ū
˜ Γ2 ) − ψ||2Φ + ||lΓ (ṽΓ1 + ū
˜ Γ1 , ṽΓ2 + ū
˜ Γ2 ) − φ||2Φ +
||l(ṽΓ1 + ū
2
X
p=1
||Qγp Πγp Γp ṽΓp ||2Vγp ,
(3.13)
where || · ||Φ is the Hilbert norm, ṼΓ1 , and ṼΓ2 are finite dimensional subspaces of VΓ1 and VΓ2 , respectively,
˜ Γ1 , ū
˜ Γ2 ) are the finite dimensional approximations of (ūΓ1 , ūΓ2 ) (it is a pair of the Cauchy Data of
and (ū
the particular solution (ūN1 , ūN2 )). This finite dimensional approximation is discussed in Section 3.2.
Therefore, at each time level, the unique pair of the densities/traces (vγ1 , vγ2 ) ∈ Vγ1 × Vγ2 of the approximation (3.5)-(3.7) to the solution (3.1) of (2.5), ( 2.2)-( 2.3) in the composite domain Ω is obtained as the
approximation:
(vγ1 , vγ2 ) ≈ (ṽγ1 , ṽγ2 ) = (Πγ1 Γ1 ṽΓ1 , Πγ2 Γ2 ṽΓ2 ).
(3.14)
3.2. System of Boundary Equations: Discretization of the Cauchy Data
As before, we will assume that the solution is given at a fixed time level and we will suppress the explicit
dependence on time for the clarity of the presentation.
12
γin
γex
Ω
sj*,k*
d j*,k*
(j*,k*)
Fig. 3.2: Example (a sketch) of the geometry with the curve boundary Ω; point (x?j , yk? ) is in the set γex ∈ M ,
dj∗,k∗ is the distance from this point to the boundary of the domain Ω, and sj ? ,k? is the corresponding arc
length.
1. As we already showed in Sections 3 - 3.1, at each time level, the problem of finding the approximation
to the unique solution uΩ := (uΩ1 , uΩ2 ) of the model (2.5), (2.2) - (2.3) in composite domain
Ω ≡ Ω1 ∪ Ω2 reduces to the problem of finding the unique pair of the densities/traces (vγ1 , vγ2 ) ≡
(uγ1 , uγ2 ) ∈ Vγ1 × Vγ2 (3.14). This pair of the densities/traces satisfies the generalized PoincaréSteklov interface equations (3.8), as well as the boundary and interface conditions (3.6) - (3.7).
2. After that, the approximation of the solution (uΩ1 , uΩ2 ) of (2.5), ( 2.2)-( 2.3) in Ω̄ is reconstructed
from the approximated densities (ṽγ1 , ṽγ2 ) ∈ Vγ1 × Vγ2 using the pair of difference potentials
(PN1 γ1 ṽγ1 , PN2 γ2 ṽγ2 ), see equation (3.5).
In order to solve for the unknown densities/traces in Step 1 above, we will consider a weak formulation
Def. 3.3, Section 3.1, and we will employ a spectral approach for the approximation (3.14) of (vγ1 , vγ2 ) as
elaborated below.
Consider a set of basis functions on the curve boundaries Γ1 and Γ2 :
φ1 (s), ..., φL (s), and φ?1 (s), ..., φ?L (s)
(3.15)
where s is the arc length and L is the total number of the basis functions. We will consider here the
same basis functions on all parts of the boundaries, although in general, the basis functions can be selected
differently for the different parts of the boundaries. In general, this choice will depend on the smoothness of
the boundary of the domains and on the smoothness of the solution.
Next, we will assume that for every sufficiently smooth single-valued periodic function f (s) defined on the
boundaries Γ1 and Γ2 , the sequence
L := 0min1
up` ,up`
Z
[|f (s) −
L
X
`=1
u0p` φ` (s)|2 + |f 0 (s) −
L
X
u1p` φ?` (s)|2 ]ds
(3.16)
`=1
tends to zero with increasing L: lim L = 0 as L → ∞.
Therefore, we will employ the following approximation to construct finite-dimensional ṽΓp ∈ ṼΓp :
ṽΓp :=
L
X
u0p` φ` ,
`=1
L
X
`=1
u1p` φ?` ,
(3.17)
that discretize the elements vΓp ∈ VΓp from the space of continuous Cauchy Data on the boundary Γp ,
13
p = 1, 2:
vΓp := (vp0 (s), vp1 (s)),
(3.18)
ṽΓp :≈ vΓp .
(3.19)
and we have
In (3.17), φ` ≡ (φ` , 0) is the set of basis functions for the first component of the Cauchy data v 0 (s), and
φ?` ≡ (0, φ?` ) is the set of basis functions for the second component of the Cauchy data v 1 (s), ` = 1, ...L. In
our numerical experiments, we will consider the same set of basis functions for both components φ` ≡ φ?` .
The coefficients (u0p` , u1p` ) with p = 1, 2 and ` = 1, ...L, are the unknown expansion coefficients that need to
be determined.
To obtain the approximation of the discrete densities/traces vγp , (p = 1, 2) at the points (xj , yk ) ∈ γp , we
will consider the following Taylor expansion:
vγp ≡ vNp (xj , yk )|γp = vp0 (sj,k ) + dj,k vp1 (sj,k ) + O(d2j,k ).
(3.20)
Here, sj,k is the value of the arc length s at the point where the continuous boundary Γp intersects the normal
constructed from the point (xj , yk ) ∈ γp to Γp . The parameter dj,k is the shortest distance from (xj , yk ) ∈ γp
to the intersection point sj,k of the normal with Γp . dj,k is taken with the plus sign if (xj , yk ) ∈ Ωp , and
with a minus sign if (xj , yk ) ∈
/ Ωp , see Figure 3.2. Thus, we have
vγp ≈ vp0 (sj,k ) + dj,k vp1 (sj,k ).
(3.21)
Let us recall the operator Πγp Γp , Def. 3.2, Section 3.1 that assigns vγp , (p = 1, 2) to every vΓp ≡ (vp0 (s), vp1 (s))
from the space of continuous Cauchy data. We will construct such an operator Πγp Γp according to the above
Taylor formula (3.21), hence:
vγp ≈ Πγp Γp vΓp ,
Πγp Γp vΓp :=
vp0 (sj,k )
+
(3.22)
dj,k vp1 (sj,k ).
(3.23)
For example, if vΓp = (vp0 (s), 0), then we have that Πγp Γp vΓp = vp0 (sj,k ). Similarly, if vΓp = (0, vp1 (s)),
then the action of the operator Πγp Γp is given as Πγp Γp vΓp = dj,k vp1 (sj,k ). Thus, we have the following
finite-dimensional approximation for the discrete density vγp ∈ Vγp :
vγp ≈ ṽγp = Πγp Γp ṽΓp =
L
X
`=1
u0p` (Πγp Γp φ` ) + u1p` (Πγp Γp φ?` ) ,
(p = 1, 2)
(3.24)
where ṽγp is as defined previously in (3.12), Section 3.1.
It follows that the approximation of the density/trace vγp is the function of the unknown coefficients u0p`
and u1p` which need to be determined. Once the expansion coefficients u0p` and u1p` are obtained, the density
ṽγp ≈ vγp ≡ uγp is reconstructed using formula (3.24).
3.3. System of Boundary Equations: Discrete Variational Formulation
As we showed in Section 3, densities/traces (vγ1 , vγ1 ) range over the solutions of the Boundary Equation
(3.8) on each discrete grid boundary set (γ1 , γ2 ). Therefore, using the above approximation (3.24) for vγp in
the system of Boundary Equations (3.8), we will obtain the system of linear equations with |γp | equations
for 2L unknowns u0p := (u0p1 , ..., u0pL ), u1p := (u1p1 , ..., u1pL ):
Bp0 u0p + Bp1 u1p = 0,
(p = 1, 2).
(3.25)
Here, |γp | is the total number of points in the set γp (p = 1, 2) and (u01` , u11` ), (u02` , u12` ) are the unknown
expansion coefficients of (ṽΓ1 , ṽΓ2 ) (3.19). Matrices Bp0 and Bp1 are defined as Bp0 := (b0p1 , ..., b0pL ) and
Bp1 := (b1p1 , ..., b1pL ). Here, columns b0p` and b1p` are |γp | dimensional vectors whose components are computed
as the values at the points of the set |γp | of the grid functions Qγp [Πγp Γp φ` ] and Qγp [Πγp Γp φ?` ], respectively.
For more a detailed discussion on the computation of these matrices please see Appendix Section 7.
14
Notice that the linear system (3.25) will be the overdetermined linear system since we have to have
|γp | > L for the accurate resolution of the density vγp . Finally, let us define
Gp (ṽΓp ) := ||Qγp [Πγp Γp ṽΓp ]||2Vγp = ||Bp0 u0p + Bp1 u1p ||2Vγp ,
(p = 1, 2)
(3.26)
Therefore, at each time level ti+1 we have the following discrete variational formulation of (3.13):
Definition 3.4. Find the unique weak solution pair (ṽΓ1 , ṽΓ2 ) ∈ ṼΓ1 × ṼΓ2 for which the constants (u01` ,
u11` ) and (u02` , u12` ) with ` = 1, ..., L minimize the functional:
||l(ṽΓ1
˜ Γ1 , ṽΓ2 + ū
˜ Γ2 ) − ψ||2Φ + ||lΓ (ṽΓ1 + ū
˜ Γ1 , ṽΓ2 + ū
˜ Γ2 ) − φ||2Φ +
+ ū
˜ Γp is equal to ū
˜ Γp :=
Here, ū
P
L
`=1
ū0p` φ` ,
2
X
Gp (ṽΓp ).
(3.27)
p=1
1 ?
0
1
ū
φ
`=1 p` ` , and the coefficients (ūp` , ūp` ) are obtained using
PL
˜ Γp .
the values of the particular solution ūNp at the points of the set γp , in other words using ūγp ≈ Πγp Γp ū
The values ūγp are known values since the particular solution ūNp is constructed as the solution of the
simple auxiliary problem ((DAP, Def. 2.3), Section 2.1) (for more details see the Remark and Def. 2.8 after
Proposition 2.7 in Section 2.1, as well as discussion in Section 5). Formulation (3.27) is the well-known
least-square problem.
4. Algorithm
In this section, we will give brief summary of the important steps of the Algorithms Composition Scheme
for the model problem (2.5), (2.2) - (2.3) in the composite domain Ω̄ = Ω̄1 ∪ Ω¯2 . As it was derived in
Sections 3.1 - 3.3, our algorithm will be based on the discrete variational formulation (3.27). Let us note
a few important points of the method we developed. Firstly, the order of the operations of the proposed
framework does not increase with the choice of the numerical discretization used with it. Moreover, the
overall complexity of the method reduces to the several solutions of simple auxiliary problems on regular
Cartesian meshes (no need for the generation and storage of the unstructured meshes; no need for the design
of the schemes on the unstructured meshes). Finally, the selection of the auxiliary problems and meshes for
each domain Ωp is totally independent of each other, and is done based on the idea of the simplicity and
efficiency of the resulting numerical scheme.
We have the following steps at each time level:
1. Construct a discrete functional Gp (ṽΓp ) ≡ ||Bp0 u0p + Bp1 u1p ||2Vγp for each domain Ωp , (p = 1, 2).
This functional is the weak formulation of the Boundary Equation (3.8) (note that the minimum
of Gp (ṽΓp ) gives an approximation to the Cauchy data of the general solution to the homogeneous
equation in each subdomain).
To build Gp (ṽΓp ):
• Select the auxiliary domain denoted here as Ω0Gp , and place the original domain Ωp into the
auxiliary domain for the computation and construction of Gp (ṽΓp ), (p = 1, 2). The choice of
the auxiliary domain Ω0Gp should be convenient for computation, so we will choose it to be a
square. Select Cartesian mesh 2np × 2np for each of the auxiliary domains Ω0Gp , with np being
a positive integer and (p = 1, 2).
• Recall that ṽΓp ∈ ṼΓp is the finite dimensional approximation of the continuous Cauchy data
vΓp ∈ VΓp (see formula (2.6) in Section 2, as well as (3.19) in Section 3.2), and it is constructed
using the set of the basis functions φ` (s) for the first component of the Cauchy data, as well
?
as the set of the basis functions
second component
of the Cauchy data (formula
Pφ` (s) for the
PL
L
0
1 ?
(3.17), Section 3.1) : ṽΓp :=
`=1 up` φ` ,
`=1 up` φ` .
• Construct matrices Bp0 and Bp1 for each domain. This construction reduces to the computation
of the difference potentials (or to the solution of the simple auxiliary problems).
Remark: However if time step will not change during the simulations, and if the geometry,
the grid, and the system of basis functions (3.15) will not depend on time, then it will suffice
to precompute matrices Bp0 and Bp1 once and to store them.
See details in Section 3.3 and in Appendix Section 7.
15
2. Next, choose the new auxiliary domain denoted as Ω0ūp and place the original domain Ωp into the
auxiliary domain for the computation of the particular solution ūNp , (p = 1, 2). As before, the
choice of the auxiliary domain Ω0ūp should be convenient for computation, so we will choose it to be
a square. Select Cartesian mesh 2mp × 2mp for the auxiliary domain Ω0ūp , where mp is a positive
integer. Find any particular solution ūNp , (p = 1, 2) as the solution of the simple auxiliary problem
(see Remark and see Def. 2.8 after Proposition 2.7 in Section 2.1, as well as the discussion at the
beginning of Section 5). After that, calculate their Cauchy data on Γp , (p = 1, 2).
3. Solve the joint discrete variational problem: find Cauchy data ṽΓp ∈ ṼΓp , (p = 1, 2) that minimizes
the functional (3.27), Section 3.3:
˜ Γ1 , ṽΓ2 + ū
˜ Γ2 ) − ψ||2Φ + ||lΓ (ṽΓ1 + ū
˜ Γ1 , ṽΓ2 + ū
˜ Γ2 ) − φ||2Φ +
||l(ṽΓ1 + ū
2
X
Gp (ṽΓp ).
p=1
Remark: Let us note that if the Dirichlet to Neumann map is well defined for the problem, then
one can solve the overdetermined linear system (3.25) for the unknowns u1p := (u1p1 , ..., u1pL ) in terms
of u0p := (u0p1 , ..., u0pL ) using the least squares method:
Bp0 u0p + Bp1 u1p = 0,
(p = 1, 2),
This step can be viewed as the construction of the Dirichlet to Neumann map: to each set u0p assign
a set u1p that minimizes Gp (ṽΓp ). In this case, the general solution in each Ωp will be approximated
by a difference potential whose density will depend only on u0p .
4. Using the obtained Cauchy data ṽΓp , construct ṽγp ≈ Πγp Γp ṽΓp , Section 3.2. After that, extend the
computed densities ṽγp from the grid boundaries γp to the interior of each domain Ωp by computing
difference potentials PNp ṽγp in each domain Ωp .
At each time level reconstruct the desired approximation to the model (2.5), (2.2) - (2.3) in the
composite domain Ω̄ using the discrete generalized Green’s formula:
uNp = PNp γp ṽγp + ūNp ,
(p = 1, 2)
(see Proposition 3.1 and equation (3.5) in Section 3.)
Remark: Let us emphasize the important flexibility of the algorithm to choose different auxiliary
domains Ω0Gp and Ω0ūp for the computations of the projection and of the particular solution. This
provides an opportunity to consider different grids and combine different discretizations if necessary.
5. Numerical Examples
In this section, we demonstrate the performance of the proposed Algorithms Composition Approach on
several test problems. In the numerical experiments below, we consider the set of the basis functions on
Γp , (p = 1, 2) that is defined as:
2π 2π s , φp,3 = sin
s , ...
|Γ|
|Γ|
2π
2π
φp,2N (s) = cos
N s , φp,2N +1 = sin
Ns
|Γ|
|Γ|
φp,1 (s) = 1,
φp,2 (s) = cos
(5.1)
We assume that the sets φ ≡ φ? in the numerical examples below. In all of the tests, we will consider the
heat equation (2.5) in the composite domain Ω̄ ≡ Ω̄1 ∪ Ω̄2 as our model problem. Here, the heat equation
serves as a simplified model for more realistic systems of the materials, fluids, or chemicals with different
properties in different parts of the domains (for example, the ocean-atmosphere models, chemotaxis models,
or blood flow models [2, 36, 43, 42]). We consider equation (2.5) with the known analytical expressions for
the exact solutions. This allows us to study the errors in the approximate solutions that will depend on the
size of the auxiliary domains, mesh sizes, total number of the basis functions on Γp , etc.
Similar to [48], we will first define three functions
u(1) (x, y) := cos(x) cos(y),
16
(5.2)
u(2) (x, y) := sin(c1 x) sin(c2 y)P5 (ρ/0.9),
(5.3)
and
u(3) (x, y) := max
1 − ρ2 ,0 .
1 + 4ρ2
(5.4)
Here, the Cartesian coordinates (x, y) of points are related to their polar coordinates (r, θ) as (x, y) =
(r cos(θ), r sin(θ)). The interface boundary Γ ≡ Γ2 of the domain Ω2 is parametrically defined in polar
coordinates (r, θ) by the relation r(θ) ≡ rΓ (θ) = 1 + 0.22 sin(kθ θ), where kθ is the parameter. ρ = r/rΓ (θ)
in (5.3) - (5.4). See Figures 5.1 - 5.4 for the examples of the domains that are used in the numerical tests
below.
The function P5 (x) is continuous: identically equal to 1 for x ≤ 0; vanishes for x ≥ 1 and, on the interval
0 ≤ x ≤ 1, it is a unique ninth-degree polynomial whose derivatives up to the fourth order vanish at the
endpoints of 0 ≤ x ≤ 1:

 1, x ≤ 0,
1 − 126x5 + 420x6 − 540x7 + 315x8 − 70x9 , 0 ≤ x ≤ 1,
P5 (x) :=
(5.5)

0, x ≥ 1.
1
b
Let us note that due to the multiplier P5 (ρ/0.9), the function
u(2) (5.3) vanishes outside of Ω2 and in a
b
(2)
b1
neighborhood of Γ ≡ Γ2 . At the same time, u exhibits bstrong
oscillations deep inside Ω2 .
11
11b1 b1
0
1 b1
1
1bb0b1
0
1
b 1bb1
1
1
1
δΩb 1
1
0
11 11
1bb11 b1
1
1
11 1 1
1
1 1
bb1
1 1
1
1
1b1
11
b1
1
1
1 1 1
b1
1
1 1 11
b1
1
KKK
K1
K
1
K
K
K KK
1
1
K
K
K
KΩ1 , Ω2 , and
Figure
2: 2:Sketch
withboundary
boundary
twoKsubdomains
Ω1 , the
Ω2 ,interface
and theΓ interface Γ
1
Sketchofofdomain
domain ΩΩwith
∂Ω,
two subdomains
11∂Ω,
K
gure
2:Figure
ofof
domain
Ω
with
boundary
∂Ω,
twosubdomains
subdomains
Ω
Ωand
and
Figure
2:2:Sketch
Sketch
ofof
domain
Ωwith
withboundary
boundary
∂Ω,
two
Ω
, the
Ω
,interface
theΓ the
interface
Γ Γ
Figure
Sketch
domain
Ω
∂Ω,
two
subdomains
Ω
,
Ω
,
Γ interface
K
1 ,2the
2 ,interface
Figure
2:
Sketch
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
1and
1
2
1
2
Figure
2:Sketch
Sketch
of
domain
Ωwith
with
boundary
∂Ω,
two
subdomains
Ω
,Ω
, and
the
interface
Γ
K
1Ω
2and
gure
2:
Sketch
of
domain
Ω
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Figure
2:
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
,
the
interface
Γ
1
2
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
1
2
Figure 2: Sketch of domain Ω with boundary ∂Ω, two subdomains Ω , Ω , and
1 the2 interface Γ
"
"
""
2
00
b
22 2
" "
""
""
"
1.5
1.52 2 2
1.5
1.5
""
!
!
1.5
0.5
0.51
1 !!
1
!
0.5
!
1!
0.5
0
0.5
−0.5
−0.5
−0.5 0
! !
"
!
"
!
0
"
"
!
!
0.5
0
"
"
!
!!
0.5
00
0.5
0
"
!
2
1
1 1.5
1.5
1 1.5
1
"
"
!
0
−0.5
0
−1
−1
−0.5
−1 −0.5
"
"
−0.5 −0.5
−1.5
−1
−1.5
!
−1
−1.5−1
−2
−2−1
−1.5
!!
−1
−1.5
−2 −1.5
−2
−1.5
−2
−1.5
−2
−1.5
−1.5 −1.5
−2
−2 −2
−1
−0.5
−1−1
−0.5!
−0.5
−1.5
0
!
0.5
0 0
!
−1
−0.5
1
0.5 0.5
0
1.5
1
2
1 1.5
0.5
1.5 2
2
1
1.5
1
1.5
2
Figure
2:2:
Sketch
ofdomain
domain
with
boundary
∂Ω,
two
subdomains
Ω221,Ω
,and
Ω
and
the
interface
Γ Γ
Figure
2:
Sketch
of
ΩΩΩthe
with
boundary
∂Ω,
two
subdomains
and
the
Γ
11,2,,Ω
22,due
Figure
Sketch
Ω
with
boundary
∂Ω,
two
subdomains
Ω
Ω
,interface
and
interface
Figure
2:
Sketch
ofdomain
domain
with
boundary
∂Ω,
two
subdomains
Ω11level.
,Ω
the
Γ interface
where
u denote
theof
solution
to
difference
scheme
(2.15)
at each
time
Hence,
to the
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
K
Figure
Sketch
ofofdomain
ΩΩ
with
boundary
∂Ω,
subdomains
Ω
,1 Ω
ΩHence,
and
the
interface
Γ Γ
1(2.15)
2,,2on
where
u2:denote
thesolution
solution
to
the
difference
scheme
atat
time
level.
Hence,
due
to interface
where
uuniqueness
denote
the
scheme
(2.15)
time
level.
due
to due due
au
argument,
PN to
v
+
fNdifference
coincides
with
the(2.15)
solution
ueach
of
the
scheme
N Hence,
: the
Figure
2:
Sketch
domain
with
boundary
∂Ω,
two
subdomains
Ω
,
and
Γ
γthe
γto
Neach
where
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
to
1
2
K
ere
u
denote
the
solution
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
t
K
uu denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
a where
uniqueness
argument,
PNvγto
vto
+
fdifference
coincides
with
the
solution
uΩthe
of
the
scheme
(2.15)
on
N:
γ+
N
N
0
where
u =denote
the
solution
the
scheme
(2.15)
each
time
level.
Hence,
due
to
denote
the
solution
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
awhere
uniqueness
argument,
P
f
coincides
with
the
solution
u
of
scheme
(2.15)
on
N
:
Fig.
5.1:
Test
Problem
1:
Example
of
the
auxiliary
domain
Ω
,
domains
and
Ω
and
the
boundary
Γ
of
udenote
P
v
+
f
.
�
γ
N
γ
N
N
1
2
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
γ
N
N
γ
N
re
u
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
γ toto
Nγ v
Nfthe
N each
where
denote
the
solution
difference
scheme
(2.15)
at K
Hence,
due
to
K
where
u
solution
the
difference
scheme
(2.15)
at
time
Hence,
due
to
aniqueness
uniqueness
argument,
P
coincides
with
the
solution
utime
of
the
scheme
(2.15)
on
Non
: to
Sketch
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,of
Ωtime
,level.
and
the
interface
γv+
N
N
N
aua=
uniqueness
argument,
P
+
fN
coincides
with
the
solution
uat
of
the
scheme
(2.15)
on
here
uFigure
denote
the
the
difference
scheme
(2.15)
at
each
time
level.
due
to
1level.
2the
where
uNdenote
denote
the
solution
to
difference
scheme
(2.15)
each
level.
P
v
ffN
coincides
with
the
solution
u
scheme
(2.15)
N
vu2:
+
ffthe
.. solution
�:Γdue
γ+
N
γto
N
Keach
Ω2 ;of
parameter
=
0the
N
γargument,
N
N
γγγN
N
uniqueness
argument,
P
coincides
with
the
solution
u
of
the
scheme
(2.15)
onHence,
N
:N N
=P
P
vγ+
+
�
auuwhere
uniqueness
argument,
P
v
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
:
γkθ+
γγγv+
N
N
γ domain
Na
N
γγthe
N
N
N
N
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
=
P
v
f
.
�
N
N
N
γ
N
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
a
uniqueness
argument,
P
v
+
f
coincides
with
solution
u
of
the
scheme
(2.15)
on
N
γ
N
γ
N
N
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
here
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
niqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
γ
N
γ
N
N
1of
u=
=
v
+
ffargument,
� to
uwhere
=
v+
+
f(2.25)-(2.26)
.the
�=
NNN
γγcomment
N
N
γto
N
Nu
Figure
2:
domain
Ω+γtosolution
with
boundary
∂Ω,space
two
subdomains
Ωthe
,2 solution
Ωlevel.
,uand
the
interface
γN
N
N
γP
N
u
denote
solution
the
difference
scheme
at
each
time
Hence,
due
γ
N
N
γSketch
uniqueness
P
v
+
fbriefly
coincides
with
the(2.15)
solution
u
on
N
:::
uniqueness
argument,
P
vγγγthe
fbriefly
coincides
with
the
solution
of
the
scheme
(2.15)
on
N�
1the
2scheme
v
+
.of
uP
=
P
+..+
fNNfof
. let
�
N
N
Remark:
Finally,
let
us
the
accuracy
of
the
approximation
u
N
NN
γv
N
γf
P
v
+
fvN
uabout
ofabout
(2.1)
- space
(2.3).
One
would
expect
that
uaNP
=
P
v
�
γ
N
γu
N
Remark:
Finally,
us
comment
the
accuracy
of
the
approximation
=
γ
N
N
γ
N
γ
N
γ
N
=
P
.
�
N2: v2:
N
γ5.1.
N
Finally,
letof
us
comment
briefly
about
thewith
space
accuracy
of the
approximation
uN
=
Figure
Sketch
Ω+
with
boundary
∂Ω,
subdomains
Ω
Ωscheme
and
the
interface
auNPRemark:
uniqueness
argument,
P
coincides
with
the
solution
u
of
the
(2.15)
on
N��
:Γ
1the
2 ,2and
u
P
+
fγSketch
. the
Figure
domain
Ω
with
boundary
∂Ω,
two
subdomains
,Ω
Ω
,Hence,
and
the
interfac
First
Example
γ
N
N
uN
=
P
v
�
uniqueness
argument,
Pdomain
vNγthe
+solution
fsolution
coincides
the
solution
uorder
of
(2.15)
on
N
γSketch
N
γ
=
+
f2:
.Finally,
Figure
of
domain
with
∂Ω,
two
subdomains
Ωapproximation
,,1approximation
the
interface
γ
N
γfu
N
γv
N
γN
Nfbriefly
N
a=
uniqueness
argument,
P
v
+
fbriefly
coincides
with
the
solution
uthe
of
the
scheme
(2.15)
on
N
:Γ
1+
2 ,scheme
uRemark:
will
converge
to
the
continuous
solution
uboundary
in
the(2.1)
discrete
Holder
norm
ofthe
qΩ
� that
(with
γ
N
γ
Finally,
let
us
comment
about
the
space
accuracy
of
the
u
=
=
P
v
+
fN
.of
where
denote
solution
to
the
difference
scheme
(2.15)
at would
each
time
level.
due
to=u
Remark:
Finally,
let
us
comment
about
the
space
accuracy
of
the
u
γΩ
γthe
N
N
N
P
v
+
of
(2.25)-(2.26)
to
u
of
(2.3).
One
expect
the
solution
=
P
v
+
f
.
�
P
vv
+
f
of
(2.25)-(2.26)
to
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
N
Remark:
let
us
comment
briefly
about
the
space
accuracy
of
approximation
u
=
γN
N
N
γN
N
γ
γ
NP
N
γ
N
N
N
N
γv
γγ
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
approximation
u
=
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
=
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
would
expect
solution
N
In
the
first
example,
we
will
construct
our
test
problem
withOne
the analytical
solution that
that isthe
given
as,
N
N
p−�
γ
NNγ =
N
u
P
v
+
f
.
�
γ
N
γ
N
the
arbitrary
0+
<
� the
<
1)continuous
with
the
rate
O(h
)uuin
as
h(2.1)
→
0.
Here,
psolution
isaccuracy
the
order
of
the
accuracy
of
=
P
v
+
f
.
�
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
u
=
P
v
f
.
�
u
will
converge
solution
u
the
discrete
Holder
norm
of
order
q
+
�
(with
a
uniqueness
argument,
P
v
+
f
coincides
with
the
u
of
the
scheme
(2.15)
on
N
:
Remark:
Finally,
let
us
comment
briefly
about
the
space
of
the
approximation
u
=
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
of
(2.3).
One
would
expect
that
the
solution
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
γ
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(with
N
N
γ
N
γ
N
N
N
γ
N
γ
N
N
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
of
(2.1)
(2.3).
One
would
expect
that
the
solution
γ
N
γ
N
N
γ
N
N
N
γ
γ+ N
γ
N
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u=
Remark:
Finally,
let
us
briefly
about
the
space
accuracy
of
the
uN
P
v
of
(2.25)-(2.26)
to comment
the
solution
u uof
(2.1)
- the
One
would
expect
that
the
solution
γNγ+
+
fN
of
to
the
solution
of
(2.1)
-L(2.3).
(2.3).
One
would
expect
that
the
solution
uRemark:
will
to (2.25)-(2.26)
the
continuous
solution
in
the
discrete
Holder
norm
ofof
order
qapproximation
+
� that
(with
let
uscontinuous
comment
briefly
about
space
accuracy
the
approximation
=N
γP
N
γ Finally,
Nconverge
−t
−t
(1)
where
denote
the
solution
torate
the
scheme
(2.15)
atoperator
each
time
level.
Hence,
due
γfv
fuwill
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
the solutio
N
p−�
the
approximation
of
the
differential
operator
the
discrete
L
(we
γNwhere
edifference
uΩu
(x,
y)
=
ethe
u
,-by
(x,p
y)
∈
Ω
N
∆t
∆t,h
γ vN
1 , order
p−�
the
arbitrary
0
<
�
<
1)
with
the
O(h
)
as
h
→
0.
Here,
is
the
of
the
accuracy
of
u
will
converge
to
continuous
solution
in
the
discrete
Holder
norm
of
order
q
+
�
(with
u
converge
to
the
continuous
solution
u
in
discrete
Holder
norm
of
order
q
+
�
(with
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
u
=
P
v
+
f
.
�
u(x,
y,
t)
=
(5.6)
N
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
N
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
P
v
+
f
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
N
p−�
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
γ
N
N
γ
N
γv
u
will
converge
to
continuous
solution
uinabout
in
the
discrete
norm
of
order
q(with
+
�Hence,
(with
N
N
γ
N
N
−t
−t
(1)
(2)Holder
u
will
to
the
continuous
solution
u
in
the
discrete
Holder
order
qorder
+
�that
(with
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
uN
=N
γ the
N
γthe
N
N
where
denote
the
solution
to
the
scheme
(2.15)
at
each
time
Hence,
t
γwill
u
tothe
the
continuous
solution
u
the
discrete
norm
qlevel.
+
� that
N
Remark:
Finally,
let
us
comment
briefly
the
space
accuracy
of
the
approximation
uon
N
P
+
fu
of
to
solution
uof
of
(2.1)
-(2.3).
(2.3).
One
expect
that
the
solution
where
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
du
the
0converge
<
�(2.25)-(2.26)
<the
1)
with
the
rate
h
→
0.
Here,
psolution
isHolder
the
order
of
the
accuracy
of
N
v
+
fu
of
to
solution
u
of
-the
(2.3).
One
expect
the
solut
eis
udifference
(x,
y)
=
e(2.1)
(u
+
u
),
(x,
y)norm
∈
Ωwould
, of
γ
N
N(2.25)-(2.26)
assume
that
the
boundary
condition
(2.2)
approximated
by
(2.26)
with
the
same
orof
higher
Ω)
2would
γconverge
+
fconverge
of
(2.25)-(2.26)
to
the
solution
uas
(2.1)
-discrete
One
would
expect
the
solution
γvγthe
N
p−�
aγNwill
uniqueness
argument,
P
+
fO(h
coincides
with
u
oforder
the
scheme
p−�
to
continuous
solution
u)u
in
the
Holder
norm
of
order
q(2.15)
+
�due
(wit
γarbitrary
N
γvthe
N
γv
N
N
γP
approximation
of
the
continuous
differential
operator
L0.
by
the
discrete
operator
L∆t,h
(we
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
Here,
p
is
the
order
of
the
accuracy
of
p−�
∆t
the
arbitrary
0
<
�
with
the
rate
O(h
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
p−�
p−�
v
+
f
of
(2.25)-(2.26)
to
solution
of
(2.1)
(2.3).
One
would
expect
that
the
solution
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
2
a
uniqueness
argument,
P
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(with
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(with
γ
N
N
∆t
uNvarbitrary
=
P
vHence,
+
f�argument,
. the
∆t,h
γthe
N
γ
N
Remark:
Finally,
let
usto
comment
about
the
space
of
the
approximation
usolution
=(with
he
0f0γN
<
�=
<
1)16
with
the
rate
O(h
)u
as
h(2.1)
→
0.
Here,
pdiscrete
is
the
order
of
the
accuracy
of
N
P
v
+
of
(2.25)-(2.26)
to
solution
uof
of
-discrete
(2.3).
One
would
expect
that
the
the
<
1)
with
the
rate
O(h
) the
→
0.
Here,
paccuracy
is
the
the
accuracy
of
than
p).
for
the
central
finite
difference
scheme
that
is
here,
weorder
will
expect
O(h
)order
N
γof
N
approximation
of
continuous
differential
operator
Ldiscussed
by
the
operator
Lorder
(we
N�+
aγconverge
uniqueness
P
v
+
fbriefly
coincides
with
the
solution
uuN
of
the
scheme
on
N
u
will
converge
the
continuous
solution
uhas
in
the
discrete
Holder
norm
ofL
q(2.15)
+
γN
N
∆t
∆t,h
γ(5.3).
N
N
N
+
fN
(2.25)-(2.26)
the
solution
uas
(2.1)
-∆t
(2.3).
One
would
expect
that
the
p−�
γ
N
auNN
uniqueness
argument,
Pthe
+
fdifferential
coincides
with
the
solution
of
the
scheme
on
will
the
continuous
solution
u
in
the
Holder
norm
of
order
q(2.15)
�� (w
γarbitrary
with
cto
c2to
=
incondition
formula
Thus,
solution
u(x,
y,
t)by
will
exhibit
strong
oscillations
inside
theq
converge
to
the
continuous
solution
in
the
discrete
Holder
norm
of
order
+
� solutio
(with
γγrate
N
γv
N
N
assume
that
the
boundary
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
γ will
=
P
v
+
f
.
the
approximation
ofof
the
continuous
operator
L
the
discrete
operator
(we
1N
arbitrary
0
<
�
<
1)
with
the
O(h
)
h
→
0.
Here,
p
is
the
order
of
the
accuracy
o
p−�
∆t,h
γ
N
N
γ
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
p−�
the
approximation
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
∆t
∆t,h
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(with
∆t
∆t,h
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
u
=
P
v
+
f
.
�
p−�
the
arbitrary
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
N
the
arbitrary
0
<
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
the
approximation
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
γ
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
N
N
γ
N
he
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
γ
N
N
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(with
2
u
=
P
v
+
f
.
�
γp).
∆t
domain
Ω
,
but
none
in
the
domain
Ω
.
We
consider
four
test
problems,
Figures
5.1
5.4
with
k
=
0
in
∆t,h
p−�
∆t
N
p−�
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
∆t,h
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
2
1
θ
γ
N
N
γ
N
than
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
N
will
converge
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
qaccuracy
+
assume
the
boundary
condition
(2.2)
isis
approximated
by
(2.26)
with
the
same
ororder
higher
order
uNarbitrary
=
P
v0that
+<
. the
2 ) � (wit
<
�ffor
<
1)
with
the
rate
O(h
) )as
as
hhthat
→
0.
Here,
piskis
is
the
of
the
he
arbitrary
�to
<
1)
with
the
O(h
)about
h
→
0.
Here,
ppthe
the
order
of
the
accuracy
of
p−�
γ0that
Nthan
N
γHence,
N
assume
that
the
boundary
condition
(2.2)
is
approximated
by
with
the
same
or
higher
assume
the
boundary
condition
(2.2)
approximated
by
(2.26)
with
the
same
or
higher
order
approximation
of
continuous
differential
operator
L(2.26)
by
discrete
operator
L
2order
the
(DPM)
method,
the
reader
can
consult
[7,
4].
p).
central
finite
scheme
is
discussed
here,
we
will
expect
p−�
∆t
∆t,hof
Remark:
Finally,
let
us
comment
the
space
accuracy
of
approximation
u(w
the
Test
Problem
1,
Figure
5.1;
k(2.2)
=
2briefly
the
Test
Problem
2,
Figure
5.2;
with
=
3 will
in
the
Test
Problem
the
arbitrary
0for
<
�the
<
1)
with
the
rate
O(h
as
→
0.
Here,
the
order
of
the
accuracy
of
uthan
will
converge
to
the
continuous
solution
uof
in
the
discrete
Holder
norm
ofthe
order
q order
+
�L
(with
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
θdifference
θ we
than
p).
Hence,
the
central
finite
difference
scheme
that
is
discussed
here,
expect
O(h
)O(h
assume
that
the
boundary
condition
(2.2)
isin
approximated
by
(2.26)
with
same
or
higher
N
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
2accuracy
N
∆t
∆t,h
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
assume
that
the
boundary
condition
is
approximated
by
(2.26)
with
the
same
or
higher
order
∆t
∆t,h
p−�
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
(we
2
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
(2.1)
(2.3).
One
would
expect
that
the
solution
2
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
N
∆tby
∆t,h
γRemark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
ofthe
the
approximation
u
Nγ than
N
eapproximation
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
N
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
3,
Figure
5.3;
and
with
k
=
5
in
the
Test
Problem
4,
Figure
5.4.
p−�
he
approximation
of
the
continuous
differential
operator
L
the
discrete
operator
L
(we
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(
2
θ
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
∆t
∆t,h
ume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
same
or
higher
orde
∆t
∆t,h
Remark:
Finally,
let
comment
briefly
the
space
accuracy
of
the
approximation
u=
2
the
arbitrary
0 boundary
<
� of
<norm
1)us
with
the
rate
O(h
)about
as
h(2.1)
→
0.
Here,
pOne
is
the
order
of
the
accuracy
of
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
ofhigher
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
Passume
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
would
expect
that
the
soluti
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
∆t
∆t,h
that
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
order
the
(DPM)
method,
the
reader
can
consult
[7,
4].
γ
Nthe
N
the
approximation
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
han
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
rate
in
the
maximum
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
γ
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.3).
One
would
expect
that
the
solutio
∆t
∆t,h
assume
that
the
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
We
select
u(x,
y,
t)
in
(5.6)
to
be
the
exact
solution
for
the
heat
equation
(2.5),
we
set
time
step
dt
=
1.e−6
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(with
γ
N
N
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solutio
N
γ
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
γ the
Nγapproximation
N
2
esume
approximation
ofthe
the
continuous
differential
operator
L
by
the
discrete
operator
L
(w
2order
that
boundary
condition
(2.2)
is
(2.26)
with
the
same
orLhigher
higher
the
(DPM)
method,
the
reader
can
consult
[7,
4].
∆t
∆t,h
ume
that
the
boundary
condition
is
approximated
by
(2.26)
with
the
same
or
higher
or
the
(DPM)
method,
reader
can
consult
[7,
4].
the
of
the
continuous
differential
operator
Lby
by
the
discrete
operator
(we
Hence,
for
the
central
finite
difference
scheme
is
discussed
here,
we
will
expect
O(h
rate
in
the
maximum
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
ofO(h
assume
that
the
boundary
condition
(2.2)
isapproximated
approximated
by
(2.26)
with
the
same
or
higher
P
v
+
fHence,
of
(2.25)-(2.26)
to
the
solution
u)uscheme
of
(2.1)
-is
(2.3).
One
would
expect
that
the
solu
∆t
∆t,h
than
p).
for
central
finite
difference
scheme
that
discussed
here,
we
will
expect
)�order
22))
for
all
tests
innorm
Section
5.1,
we
useconsult
discrete
norm
(3.9)
with
αthat
=
0.5,
and
we
consider
time
interval
0.01].
p−�
the
(DPM)
method,
the
reader
can
[7,
4].
unthan
will
converge
tothe
the
continuous
solution
in
the
discrete
Holder
norm
of[0,
order
q+
+
�(wit
(w
γp).
Np).
N
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
order
Hence,
the
central
finite
difference
that
is
discussed
here,
we
will
expect
O(h
ate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
γu
N
the
(DPM)
method,
the
reader
can
consult
[7,
4].
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
(wit
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
N
the
(DPM)
method,
the
reader
can
consult
[7,
4].
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
�
2
N
2 ))
In
allfor
experiments
inin
this
Section,
we
will
choose
the
auxiliary
domains
sume
that
the
boundary
condition
(2.2)
isthe
by
(2.26)
with
the
same
or
higher
ord
han
p).
Hence,
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
assume
that
the
boundary
condition
(2.2)
is4].approximated
approximated
by
(2.26)
with
the
same
or
higher
order
2O(
p−�
the
(DPM)
method,
the
reader
can
consult
[7,
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a parabolic
model
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
than
p).
Hence,
for
the
central
finite
difference
scheme
is
discussed
here,
we
will
expect
O(h
3
Algorithms
Composition
Approach
based
on
theh
Difference
Potentials
Method
rate
in
the
maximum
norm
in
space.
For
detailed
discussion
and
the
proof
of
the
accuracy
of
in
the
maximum
norm
space.
For
the
detailed
discussion
and
the
proof
ofof
the
accuracy
p−�
uenrate
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
order
q
+
�)ofof(oo
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
he
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
→
0.
Here,
p
is
the
order
of
the
accuracy
in
the
maximum
norm
in
space.
For
detailed
discussion
and
the
proof
of
the
accuracy
p−�
he
(DPM)
method,
the
reader
can
consult
[7,
4].
N
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
the
arbitrary
0the
<norm
�central
<-central
1)
with
thedifference
rate
O(h
) on
as
hthe
→ Difference
0.∆t
Here,
p here,
is
the
order
of
theexpect
accuracy
∆t,h2 )
0
0based
under
consideration
(2.1)
(2.3).
than
p).
Hence,
for
the
finite
difference
scheme
that
is
discussed
we
will
expect
O(h
3
Algorithms
Composition
Approach
Potentials
Method
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
an
p).
Hence,
for
finite
scheme
that
is
discussed
here,
we
will
O(h
te
in
the
maximum
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
ΩFor
≡
Ωbased
:=
2]
[−2,
2],
3(DPM)
Algorithms
Composition
Approach
on
the
Difference
Potentials
Method
rate
the
maximum
norm
in
space.
the
detailed
discussion
and
the
the
accuracy
ofof
the
method,
the
reader
can
consult
[7,
4].
GFor
ū p−�
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
(DPM)
method,
the
reader
can
consult
[7,
4].
3in
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
rate
in
the
maximum
norm
in
space.
the
detailed
discussion
proof
ofMethod
the
accuracy
of
the
(DPM)
method,
the
reader
can
consult
[7,
4].
he
arbitrary
0the
< boundary
�introduction
<
1)
with
the
rate
O(h
)[−2,
as
h×the
→
0.LL
Here,
p
is
the
order
of
the
accurac
approximation
of
the
continuous
differential
operator
by
the
discrete
operator
L
(
the
(DPM)
method,
reader
can
consult
[7,
4].
the
approximation
of
the
continuous
differential
operator
by
the
operator
L
(w
∆t
assume
that
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
∆t,h
∆t
∆t,h
the
approximation
of
the
continuous
differential
operator
L
discrete
operator
L
3
Algorithms
Composition
Approach
based
on
Difference
Potentials
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
We
start
the
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
∆t
∆t,h
rate
inmaximum
themethod,
maximum
norm
in
space.
For the
the
detailed
discussion
andand
the the
proof
ofMethod
theofaccuracy
of (w
17
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
he
(DPM)
method,
the
reader
can
consult
[7,
4].
te
in
the
norm
in
space.
For
detailed
discussion
proof
the
accuracy
the
(DPM)
the
reader
can
consult
[7,
4].
2ord
the
(DPM)
method,
the
reader
can
consult
4].
(DPM)
method,
the
reader
can
consult
[7,
4].
system
(2.1)
- boundary
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in domain
Ω)by
infor
some
composite
assume
that
the
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
he
approximation
of
the
continuous
differential
operator
L
the
discrete
operator
Lorde
under
consideration
(2.1)
-central
(2.3).
assume
that
the
boundary
condition
(2.2)
is[7,
approximated
by
(2.26)
with
same
or
higher
than
p).
Hence,
for
the
finite
difference
scheme
that
isApproach
discussed
here,
we
will
expect
O(h
)ord
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
Below
we
develop
and
numerically
test
the
Algorithm
Composition
aafor
parabolic
model
∆t
∆t,h
assume
that
the
boundary
condition
(2.2)
is
approximated
by
same
or
higher
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
athe
parabolic
model
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
parabolic
model
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
parabolic
model
the
(DPM)
method,
the
reader
can
consult
[7,
4].
2
e
(DPM)
method,
the
reader
can
consult
[7,
4].
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
2o
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
under
consideration
(2.1)
(2.3).
under
consideration
(2.1)
-central
(2.3).
than
p).
Hence,
for
the
finite
difference
scheme
that
is
discussed
wesame
will
expect
O(h
han
p).
Hence,
for
the
difference
scheme
that
is
here,
will
expect
O(h
rate
in
the
maximum
norm
in finite
space.
For
the
detailed
discussion
and the
of
the
accuracy
of
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
under
consideration
(2.1)
- -central
(2.3).
assume
that
the
boundary
condition
(2.2)
isbased
approximated
by
(2.26)
the
or
higher
Algorithms
Composition
Approach
on
the
Difference
Potentials
Method
under
consideration
(2.1)
-central
(2.3).
than
p).
Hence,
for
the
finite
difference
scheme
that
isdiscussed
here,
we
will
expect
O(h
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
forwith
aproof
parabolic
model
two
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
1- (2.3).
2(2.8)
1its time-discrete
2
1illustration
2on in
system
(2.1)
- the
(2.3)
(or
formulation
in
form
domain
Ω) in
some
composite
under
consideration
(2.1)
3
Algorithms
Composition
Approach
based
the
Difference
Potentials
Method
We
start
introduction
of
the
scheme,
and
the
of
the
ideas,
by
considering
the
Algorithms
Composition
Approach
based
on
the
Potentials
Method
Below
we
develop
and
numerically
test
the
Algorithm
Composition
for
aconsidering
parabolic
model
3in
Algorithms
Composition
Approach
based
ondiscussion
the
Difference
Potentials
Method
We
start
the
introduction
ofin
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
We
start
the
introduction
of
the
scheme,
and
the
illustration
ofDifference
theApproach
considering
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
the
the
(DPM)
method,
the
reader
can
consult
[7,
4].
rate
in
maximum
norm
in
space.
For
the
detailed
discussion
and
of
the
accuracy
ate
the
maximum
norm
space.
For
the
detailed
and
the
proof
ofthe
the
accuracy
rate
inthe
the
maximum
norm
in
space.
For
the
detailed
discussion
the
proof
of
the
accuracy
under
consideration
(2.1)
-central
(2.3).
han
p).
Hence,
for
the
finite
difference
scheme
that
is
discussed
here,
we
will
expect
Oo
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
2
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
1
1
2
2
domain
- an arbitrary
bounded domain
R Algorithm
with the boundary
∂Ω - andApproach
such that it
ofMethod
Below
we Ωdevelop
and numerically
test inthe
Composition
forconsists
a two
parabolic
model
−2
−2
−2
−1.5
−1
−2−1.5
−2
−1.5
−2
−2
−1.5
−1
−1
−0.5
−1
−0.5
0
0
−0.5
−0.5
0.5
0
0
0.5
0.5
0.5
1
1.5
1
1.5
1
1.5
2
2
2
2
1
2
1
N1
We start
introduction
of the(2.8)
scheme,
and the illustration
of the ideas,
by considering
system
(2.1)the
- (2.3)
(or formulation
in its time-discrete
form in domain
Ω) in some
composite the
1
1b1
1
1
1
100
11b1bb1
1bb1
1bbb0001
1
δ
Ω
b 10
1 11
11
1 bb1
1
b 10
1
1 1 1
1
1
1
1 1
1
b1 b1
1
1
1
11 11
b 1b 1
1
1
1 1
b1
1
11
1 1
b1
1
KK K
1
K
1
K
K
K
K K
1
1
K
K
K
K
Figure
2: 2:Sketch
withboundary
boundary
∂Ω,
two
subdomains
Ω1 , the
Ω2 ,interface
and theΓ interface Γ
1
Figure
Sketchofofdomain
domain ΩΩwith
∂Ω,
two subdomains
Ktwo
1
K ΩΩ11, ,ΩΩΩ21,2,and
Figure
2: 2:Sketch
Sketch
ofof
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω21,, and
Ω12,,interface
and
theΓ interface
Γ
igure
2:
of
domain
Ω
with
boundary
∂Ω,
subdomains
Ω
Ω
,interface
and
theΓ interface
Γ
K
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
and
the
1
Figure
Sketch
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
the
2the
Figure
2:
Sketch
of domain
Ωwith
with
boundary
∂Ω,
two
subdomains
Ω
, 1and
interface
Γ
K
1, Ω
2Ω
gure
2:
Sketch
of
domain
Ω
boundary
∂Ω,
two
subdomains
,
Ω
,
and
the
interface
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
2
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
2
Figure
2:Sketch
Sketch
ofdomain
domain Ω
ΩΩwith
with
boundary
∂Ω,
two
subdomains
Ω 1, Ω
,Ω
and
the
Γ interface
1 ,Ω
2, interface
Figure
ofof
domain
boundary
∂Ω,
two
subdomains
Ω
and
the
interface
Γ ΓΓ
Figure
2:2:2:uSketch
of
with
boundary
∂Ω,
subdomains
Ω
Ω
,and
and
the
Figure
Sketch
domain
with
boundary
two
subdomains
Figure
2:
Sketch
of
domaintoΩ
Ωthe
with
boundary
∂Ω,
two
subdomains
Ω11,level.
ΩΩ22,1 ,and
interface
Γ interface
11,2the
22
where
denote
the
solution
difference
scheme
(2.15)
at each
time
Hence,
due
tothe
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
K
Figure
2:denote
Sketch
ofofdomain
ΩΩf with
boundary
∂Ω,
subdomains
Ω1,(2.15)
,1ΩΩ2,,2and
and
theinterface
interfaceΓΓ Γ
where
udenote
thesolution
solution
difference
scheme
(2.15)
at subdomains
each
time
level.
Hence,
a uniqueness
argument,
PN γto
vγ the
+
coincides
with
the
solution
uN each
of
the
scheme
ondue
NHence,
: to
Figure
2:
Sketch
domain
with
boundary
∂Ω,
two
Ω
the
where
u
the
solution
to
the
difference
scheme
(2.15)
at
time
level.
Hence,
due
to
N
1
2
where
u
denote
the
to
the
difference
scheme
(2.15)
at
each
time
level.
due
K
ere
u
denote
the
solution
to
the
difference
scheme
atN
each
time
level.
Hence,
due
uu denote
the
solution
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
a awhere
uniqueness
argument,
PNvto
vto
+
fdifference
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N : to
K
γto
γ2:
Ndifference
0 (2.15)
where
u
denote
the
solution
the
scheme
(2.15)
each
time
level.
Hence,
due
to
where
denote
the
solution
the
scheme
(2.15)
at
each
time
level.
Hence,
due
to
uniqueness
argument,
P
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
Fig.
5.2:
Test
Problem
Example
of
the
auxiliary
domain
Ω
,
domains
Ω
and
Ω
and
the
boundary
Γ
of
u
=
P
v
+
f
.
�
1
2
γ
N
γ
N
N
γ
N
N
γ
N
b
1
re
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
b
1
K
γ
K
N
γ
N
N
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:due
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
b
1
aa=
uniqueness
argument,
Pγγ2v
v+to
+
fNcoincides
coincides
with
the
solution
uat
of
scheme
(2.15)
N�:due
N
uP
solution
the
difference
(2.15)
at
each
time
level.
toto
1scheme
2 the
uuniqueness
P
+
b1
KuuuofNNthe
γ+
Nv
γγγ
Ncoincides
N
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
each
time
level.
Hence,
due
to
the
domain
k
=
uniqueness
argument,
PθNN
coincides
with
the
solution
uNthe
of
scheme
(2.15)
on
N
N
NNdenote
γγv
Nthe
2; P
uniqueness
argument,
P
ffN
with
the
solution
of
the
(2.15)
onHence,
N:on
:N
N
u
=P
vγargument,
+
ffargument,
.. Ω
�
awhere
f
with
the
solution
of
the
scheme
(2.15)
on
:
γ+
Ncomment
γγv
N
γ+
N=
N
γ
N
Remark:
Finally,
let
us
briefly
about
the
space
accuracy
approximation
u
=
N
a
uniqueness
P
v
+
f
coincides
with
the
solution
of
the
scheme
(2.15)
on
N
u
v
f
.
�
b
1
γ
N
γ
N
N
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
γ
N
N
γ
N
aNaaFigure
uniqueness
argument,
P
v
+
f
coincides
with
solution
u
of
the
scheme
(2.15)
on
N
:t
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
here
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
niqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
1
γ
N
γ
N
N
1
2
=
P
v
+
f
.
�
u
=
P
v
+
f
.
�
γ
N
γ
N
N
γ
N
γ
N
N
b
1
1
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
γ
N
γ
N
γ
N
N
γ
N
b
1
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
1
γ+solution
N
N
NN
uRemark:
=
v
+
f(2.25)-(2.26)
. .let
bb1
uniqueness
argument,
P
f
coincides
the
solution
u
of
the
scheme
Finally,
letus
uscomment
comment
briefly
the
space
accuracy
of
the
approximation
uN = on N�
00would
1 solution
2uN = ��(2.15)
v
+
.
P
vP
+
ffN
of
toγvthe
u ofabout
(2.1)
(2.3).
One
expect
that
the
γN
N
N
γ+
N
u=
=
P
v
f
.
�
1
000
γ
N
γ
N
Remark:
Finally,
briefly
about
the
space
accuracy
of
the
approximation
γP
N
bb1
γ
N
γ
γ
NP
N
γ
N
b-with
1
u
=
v
+
f
b
1
γ
N
N
γ
N
Figure
Sketch
Ω
with
boundary
∂Ω,
subdomains
Ωthe
,1Ω,Ωscheme
and
the
interface
1
uniqueness
P
vΩ
+
ffNNbriefly
coincides
with
the
solution
u
ofΩ
the
scheme
(2.15)
on
N
:Γ
Remark:
Finally,
letof
usdomain
comment
briefly
about
the
space
accuracy
ofof
the
approximation
u(2.15)
δΩ
1,the
2, ,scheme
u
=
P
v
fSketch
. .of
Figure
Sketch
domain
with
boundary
∂Ω,
two
subdomains
Ω
, Hence,
and
the
bbtwo
1
N
γ
N
uN
=
P
vf2:
�
uniqueness
argument,
P
vγthe
+
fsolution
coincides
with
the
solution
u
of
(2.15)
on
N
Figure
2:
of
domain
Ω
with
∂Ω,
subdomains
interface
Γ�
γ
2and
=
P
vNγ2:
+
f+
.Finally,
0solution
γ
N
N
γ
N
aP
uniqueness
argument,
P
+
coincides
with
u
of
on
N
1
N
γN
N
Nthe
uaP
=
P
vγN
+argument,
fN
�:�
u
will
converge
to
the
continuous
u1
in
discrete
Holder
norm
order
qΩ
+
�2that
(with
Remark:
Finally,
let
us
comment
about
accuracy
of
approximation
u=
=interface
γN
γ
N
where
denote
the
solution
to
the
difference
(2.15)
at of
each
time
level.
due
to=
γsolution
N
γγv
N
b1
Remark:
Finally,
let
us
comment
briefly
the
space
accuracy
of
the
approximation
u
1
N
v
+
fu
of
(2.25)-(2.26)
to
the
solution
uabout
ofthe
(2.1)
- space
(2.3).
One
would
expect
the
solution
γ
N
γ
1
N
v
+
of
(2.25)-(2.26)
to
uboundary
of
(2.1)
-scheme
(2.3).
One
would
expect
that
the
solution
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
uthe
=
N
=
P
v
+
f
.
γ
NN
N
N
N
1
γ
γ Remark:
γ
N
N
γ
N
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
the
approximation
u
=
N
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
N
p−�
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
N
u
=
P
v
+
f
.
�
γv P
N
the
arbitrary
0.(2.25)-(2.26)
�.the
<
1)continuous
with
the
rate
)uin
h
→
0.- (2.3).
Here,
psolution
is One
the
order
of
the
accuracy
of
γ
N
N
γNγ+
N
1
1
=
P
f(2.25)-(2.26)
�
u
=
v
f<
b1
P
+
fof
of
to
the
solution
uas
of
(2.1)
--discrete
(2.3).
would
expect
that
the
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
u
1
aNγγNwill
uniqueness
P
vthe
+
fO(h
coincides
with
the
ueach
of
the
(2.15)
on
N=
:N�to
u
will
to
solution
the
discrete
Holder
norm
ofexpect
order
qscheme
+
�approximation
(with
Remark:
let
us
comment
briefly
about
the
space
of
the
approximation
uN
P
+
fN
of+
(2.25)-(2.26)
to
the
solution
of
(2.1)
One
would
expect
that
the
solution
where
denote
the
solution
the
scheme
(2.15)
at
time
level.
Hence,
due
γ
to
the
continuous
solution
u
in
the
Holder
norm
of
order
qsolution
+solution
� solution
(with
N
N
N
γv
γconverge
NP
N
γvfu
N
0 accuracy
1
N=
P
v
+
of
(2.25)-(2.26)
to
the
solution
of
(2.1)
(2.3).
One
would
expect
that
the
γsolution
N
γto
Ndifference
N
γN
N
N
γ
1
N
γ+
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
usolutio
b-1
Nu
Remark:
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u=
1
P
v
+
fFinally,
ofargument,
(2.25)-(2.26)
to
solution
of
(2.1)
(2.3).
One
would
expect
that
the
solution
fconverge
to
the
uuuuu
of
(2.1)
-scheme
(2.3).
One
would
that
the
γ
γFinally,
N
N
N
γthe
NN
N
γ approximation
γv
u
will
converge
to
the
continuous
solution
in
the
discrete
Holder
norm
of
order
q
+
�ofthat
(with
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
where
u
denote
the
solution
to
the
difference
(2.15)
at
each
time
level.
Hence,
due
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
the
p−�
N
∆t
∆t,h
p−�
γ
N
γ
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
uuniqueness
will
converge
the
continuous
solution
in
the
Holder
norm
of
order
q(with
�Hence,
(with
uN
will
converge
to
the
continuous
solution
the
discrete
Holder
norm
of
order
qorder
+
�that
(with
1
Remark:
let
us
comment
briefly
about
the
space
accuracy
of
the
=
denote
solution
to
difference
scheme
(2.15)
at
each
time
Hence,
due
=
P
v0Finally,
+
fthe
.solution
1
1
bb1
N
P
v
+
fuof
of
to
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
the
arbitrary
<
<
1)
with
the
O(h
)in
as
h
→
0.
Here,
pOne
is
order
of
the
accuracy
of
P
vv
++
fN
(2.25)-(2.26)
solution
u)uu
of
(2.1)
-discrete
(2.3).
One
would
expect
that
the
solution
N
γ(2.25)-(2.26)
N
N
γof
N
aNNwhere
argument,
P
vthe
+rate
fthe
coincides
with
the
solution
uthe
of
the
scheme
on
N=
:N
1
1
u
will
converge
to
continuous
solution
in
discrete
Holder
norm
of
order
qapproximation
+
�+
γ
will
converge
to
the
continuous
solution
uin
in
the
discrete
Holder
norm
of
q(2.15)
+
�Hence,
(with
N
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u�u
1
p−�
γ
N
where
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
due
tot=
γthe
Nto
γto
N
N
γ+
γ
N
P
fu
(2.25)-(2.26)
the
solution
u
of
(2.1)
---1
(2.3).
would
expect
the
solution
u
will
converge
to��boundary
the
continuous
solution
up−�
the
discrete
Holder
norm
of
order
qlevel.
�+that
(with
N
Remark:
Finally,
let
us
comment
briefly
about
the
space
of
the
approximation
u(wit
1
bb1
1
1
where
denote
the
to
the
difference
scheme
(2.15)
at
each
time
level.
du
γN
N
N
v
fu+
(2.25)-(2.26)
to
the
solution
of
(2.1)
(2.3).
One
would
expect
the
soluti
assume
that
the
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
the
0argument,
<
<
1)the
with
the
rate
O(h
as
hthe
→
0.
Here,
psolution
isaccuracy
the
order
of
the
accuracy
of
γ arbitrary
1
γuniqueness
p−�
PaN
v
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
P
v
+
f
coincides
with
the
u
of
the
scheme
(2.15)
on
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
γ
N
N
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
p−�
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
γ
N
γ
N
N
γ
the
arbitrary
<<
1)
with
the
rate
O(h
as
h(2.1)
→
Here,
is
the
order
of
the
accuracy
of
∆t0.
∆t,h
p−�
1p−�
b∆t
1
P
v
+
of
(2.25)-(2.26)
to
the
solution
of
-discrete
(2.3).
One
would
expect
the
solution
the
approximation
the
continuous
differential
operator
L
by
discrete
operator
(we
2order
1
the
arbitrary
<
1)
with
the
O(h
)u
hthat
→
0.
Here,
is
the
order
the
accuracy
of
aarbitrary
uniqueness
argument,
P
vv
+
fsolution
coincides
with
the
uof
of
the
scheme
onNN:
uuniqueness
will
converge
to
continuous
solution
in
the
Holder
norm
of
order
qL
+
�q(2.15)
1
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
+
�solutio
(with
1
γP
N
N
∆t,h
uNv
=
v
+
.of�the
�
Remark:
Finally,
let
us
comment
briefly
about
accuracy
the
approximation
u
=on
γγrate
N
γ
N
Nof
γwill
P
+
f0Hence,
of
(2.25)-(2.26)
to
the
solution
of
(2.1)
-space
(2.3).
would
expect
that
the
solution
he
<
�0f�argument,
<
1)
with
the
rate
O(h
)uu)as
as
h
→
0.
Here,
pthe
iswe
the
order
of
the
accuracy
of
N
1
aarbitrary
uniqueness
P
+
fis
coincides
with
the
solution
u
of
the
scheme
the
0γargument,
<
1)
with
the
O(h
)p−�
as
h
→
0.
Here,
psolution
ispOne
the
order
of
the
accuracy
of
than
p).
for
the
central
finite
difference
scheme
isdiscrete
discussed
here,
will
expect
O(h
)that
N
N
N
γfof
N
u
converge
to
the
continuous
u
in
the
Holder
norm
order
q(2.15)
+(with
�+
(with
N
N
N
γγrate
N
N
the
approximation
of
the
continuous
differential
operator
L
bypwith
the
discrete
operator
L
(we
γ vfγN
+
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
N
1
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
∆t
∆t,h
1
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
�
(w
γ
N
N
γ
N
N
assume
that
the
boundary
condition
(2.2)
approximated
by
(2.26)
the
same
or
higher
order
γ
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(with
u
=
P
v
+
f
.
p−�
∆t
∆t,h
the
approximation
of
the
continuous
differential
operator
L
by
the
operator
L
(we
p−�
N
arbitrary
0the
<
�boundary
<
1)
with
the
rate
O(h
)uuas
as
h
→
0.
Here,
pdiscrete
isthe
the
order
of
the
accuracy
the
approximation
of1)
the
continuous
differential
operator
L-0.
by
the
discrete
operator
Lthat
(we
γfv
N
N
γconverge
N
∆t
∆t,h
assume
that
the
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
∆t
∆t,h
u
will
to
the
continuous
solution
in
the
discrete
Holder
norm
of
order
q
+
�
(with
1
p−�
1
bHere,
1
rate
in
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
accuracy
of
u
=
P
+
f
.
the
arbitrary
�
<
with
the
rate
O(h
)
h
→
p
is
the
order
of
the
accuracy
of
P
v
+
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
the
solution
N
the
arbitrary
0
<
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
will
converge
to
the
continuous
solution
in
the
discrete
Holder
norm
order
q
+
�
(with
the
approximation
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
γ
N
N
γ
N
1
γ
N
N
he
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
2
u
=
P
v
+
f
.
�
γp).
assume
that
the
boundary
condition
(2.2)
isis
approximated
by
(2.26)
with
the
same
orapproximation
higher
order
N
∆t,h
the
arbitrary
<
�the
< let
1)continuous
with
therate
rate
O(h
as the
his→
→
0.
pthe
isiswill
the
order
the
accuracy
of �
p−�
∆tHere,
∆t,h
K∆t
γ
N
N<
γ+0
N
Remark:
Finally,
us
comment
briefly
about
the
space
accuracy
of
the
p−�
than
Hence,
for
central
finite
difference
scheme
that
discussed
here,
we
expect
O(h
)the
N
will
converge
to
the
solution
u))) as
in
discrete
Holder
norm
ofofexpect
order
qaccuracy
+
� =
(wit
assume
that
the
boundary
condition
(2.2)
approximated
by
(2.26)
with
same
or
higher
order
=
P
v
f
.
2u
Nhe
arbitrary
0
�
<
1)
with
the
O(h
h
0.
Here,
p
the
order
of
the
K
p−�
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
γ
Napproximation
N
γ
N
K
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
arbitrary
0
<
�
<
1)
with
the
rate
O(h
as
h
Here,
p
is
the
order
of
accuracy
o
the
(DPM)
method,
the
reader
can
consult
[7,
4].
p−�
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
O(h
)
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
2
KK
the
arbitrary
0boundary
<<of
�of
<
1)
with
the
rate
O(h
))about
as
h the
→
0.
Here,
the
order
of
the
accuracy
ofu(w
uapproximation
will
converge
the
continuous
solution
uof
in
the
discrete
Holder
norm
of
order
q order
+
�)Lsolution
(with
∆t
K
∆t,h
the
the
continuous
differential
operator
L
byOne
the
discrete
operator
(we
1
Remark:
Finally,
let
us
comment
about
space
accuracy
of
the
approximation
1approximated
K
assume
that
the
boundary
condition
(2.2)
isbriefly
approximated
by
(2.26)
with
same
or
higher
the
approximation
of
the
continuous
differential
operator
LHere,
by
the
discrete
operator
L
(we
2
N
the
arbitrary
0for
�to
<
1)
with
the
rate
O(h
as
h
→
0.
ppwith
isis
the
order
of
the
accuracy
of
than
p).
Hence,
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
∆t
∆t,h
N
ssume
that
the
condition
(2.2)
is
by
(2.26)
the
same
or
higher
∆t
∆t,h
p−�
Remark:
Finally,
let
us
comment
briefly
the
space
accuracy
of
the
approximation
u
the
approximation
the
continuous
differential
operator
L
by
the
discrete
operator
(we
2L
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
K
2order
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
(2.1)
(2.3).
would
expect
that
the
K
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
N
1
∆t
∆t,h
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
γ
N
N
K
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
γ
N
e
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
o=
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
p−�
K
approximation
of
the
continuous
differential
operator
by
the
discrete
operator
L
(
2higher
1
he
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
K
∆t
∆t,h
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
2
∆t
∆t,h
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
ume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
ord
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
K
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
1 with
2of the
∆t
the
approximation
of
thecontinuous
continuous
differential
L
by
the
discrete
operator
L
(we
P
v
+
fthe
to
the
solution
of
(2.1)
--K
(2.3).
One
would
expect
that
the
assume
that
boundary
condition
(2.2)
is∂Ω,
by
(2.26)
with
the
same
or
higher
order
the
method,
the
reader
consult
[7,
4].
1 (2.26)
2 One
1approximated
K(2.3).
han
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
O(h
)solut
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
the
accuracy
∆t
∆t,h
γ(DPM)
NP
N
assume
the
boundary
condition
(2.2)
by
same
or
higher
order
+
fN
of(2.25)-(2.26)
(2.25)-(2.26)
to
the
solution
uon
of
(2.1)
(2.3).
would
expect
that
the
solutio
γ
u
will
converge
to
the
solution
uu
in
the
discrete
Holder
norm
of
order
qof
+
�∆t,h
(with
rate
in
the
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
Figure
2:
Sketch
of
domain
Ωcan
with
boundary
∂Ω,
two
subdomains
Ω
,proof
Ω
,interface
and
the
interface
Γ
γthat
N
P
v
+
fof
of
(2.25)-(2.26)
to
the
solution
One
expect
that
the
solution
N
γv
Figure
2:
Sketch
ofdomain
domain
Ω
with
boundary
two
subdomains
Ω
,(2.26)
Ω
and
the
Γexpect
3
Algorithms
Composition
Approach
based
the
Difference
Potentials
1would
2Method
rate
maximum
norm
in
space.
For
the
detailed
discussion
of
the
accuracy
of
γ
N
Nmaximum
1and
2 ,the
γin
e
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(w
2
the
(DPM)
method,
the
reader
can
consult
[7,
4].
∆t
∆t,h
me
that
the
boundary
condition
(2.2)
is
approximated
by
with
the
same
or
higher
ord
igure
2:
Sketch
of
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
ssume
that
the
boundary
condition
(2.2)
is
approximated
(2.26)
with
the
same
or
higher
order
K
2Γ
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
the
(DPM)
method,
the
reader
can
consult
[7,
4].
1
2
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
2Γ
gure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
∆t
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solu
∆t,h
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)�ofO(h
1
2
an
p).
Hence,
for
the
central
finite
difference
scheme
that
discussed
here,
we
will
expect
2with
p−�
the
(DPM)
method,
the
reader
can
consult
[7,
4].
that
the
boundary
condition
(2.2)
by
(2.26)
same
or
higher
order
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω111p,Holder
,Holder
Ω
,and
and
the
interface
Γ
1the
2we
γp).
N
N
than
Hence,
the
central
finite
difference
scheme
is
discussed
here,
we
will
expect
O(h
)
uthan
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(w
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
the
(DPM)
method,
the
reader
can
consult
[7,[7,
ate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
1
γ
2
than
p).
Hence,
for
the
central
finite
difference
that
is
discussed
here,
will
expect
O(h
)
uassume
will
converge
to
the
continuous
solution
u
in
discrete
norm
of
order
q
+
(wit
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
Ω
,
the
interface
Γ
1
2
N
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
is
the
order
of
the
accuracy
of
2
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
u
will
converge
to
the
continuous
solution
the
discrete
norm
order
q
+
�
(with
N
the
(DPM)
method,
the
reader
can
consult
4].
K
1
2
N
2)
2
sume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
orde
2
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
the
(DPM)
method,
the
reader
can
consult
[7,
4].
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
han
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
n
p).
Hence,
for
the
central
finite
difference
scheme
isHere,
discussed
we
will
expect
p−�
than
p).
Hence,
for
the
central
finite
difference
scheme
discussed
we
expect
3the
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of+
p−�
1here,
2,interface
than
p).
Hence,
for
the
central
finite
difference
that
discussed
here,
will
expect
O(h
)�of)of
will
converge
the
continuous
solution
u∂Ω,
in
the
discrete
Holder
norm
of
order
qO(h
(2o
eNrate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
the
accuracy
Figure
2:
Sketch
of
domain
Ω
with
boundary
two
subdomains
Ω
,here,
Ω
,operator
and
the
interface
Γ
where
uthe
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
Hence,
due
to
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
ofof
the
accuracy
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
Ω
,the
Ω
and
the
interface
ΓO(h
he
(DPM)
method,
the
reader
can
consult
[7,
4].
Ksubdomains
Figure
2:
Sketch
domain
Ω
with
boundary
∂Ω,
subdomains
1level.
2we
the
arbitrary
0Sketch
<
�<
<
1)
with
the
rate
O(h
)two
as
h
→
Here,
is
the
order
of
the
accuracy
rate
in
maximum
norm
in
space.
For
discussion
the
proof
ofwill
the
accuracy
Figure
2:
of
domain
Ω
with
boundary
∂Ω,
Ω
Ω
,pand
the
Γthe
the
arbitrary
0argument,
<
�of
<
1)
with
the
rate
O(h
)(2.15)
assubdomains
0.is
is
order
of
the
accuracy
1
2proof
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
L
(we
1 , and
2pp
arbitrary
0to
�(2.1)
<
1)
with
the
rate
O(h
h(2.15)
→
0.
Here,
is
the
order
accuracy
of
where
u
denote
the
solution
to
the
difference
scheme
at
each
time
level.
Hence,
due
to
2)
∆t
∆t,h
a
uniqueness
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
where
u
denote
the
solution
to
the
difference
scheme
at
each
time
level.
Hence,
due
to
under
consideration
(2.3).
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
γ
N
γ
N
N
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
K
an
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
p−�
Composition
Approach
based
on
the
Difference
Potentials
Method
3
Algorithms
Composition
Approach
based
on
Difference
Potentials
Method
ate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
ot
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of(2.15)
the
accuracy
of(w
the
(DPM)
method,
the
reader
can
consult
[7,
4].
where
umaximum
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
due
to
where
u
denote
solution
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
3uAlgorithms
Algorithms
Composition
Approach
based
the
Difference
Potentials
Method
in
the
norm
in
For
the
detailed
discussion
and
the
proof
of
the
accuracy
rate
the
norm
in
For
discussion
and
the
proof
of
the
accuracy
of(we
aassume
argument,
P
vspace.
+
fwith
coincides
with
the
solution
u
of
the
scheme
on
NLaccurac
:Ldue
γto
(DPM)
method,
the
reader
can
consult
[7,
4].
N
γ
N
N
the
(DPM)
method,
the
reader
can
consult
[7,
4].
0→
he
arbitrary
0maximum
<
�the
1)
with
the
rate
O(h
)∂Ω,
as
h
0.
Here,
pthe
the
order
of
the
Figure
2:
Sketch
of
domain
Ω
boundary
two
subdomains
,time
Ω
,Hence,
and
the
interface
Γ
where
uin
denote
the
solution
to
the
difference
scheme
(2.15)
each
time
level.
Hence,
to
where
uP
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
due
to
the
(DPM)
method,
the
reader
can
consult
auniqueness
uniqueness
argument,
P
v
+
fspace.
coincides
with
the
solution
uApproach
of
the
scheme
(2.15)
on
N
:due
ere
u
denote
the
solution
to
difference
scheme
at
each
time
level.
Hence,
1is
2Hence,
the
approximation
of
the
differential
operator
L(2.26)
by
the
discrete
operator
the
approximation
of
the
continuous
differential
operator
L
by
discrete
operator
that
the
boundary
condition
(2.2)
is
approximated
by
with
the
same
or
higher
order
γ
N
γcontinuous
Nthe
N
Fig.
Test
Problem
3:
Example
of
the
auxiliary
domain
Ωdiscussion
,(2.15)
domains
Ωat
and
Ω
and
the
boundary
ΓHence,
ofHence,
K
the
approximation
of
the
continuous
differential
operator
L
the
discrete
operator
L
=
+the
fand
.<
�
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
∆t
∆t
∆t,h
∆t,h
1 by
2parabolic
Below
we
develop
numerically
test
the
Algorithm
Composition
for
aΩ
model
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
re
u
denote
the
solution
to
the
difference
scheme
(2.15)
each
time
level.
due
where
u
denote
the
solution
to
difference
scheme
(2.15)
at
each
level.
due
to
γ5.3:
N
Nargument,
γv
N
∆t
∆t,h
K
K
where
u
denote
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
rate
in
the
maximum
norm
in
space.
For
the
detailed
and
the
proof
of
the
accuracy
of
uniqueness
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
N
N
Nthe
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ωlevel.
,scheme
Ωscheme
and
the
interface
3aathe
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
te
in
the
norm
in
For
the
detailed
discussion
and
the
proof
of
the
o
uniqueness
argument,
vγγγγvvspace.
+
coincides
with
the
solution
uuuN
the
scheme
(2.15)
on
N
: accuracy
auaN=
uniqueness
argument,
P
+
fcoincides
coincides
with
solution
uof
of
the
(2.15)
on
(DPM)
method,
the
reader
can
consult
[7,
4].
1scheme
2 ,Hence,
uuniqueness
Pdenote
v
+
ff(2.3)
�:Γdue
he
(DPM)
method,
the
can
consult
[7,
4].
N
γγ
N
the
(DPM)
the
can
consult
N
N
N
2orde
the
domain
Ω
kP
=
3N
γargument,
N
Nmaximum
γ
N
uniqueness
argument,
P
+
ftest
coincides
with
thethe
solution
of
(2.15)
on
N
:or
=
P
v
+the
..boundary
�
(DPM)
method,
the
reader
can
consult
[7,
4].
P
v
+
ffApproach
with
the
solution
of
the
(2.15)
on
N
:Nhigher
2 ;the
θreader
γγ+
N
N
γ method,
where
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
due
to
NHence,
γu
N
N
γreader
N
Nthe
system
(2.1)
-the
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
inN
some
composite
he
approximation
of
continuous
differential
operator
L
by
the
discrete
operator
L
where
u
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
to
under
consideration
(2.1)
-central
(2.3).
assume
that
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
niqueness
argument,
P
v
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
than
p).
for
the
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
approximation
uparabolic
=
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
3
Algorithms
Composition
based
on
the
Difference
Potentials
Method
assume
that
the
boundary
condition
(2.2)
approximated
by
(2.26)
with
the
same
or
higher
order
∆t
∆t,h
Below
we
develop
and
numerically
the
Algorithm
Composition
Approach
for
a
model
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
N
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
γ
N
γ
N
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
Below
we
develop
and
numerically
the
Algorithm
Composition
Approach
for
a
parabolic
model
a
uniqueness
argument,
P
v
+
f
coincides
with
solution
u
of
the
scheme
(2.15)
on
N
:
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
here
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
t
K
the
(DPM)
method,
the
reader
can
consult
[7,
4].
niqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
γ
N
γ
N
N
1
2
ueNuu(DPM)
=
P
v
+
ffN
.arbitrary
NP
γ γ γvγ can
N
N it consists
N
Nconsult
Nthe that
=
P
v
+
f+
��
2 with
γ
N
γN
=
P
v
+
. let
�=� 2 2
γdevelop
Nu
γmethod,
Nargument,
u
=
P
γ
N
N
γΩ
N
the
reader
[7,
4].
domain
-N
bounded
domain
in
R
the
boundary
∂Ω
-uand
such
of
aWe
uniqueness
+Approach
fApproach
coincides
with
the
solution
of
scheme
(2.15)
on
N�
:model
γan
N
N
γSketch
N
=
P
v
+
.f.f(2.25)-(2.26)
Below
and
numerically
test
the
Algorithm
Composition
Approach
for
asolution
parabolic
Remark:
Finally,
let
us
comment
about
the
space
accuracy
ofby
the
approximation
u
3
Algorithms
Composition
based
on
the
Difference
Potentials
Method
Remark:
Finally,
us
comment
briefly
about
the
space
accuracy
of
the
approximation
uwe
=
γΩ
N
γinthe
N briefly
N
P
v
+
fv
of
to
solution
u
of
(2.1)
-with
(2.3).
One
would
expect
that
the
start
the
introduction
of
scheme,
and
the
illustration
of
the
ideas,
considering
the
γ
Nthan
N
γwe
N
N
N
γ
N
Figure
2:
of
domain
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
under
consideration
(2.1)
(2.3).
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
will
expect
O(h
under
consideration
(2.1)
(2.3).
rate
in
the
maximum
norm
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
uniqueness
argument,
P
v
+
f
coincides
the
solution
u
of
the
scheme
(2.15)
N
under
consideration
(2.1)
(2.3).
3
Algorithms
Composition
based
on
the
Difference
Potentials
Method
ssume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
o�
1
2
than
p).
Hence,
for
the
central
finite
difference
that
is
discussed
here,
we
will
expect
O(h
)
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
subdomains
Ω
,
Ω
,
and
the
interface
under
consideration
(2.1)
(2.3).
=
P
v
+
f
.
3
Algorithms
Composition
Approach
the
Difference
Potentials
Method
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
than
p).
Hence,
for
the
central
finite
difference
scheme
is
discussed
here,
we
will
expect
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
onon
NO(
:Γ
Algorithms
Composition
Approach
based
Difference
Potentials
Method
γ
N
γ
N
N
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
1
2
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
γ
u
=
P
v
+
f
.
�
N
γ
N
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
γ
N
γ
N
N
γ
NG
Nwell
uNRemark:
=
P
vffor
�
uniqueness
argument,
PΩcontinuous
vγthe
+
fsolution
coincides
with
the
solution
uN
ofapproximation
the
scheme
(2.15)
on
N
γγFinally,
N
γ
1approximation
2 Hence,
=
P
v
f
.
N
N
γ+
N
γ(ṽ
N
γand
Nbriefly
N
two
disjoint
subdomains
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
the
construction
of
),
as
as
for
the
computation
of
the
particular
solution
ū
in
Ω
(note
γ
N
γ
N
Remark:
Finally,
let
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
u
=
P
v
+
f
.
�
let
us
comment
about
the
space
accuracy
of
the
u
=
1
2
1
2
1
2
1
Γ
N
1
u
will
converge
to
the
u
in
the
discrete
Holder
norm
of
order
q
+
�
(with
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
u
=
P
v
+
of
(2.25)-(2.26)
to
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
the
approximation
u
=
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
due
to
γ
N
N
γ
N
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
under
consideration
(2.1)
(2.3).
N
N
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
N
γ
N
N
N
3
Algorithms
Composition
Approach
based
Difference
Potentials
Method
γ
γ
3
Algorithms
Composition
Approach
the
Difference
Potentials
Method
N
N
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
We
start
the
introduction
of
the
scheme,
and
illustration
of
the
ideas,
by
considering
the
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
γ
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
N
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
0space.
0For
Figure
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,the
Ω
,,approximation
and
the
interface
the
(DPM)
method,
the
reader
can
consult
[7,
4].
rate
in
maximum
norm
in
space.
the
detailed
discussion
and
the
proof
ofMethod
the
accuracy
o
in
the
norm
For
discussion
and
the
proof
of
the
accuracy
of�
=
P
vfthe
+
fmaximum
.(2.25)-(2.26)
aNurate
uniqueness
argument,
P
vin
+
fO(h
with
the
solution
u
ofdomain
the
scheme
(2.15)
on uNu�:Γ�
1the
2proof
under
consideration
(2.1)
-central
(2.3).
u
=
P
v
f.f(2.25)-(2.26)
.auxiliary
han
p).
Hence,
for
the
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O
p−�
uwhere
=
P
v2:
+
.let
rate
in
the
maximum
in
space.
For
the
detailed
discussion
and
of
the
accuracy
these
domains
Ω
and
Ωcoincides
will
coincide
with
the
boundaries
ofexpect
the
Ω
see
Figures
Algorithms
Composition
Approach
on
the
Difference
Potentials
γ
N
γ
Nbriefly
N
γ
N
N
γ
N
γ
N
N
γ
N
2uu
ΓWe
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
1the
N
N
3
Algorithms
Composition
Approach
based
on
Potentials
ūu
G
1we
1γ
2norm
2to
Finally,
us
comment
briefly
about
the
space
accuracy
of
the
arbitrary
0Finally,
<
� the
<
1)
with
the
rate
)with
as
hbased
→
0.
Here,
pthe
is
the
order
of
the
accuracy
of
=
P
v
f+
P
v
of
(2.25)-(2.26)
to
the
solution
of
(2.1)
---space
(2.3).
One
would
that
solution
Remark:
Finally,
let
us
comment
about
the
space
accuracy
of
the
approximation
converge
to
continuous
solution
in
the
discrete
Holder
norm
of
order
qby
+
�composite
(with
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
uMethod
=
P
v
+
fN
of
the
solution
of
(2.1)
(2.3).
One
expect
that
the
solution
domain
Ω
-that
an
arbitrary
bounded
domain
in
R
the
boundary
∂Ω
-Difference
and
such
that
it
consists
of
N=
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
P
v
+
f+
of
to
the
solution
u
of
--discrete
(2.3).
One
would
expect
that
the
solution
γ
NRemark:
N
γ+
N
γwill
N
N
Below
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
Remark:
let
us
comment
briefly
about
the
accuracy
of
the
approximation
u
=� solution
Nu
N
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
considering
the
γN
N
N
N
γ
u
will
converge
to
the
continuous
solution
u
in
the
Holder
norm
of
order
q
+
(with
γ
N
N
γ
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
form
in
domain
Ω)
in
some
γ
N
γ
N
N
γ
N
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
γ
N
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
γ
N
nder
consideration
(2.1)
(2.3).
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
Below
we
develop
and
numerically
test
the
Composition
Approach
for
a
parabolic
model
γ
N
N
system
(2.1)
-γmethod,
(or
formulation
(2.8)
insimplicity
its
time-discrete
form
in
domain
Ω)
in proof
some
composite
γthe
5.1
-(2.3)
5.4).
At
the
same
time,
weApproach
will
select
theoperator
auxiliary
domain
Algorithms
Composition
based
on
the
Difference
Potentials
Method
We
start
the
introduction
of
the
scheme,
the
illustration
of
the
ideas,
by
considering
the
the
(DPM)
the
reader
can
consult
[7,
4].
(DPM)
method,
the
reader
can
consult
subdomains
are
here
only
for
the
of
the
presentation,
and
that
the
discussed
u
=
P
v
+
f(2.25)-(2.26)
.numerically
�
p−�
low
we
develop
and
test
the
Composition
Approach
for
aof
parabolic
mod
the
approximation
of
the
continuous
differential
L
by
discrete
operator
L
(we
N
N
γ
N
the
(DPM)
method,
the
reader
can
consult
[7,
4].
ate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
of
the
accurac
∆t
∆t,h
2as
2u
p−�
the
arbitrary
0Finally,
<
�considered
<
1)
with
the
rate
O(h
)and
h
→
0.
Here,
pthe
ispiecewise
the
order
of
the
accuracy
u
will
converge
to
the
continuous
solution
u2Algorithm
in
the
Holder
norm
ofof
order
qthe
+
�parabolic
(with
Remark:
Finally,
let
us
comment
briefly
about
the
accuracy
the
approximation
uN
=
two
disjoint
subdomains
Ω
and
Ω
:domain
Ω
=
Ω
∪
Ω
,u
Γ
=
Ω
∩
Ω
with
smooth
boundaries
Remark:
Finally,
let
us
comment
briefly
about
the
accuracy
of
the
approximation
u
=t
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
qconsists
+
�parabolic
(with
Remark:
let
us
comment
briefly
about
the
space
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
each
time
level.
Hence,
due
P
v
+
f-v
of
tothe
solution
of
(2.1)
-scheme
(2.3).
One
would
expect
that
the
solution
where
uP
denote
the
solution
to
the
difference
(2.15)
at
each
time
level.
Hence,
due
N
v
+
f
(2.25)-(2.26)
to
the
solution
of
(2.1)
(2.3).
One
would
expect
that
the
solutio
1with
2the
1
2
1discrete
2--space
u
will
converge
the
continuous
solution
u
in
discrete
Holder
norm
of
order
q
+
�
(with
N
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
would
expect
that
the
solution
P
v
+
f
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
N
γof
N
N
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
u
=
+
f
.
�
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:N
domain
Ω
an
arbitrary
bounded
domain
in
R
boundary
∂Ω
and
such
that
it
of
domain
Ω
an
arbitrary
bounded
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
under
consideration
(2.1)
(2.3).
γconverge
2
N
the
arbitrary
0
<
�
<
1)
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
accuracy
of
Below
we
develop
and
numerically
test
the
Composition
Approach
for
a
model
γ
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
model
γ
N
u
will
converge
to
the
continuous
solution
in
the
discrete
Holder
norm
of
order
q
+
�
(with
N
N
γ
N
N
γ
N
N
γ
N
γ
N
γ
N
N
γ
γ
γ
u
will
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(with
N
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
du
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
N
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solut
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
mode
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
idea
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
We
start
the
introduction
of
the
scheme,
and
illustration
of
the
ideas,
by
considering
the
under
consideration
(2.1)
(2.3).
under
consideration
(2.1)
(2.3).
0 p−�
γ
N
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
p−�
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
p−�
ΩΩ
=
[−1.6,
1.6]
×
[−1.6,
1.6]
der
consideration
(2.1)
-solution
(2.3).
2u
the
approximation
of
the
continuous
differential
Lthe
by
the
discrete
operator
L
(we
ow
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
aqLaccuracy
parabolic
mo
Γ
:=
Γ
and
:=
∂Ω
∪
Γthe
(see
3).
We
would
like
to
also
emphasize
that
the
two
the
arbitrary
0(2.25)-(2.26)
<
1)
with
the
rate
h
→
0.
of
the
accuracy
of
3
Algorithms
based
on
the
Difference
Potentials
Method
uconverge
will
to
the
continuous
solution
u)operator
in
the
discrete
Holder
norm
order
q∆t,h
+
�it
(with
the
arbitrary
0an
<
�<
<
1)
with
the
rate
O(h
as
hof
→
0.
Here,
ppsolution
is
the
order
of
the
accuracy
of
Remark:
let
us
comment
briefly
about
space
accuracy
of
the
approximation
u�N
=
where
denote
the
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
Gin
p−�
1γ
2
2i+1
N
P
v
+
fuγ1fN
of
to
the
solution
of
(2.1)
One
would
expect
that
the
solution
p−�
he
(DPM)
method,
the
reader
can
consult
4].
P
v
+
of
(2.25)-(2.26)
to
the
solution
u
(2.1)
-discrete
(2.3).
two
disjoint
subdomains
Ω
and
Ω
:Approach
Ω
=
Ω
∪))R
Ω
∩
Ω
with
smooth
boundaries
the
0Finally,
��Composition
<
1)
with
O(h
)as
as
h
→
0.
Here,
ppiecewise
is
the
order
of
the
accuracy
two
disjoint
subdomains
Ω
and
Ω
Ω
=
Ω
,u
Γ
=
with
piecewise
smooth
boundaries
i+1
P
v
+
f∂Ω
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
-Here,
(2.3).
One
would
expect
that
solution
aunder
uniqueness
argument,
v
+
fO(h
coincides
with
the
u
of
the
scheme
on
N
domain
Ω
-converge
arbitrary
bounded
domain
with
the
boundary
∂Ω
-order
and
such
that
consists
of
aystem
uniqueness
argument,
P
v
+
fFigure
coincides
with
the
uof
of
the
scheme
on
γarbitrary
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
+
�the
(with
N
1
2:rate
1∪
1∆t
22Here,
u
will
converge
to
continuous
solution
in
the
discrete
Holder
norm
of
order
q(2.15)
+
(with
γstart
1
2γ
1
22,[7,
N
N
the
of
the
continuous
differential
operator
L
by
the
discrete
operator
(we
2
will
to
the
continuous
solution
u
in
the
Holder
norm
of
order
q(2.15)
+
�of
(wit
We
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
u
=
P
v∪the
+
.Γ
�
N
N
γ
N
γ
N
N
consideration
(2.1)
-N
(2.3).
γapproximation
under
consideration
(2.1)
-(2.3).
(2.3).
γ
N
γ
N
N
he
arbitrary
0subdomains
<
�f�<
<
1)
with
the
rate
O(h
)
as
h
→
0.
p
is
the
order
of
the
N
γ
2∪
∆t
the
arbitrary
0γthe
<
1)
with
the
rate
O(h
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
Our
goal
at
each
time
level
tP
is
to
find
an
approximations
to
u
in
domain
Ω:
∆t,h
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
N
N
N
two
disjoint
Ω
and
Ω
:
Ω
=
Ω
Ω
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
N
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
1
2
1
2
1
2
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solutio
p−�
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
nder
consideration
(2.1)
We
start
introduction
of
the
scheme,
illustration
of
the
ideas,
by
considering
the
1
2
1
2
1
2
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
We
start
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
under
consideration
(2.1)
(2.3).
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(w
γ
N
N
elow
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
mod
γ
N
γ
N
N
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
γ
subdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
p−�
∆t
∆t,h
p−�
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
We
start
the
introduction
of
the
scheme,
the
illustration
of
the
ideas,
by
considering
th
3
Algorithms
Composition
Approach
on
Difference
Potentials
Method
P
v
+
fγ1N
of
(2.25)-(2.26)
to
the
solution
u
(2.1)
-discrete
(2.3).
One
would
expect
that
the
solution
Γ
:=
∂Ω
∪
Γ<
and
∂Ω
∪
(see
Figure
like
to
also
emphasize
that
the
two
∆t
∆t,h
3
Algorithms
Approach
based
the
Difference
Potentials
Method
p−�
aN
uniqueness
argument,
P
vΩ
+
fsolution
coincides
with
the
solution
uthe
of
the
scheme
(2.15)
on
N
:
er
consideration
(2.1)
-:=
(2.3).
the
of
continuous
differential
operator
L
by
the
discrete
operator
Lthe
(we
�in
1approximation
1
2the
2N
for
the
construction
of
the
G
(ṽ
),
and
we
will
choose
the
auxiliary
domain
uassume
will
converge
the
continuous
solution
uu
in
the
Holder
norm
of
order
qsolution
+
� (we
(with
two
disjoint
subdomains
Ω
and
:For
Ω
=
Ω
∪
Ω
,as
Γh
=
∩
Ω
with
piecewise
smooth
boundaries
1will
2
2
γstart
N
uN
to
the
continuous
in
the
discrete
γthe
2Γ
N
N
∆t
∆t,h
γ
u
=
P
v
+
fintroduction
.ΓComposition
�
that
the
boundary
condition
(2.2)
is
by
(2.26)
with
the
same
or
higher
the
arbitrary
�(2.25)-(2.26)
<
1)
with
the
rate
O(h
)and
as
→
0.
Here,
ppiecewise
is
the
order
of
the
accuracy
1in
2Γ(see
1approximated
2
1
2the
will
converge
to
the
continuous
solution
u
in
the
Holder
norm
of
order
qorder
+Method
(with
N
the
arbitrary
0
<
1)
with
the
rate
O(h
)of
as
hisΩ
→
0.
palso
is
the
order
of
the
accuracy
of
system
(2.1)
-fextended
(2.3)
(or
formulation
(2.8)
its
time-discrete
form
in
domain
Ω)
in
some
composite
rate
in
the
maximum
norm
space.
the
detailed
discussion
and
the
proof
of
the
accuracy
of
u
=
P
v
+
fto
.to
We
the
scheme,
illustration
of
the
ideas,
by
considering
the
the
approximation
continuous
differential
operator
L
by
the
discrete
operator
L
γFinally,
N
Γ
:=
∂Ω
Γ
and
Γ
:=
∂Ω
∪
Γγof
Figure
3).
We
would
like
to
also
emphasize
that
the
two
We
the
introduction
of
the
scheme,
and
the
illustration
of
ideas,
by
the
P
v
+
to
the
solution
of
(2.1)
-by
(2.3).
One
would
expect
that
2its
N
arbitrary
0∪
<
�of
<
1)
with
the
rate
O(h
)operator
h
→
0.
Here,
is
the
order
of
the
accuracy
he
of
the
continuous
differential
operator
L
by
the
discrete
operator
L(we
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
2considering
i+1
N
N
γ1converge
N
∆t
∆t,h
1
2the
2 comment
γstart
N
N
p−�
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
Figure
3).
We
would
like
to
emphasize
that
the
two
∆t
∆t,h
Remark:
let
us
briefly
about
the
space
accuracy
of
the
approximation
u�of
=
γ
two
disjoint
subdomains
Ω
and
Ω
:(see
Ω
=
Ω
Ω
,with
Γ
=
Ω
Ω
with
smooth
boundaries
1approximation
1N
2
2the
We
start
the
introduction
the
scheme,
and
illustration
of
the
ideas,
by
considering
the
than
p).
Hence,
for
the
central
finite
difference
scheme
that
discussed
here,
we
will
expect
O(h
)(we
N
omain
Ω
an
arbitrary
bounded
domain
in
R
the
boundary
∂Ω
and
such
that
it
consists
of
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
time-discrete
form
in
domain
Ω)
in
some
composite
u∪
,2u(x,
y)
∈the
Ω
,∩
idea
can
be
in
a
way
to
multiple
subdomains
or
composite
domains.
will
converge
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(wit
the
approximation
of
the
continuous
differential
L
by
the
discrete
operator
L
assume
that
the
boundary
condition
(2.2)
approximated
(2.26)
the
same
or
higher
order
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
time-discrete
form
in
domain
Ω)
in
some
composite
We
start
the
introduction
of
the
scheme,
and
illustration
of
the
ideas,
by
considering
the
1-straightforward
2of
1isapproximated
1
2
u
=
P
v
+
f
.
1
∆t
N
∆t,h
arbitrary
0
<
�
<
1)
with
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
nder
consideration
(2.1)
(2.3).
assume
that
the
boundary
condition
(2.2)
is
by
(2.26)
with
the
same
or
higher
order
subdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
p−�
p−�
Ω
γ
subdomains
are
considered
here
only
for
the
simplicity
presentation,
and
that
the
discussed
N
N
γ
N
2
i+1
p−�
that
the
boundary
condition
(2.2)
is
by
(2.26)
with
the
same
or
higher
order
uassume
will
to
continuous
solution
u))time-discrete
in
the
discrete
Holder
norm
of
order
q2∆t,h
+
�order
(with
2[−1.7,
0O(h
Γ
:=
∂Ω
∪introduction
Γ
and
Γ(or
:=
∂Ω
∪
(see
Figure
3).
We
would
like
to
emphasize
that
the
two
uN
=
P
+
fof
.the
tem
(2.1)
-converge
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composi
the
method,
the
reader
can
consult
[7,
4].
the
arbitrary
0<
<
�<
<
1)
with
the
rate
as
h
→
0.
Here,
palso
the
order
of
the
accuracy
of�
the
arbitrary
<
�the
<
1)
with
the
rate
O(h
as
hthat
→
0.
Here,
1approximation
1converge
2the
2the
than
p).
Hence,
for
central
finite
difference
scheme
is
discussed
here,
we
will
expect
O(h
)of u(we
uΓΩ
:=
(3.1)
We
start
the
of
scheme,
and
the
illustration
of
the
ideas,
by
considering
γ(2.3)
N
N
γare
N
the
of
continuous
differential
operator
L
by
the
discrete
operator
L
(we
i+1
subdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
i+1
i+1
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
same
or
higher
order
Ω
=approximated
1.7]
×
[−1.7,
1.7]
domain
Ω(DPM)
-∪
an
arbitrary
bounded
domain
R
with
the
boundary
∂Ω
-is
and
such
that
it
consists
he
arbitrary
0be
�(2.3)
1)
with
rate
O(h
as
h
→
0.
Here,
pin
is
order
of
accuracy
oft
the
approximation
the
differential
operator
L
by
the
discrete
operator
L
u
will
to
the
continuous
solution
u)about
in
the
discrete
Holder
norm
of
order
qthe
+
�consists
(with
system
(2.1)
-0boundary
(2.3)
formulation
(2.8)
in
time-discrete
form
in
domain
Ω)
insome
some
∆t
system
(2.1)
-v
(or
formulation
(2.8)
in
form
domain
Ω)
in
some
composite
2
ssume
that
the
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
N
ūin
∆t
∆t,h
subdomains
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
p−�
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
approximation
of
the
continuous
differential
operator
L
the
discrete
operator
L
2composite
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
u
,its
(x,
y)
∈that
Ω
,to
Our
goal
at
each
time
level
tcontinuous
is
to
find
an
approximations
u
in
domain
Ω:
Γ
:=
∂Ω
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Metho
system
(2.1)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
composite
N
∆t
∆t,h
2the
Remark:
Finally,
let
us
comment
briefly
the
space
accuracy
of
the
approximation
u(w
1
1
2
2
2
γ
N
N
domain
Ω
an
arbitrary
bounded
domain
boundary
∂Ω
and
such
that
it
of
idea
can
extended
in
a
straightforward
way
subdomains
or
composite
domains.
wo
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
p−�
Ω
than
p).
Hence,
for
the
central
finite
difference
scheme
is
discussed
here,
we
will
expect
O(h
)
domain
Ω
an
arbitrary
bounded
domain
in
R
with
boundary
∂Ω
and
such
that
it
consists
of
idea
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
γ
N
e
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
ystem
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
1
2
1
2
1
2
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
under
consideration
(2.1)
(2.3).
2
Below
we
develop
and
numerically
test
the
Composition
Approach
for
a
parabolic
mod
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
th
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(
2
subdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
p−�
the
arbitrary
0
<
�
<
1)
with
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
∆t
∆t,h
2
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
2order
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
2some
the
approximation
of
continuous
differential
operator
L
main
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
idea
can
be
extended
in
athe
straightforward
way
to
multiple
subdomains
or
composite
domains.
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
2time-discrete
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
�
∆t
∆t,h
i+1
i+1
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
∆t
two
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
em
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
form
in
domain
Ω)
in
compos
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
mo
assume
that
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
the
(DPM)
method,
the
reader
can
consult
[7,
4].
i+1
i+1
he
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
domain
Ω
an
arbitrary
bounded
domain
in
boundary
∂Ω
and
such
that
it
consists
of
idea
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
han
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
N
1
2
1
2
1
2
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
for
the
approximation
of
the
particular
solution
ū
in
Ω
.
We
will
use
the
same
Cartesian
meshes
subdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solutio
∆t
∆t,h
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
Our
goal
at
each
time
level
t
is
to
find
an
approximations
to
u
in
domain
Ω:
2
ume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
orde
N
2
u
will
converge
to
the
continuous
solution
in
the
discrete
Holder
norm
of
order
q
+
�
(with
i+1
Our
goal
at
each
time
level
t
is
to
find
an
approximations
u
in
domain
Ω:
γ
N
N
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
two
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
N
γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solut
two
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
∩
Ω
with
piecewise
smooth
boundaries
12continuous
2
1u
2
i+1
i+1
i+1
1 the
1start
2Composition
under
consideration
(2.1)
-the
(2.3).
3
Algorithms
Approach
based
on
the
Difference
Potentials
Method
γ
N
N
Ω
-in
an
arbitrary
bounded
domain
in
with
the
boundary
∂Ω
-ofdiscrete
and
such
that
itLofO(h
consists
We
the
introduction
of
the
scheme,
and
the
of
the
ideas,
by
considering
the
in
the
maximum
norm
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
eomain
approximation
of
the
continuous
differential
L
by
the
operator
L
(w
uin
,R
(x,
y)
∈
Ωsubdomain
,illustration
1straightforward
11approximated
2operator
2(2.26)
idea
can
extended
in
ain
way
to
multiple
subdomains
or
composite
domains.
γrate
under
consideration
(2.1)
-in
(2.3).
2order
the
approximation
of
the
differential
operator
L
by
the
discrete
operator
1that
∆t
∆t,h
stem
(2.1)
-be
(2.3)
(or
formulation
(2.8)
its
time-discrete
form
in
domain
Ω)
in
some
composit
ume
that
the
boundary
condition
(2.2)
is
by
with
the
same
or
higher
ord
i+1
i+1
(DPM)
method,
the
reader
can
consult
[7,
4].
2o)
Our
goal
at
time
level
tare
is
to
find
an
approximations
to
in
domain
Ω:
where
umaximum
(here
pthe
=
1,
2)
solutions
to
(2.8)
in
each
Ωby
:with
2∪
Ω
∆t
∆t,h
i+1
rate
in
the
norm
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
assume
that
boundary
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
1
assume
that
the
boundary
condition
(2.2)
approximated
by
p2
the
approximation
of
the
continuous
differential
L
the
discrete
operator
Lthat
(we
P
v
+
fHence,
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
the
solu
than
p).
for
central
finite
difference
scheme
discussed
here,
we
will
expect
)(we
rate
the
maximum
norm
space.
the
detailed
discussion
and
the
proof
the
accuracy
of
ossume
disjoint
Ω
and
Ω
:2the
Ω
=
Ω
∪
Ω
,,We
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundari
�
p−�
Ωeach
(DPM)
method,
the
can
consult
[7,
4].
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
would
like
to
also
emphasize
that
the
two
two
disjoint
subdomains
Ω
and
Ω
::For
Ω
=
Γ
=
∩
Ω
piecewise
smooth
boundaries
P
v
+
fthe
of
(2.25)-(2.26)
to
solution
u
of
(2.1)
-≡-1111is
(2.3).
One
would
expect
that
the
solution
∆t
∆t,h
Our
goal
at
each
time
level
tcondition
is
to
find
an
approximations
to
u
in
domain
Ω:
γp).
N
N
�
than
Hence,
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
1reader
2Ω
1is
2
two
disjoint
subdomains
Ω
and
:
Ω
=
Ω
Ω
,
Ω
∩
Ω
with
piecewise
smooth
boundaries
under
consideration
(2.1)
(2.3).
main
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
1p).
2<
2in
1
2
1
2operator
2(2.26)
idea
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
n
n
n∪
n
γ1subdomains
N
N
ate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
u
:=
(3.1)
that
the
boundary
condition
(2.2)
is
approximated
by
with
the
same
or
higher
order
γu
two
disjoint
subdomains
Ω
and
Ω
Ω
Ω
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
γ∂Ω
i+1
1
2
2
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(wit
the
(DPM)
method,
the
reader
can
consult
[7,
4].
i+1
i+1
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
m
the
arbitrary
0
<
�
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
1
2
1
2
1
2
i+1
n
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
N
Ω
2
×
2
≡
2
×
2
with
n
n
=
9,
i+1
Γ
:=
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
ubdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
2 form
the
(DPM)
method,
the
reader
can
consult
[7,
4].
2, Ω
usume
will
converge
to
the
continuous
solution
uscheme
in
discrete
norm
of
order
qthe
+ �orde
(w
uan
,4].
(x,
y)
,Ω111Ω
Γ
:=
∂Ω
∪
and
Γ
:=
∂Ω
Γ2to
(see
Figure
would
like
to
also
emphasize
that
two
1assume
1that
2the
2∪
Our
goal
atΓ
time
level
tcondition
is
find
approximations
to
u
domain
Ω:
We
start
the
introduction
of
the
scheme,
of
the
ideas,
by
considering
system
(2.1)
-each
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
inHolder
domain
Ω)
in
some
composite
2the
�
uis
(x,
y)
∈the
,illustration
2the
wo
disjoint
subdomains
Ω
and
Ω
:the
Ω
=
Ω
∪
ΓΩWe
=
∩
Ω(2.26)
with
piecewise
smooth
boundaries
N
1
1start
2
2
We
the
introduction
of
scheme,
and
illustration
ofalso
ideas,
by
considering
th
that
the
boundary
condition
(2.2)
isthe
approximated
by
(2.26)
with
the
same
or
higher
the
boundary
(2.2)
approximated
by
with
the
same
or
higher
order
u
Ω
1
1Ω
2 ,∈
1discrete
2
2is
i+1
i+1
the
(DPM)
method,
the
reader
can
consult
[7,
4].
the
(DPM)
method,
reader
can
consult
[7,
Ω
main
Ω
-∪
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
-the
and
such
that
it
consists
i+1
assume
that
the
boundary
condition
(2.2)
approximated
by
(2.26)
with
the
same
or
higher
order
�
u
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)tw
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
1 p−�
2()o
than
p).
Hence,
for
the
central
finite
difference
3∂Ω
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
i+1
i+1
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
ap
parabolic
model
1
Γ
:=
∂Ω
∪
Γ
and
Γnorm
:=
∂Ω
∪
Γ
(see
Figure
would
like
to
emphasize
that
the
two
subdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
i+1
u
will
converge
to
the
continuous
solution
u
in
Holder
norm
of
order
q
+
�
(with
un
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
:=
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
Our
goal
at
each
time
level
t
is
to
find
an
approximations
to
u
in
domain
Ω:
rate
in
the
maximum
in
space.
For
detailed
discussion
and
the
proof
of
the
accuracy
of
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
would
like
to
also
emphasize
that
the
two
1
1
2
2
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Figure
3).
We
would
like
to
also
emphasize
that
the
two
u
:=
(3.1)
N
he
(DPM)
method,
the
reader
can
consult
[7,
4].
N
1
2
2
i+1
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
han
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
i+1
1
1
2
2
1
1
2
2
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundar
u
,
(x,
y)
∈
Ω
,
u
:=
(3.1)
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
is
the
order
of
the
accuracy
p−�
L
[u
]
=
G
,
(x,
y)
∈
Ω
,
and
(p
=
1,
2)
(3.2)
2
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
i+1
�
0
0
Ω
1
1
2
1
2
1
2
ptime-discrete
∆t
2
nder
consideration
(2.1)
(2.3).
subdomains
are
considered
here
only
for
the
of
the
presentation,
and
that
the
discussed
ean
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
∆t
∆t,h
dea
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
Ω
Ω
Ω
∆t
u
,
(x,
y)
∈
Ω
,
i+1
2
subdomains
are
considered
here
only
for
the
simplicity
the
presentation,
and
that
the
discussed
1
u
,
(x,
y)
∈
Ω
,
for
the
auxiliary
domains
Ω
and
Ω
in
the
Tables
5.2
5.9.
We
will
select
Cartesian
mesh
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
form
in
domain
Ω)
in
some
composite
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
2
u
∈the
1 Difference
than
p).
Hence,
for
central
difference
scheme
that
isis
discussed
here,
we
will
expect
O(h
)two
:=
∂Ω
∪subdomains
Γ
and
Γnorm
:=
∂Ω
∪
ΓApproach
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
system
(2.1)
-are
(2.3)
(or
formulation
(2.8)
form
in
domain
Ω)
in
some
composit
Ωbased
i+1
G
G
2
p−�
under
consideration
(2.1)
-central
(2.3).
p).
Hence,
for
the
central
finite
difference
scheme
that
discussed
here,
we
will
expect
Ω
than
p).
Hence,
for
the
difference
scheme
that
is
discussed
we
will
expect
O(h
) O(h
1Algorithms
2in
2space.
ufinite
:=
(3.1)
p−�
3
Algorithms
Composition
Approach
on
the
Potentials
Method
Ω
�
1
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
ofof
3(DPM)
Composition
Difference
Potentials
Method
i+1
i+1
22
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
o1idea
disjoint
Ω
and
Ω
:i+1
Ω
=
Ω
∪
Ω
,on
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundarie
the
method,
the
can
consult
[7,
4].
subdomains
are
considered
here
only
the
simplicity
of
presentation,
and
that
the
discussed
in
the
maximum
in
For
the
detailed
discussion
and
the
proof
of
the
accuracy
can
be
extended
areader
way
to
subdomains
or
composite
domains.
the
arbitrary
0considered
<
�the
<
1)
with
the
rate
O(h
)time-discrete
as
h
→
Here,
pto
ishere,
order
of
the
accuracy
u
,multiple
y)
∈
Ω
,0.
1straightforward
2
1based
2)(x,
11the
2i+1
Ω
subdomains
considered
here
for
the
simplicity
of
the
presentation,
and
that
the
discussed
subdomains
are
here
only
for
the
presentation,
and
that
the
discussed
bdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discusse
the
(DPM)
method,
the
reader
can
consult
[7,
4].
he
arbitrary
0
<
�
<
1)
with
the
rate
O(h
as
→
0.
Here,
p
is
the
order
of
the
accurac
u
:=
(3.1)
i+1
where
uwe
(here
pboundary
=
1,
2)
are
the
solutions
toan
(2.8)
in
each
subdomain
Ω
u
,
(x,
y)
∈
Ω
,
i+1
i+1
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
Ω
ate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
2
the
approximation
of
the
continuous
differential
operator
L
the
discrete
operator
L
(w
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
compo
p : by
i+1
2
assume
that
the
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
1
=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
also
emphasize
that
the
to(
Ω
Ω
∆t
2
∆t,h
idea
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
Ω
Below
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
i+1
Our
goal
at
each
time
level
t
is
to
find
approximations
to
u
in
domain
Ω:
p
i+1
1
2
2
m
m
(DPM)
method,
the
reader
can
consult
[7,
4].
2
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
u
,
(x,
y)
∈
Ω
,
u
,
(x,
y)
∈
Ω
,
domain
Ω
an
arbitrary
bounded
domain
boundary
∂Ω
and
such
that
it
consists
of
idea
can
be
extended
in
a
straightforward
subdomains
or
composite
domains.
two
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
u
:=
(3.1)
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
1
2
i+1
i+1
i+1
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
subject
to
the
appropriate
boundary
conditions
l(u
)
=
ψ
on
the
boundary
∂Ω
of
the
original
1
2
1
2
1
2
2
×
2
rate
in
the
maximum
norm
in
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
ubdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
Ω
Ω
∆t
∆t,h
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
i+1
Ω
te
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
1
2
i+1
the
(DPM)
method,
the
reader
can
consult
[7,
4].
the
(DPM)
method,
the
reader
can
consult
[7,
4].
Ω
idea
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
Our
goal
at
each
time
level
t
is
to
find
an
approximations
to
u
in
domain
Ω:
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
tw
idea
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
u
,
(x,
y)
∈
Ω
,
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
i+1
2
2
(DPM)
method,
the
reader
can
consult
[7,
4].
2
idea
can
be
extended
in
a
straightforward
subdomains
or
composite
domains.
i+1
i+1
1
2
2
u
:=
(3.1)
∆t
∆t,h
where
u
(here
p
=
1,
2)
are
the
solutions
to
(2.8)
in
each
subdomain
Ω
:
a
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
i+1
he
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L∆t,h
pi+1
under
consideration
(2.1)
-central
(2.3).
assume
that
the
condition
(2.2)
isΩapproximations
approximated
by
(2.26)
with
same
orsome
higher
orde
24].
where
uwe
(here
pboundary
=
1,
2)
are
the
solutions
to
(2.8)
in
each
subdomain
Ω
: inalso
�
he
(DPM)
the
reader
can
consult
[7,
ufor
than
p).
Hence,
for
the
finite
difference
scheme
that
is∩
discussed
here,
we
will
expect
)
system
(2.1)
-subdomains
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
domain
Ω)
some
composite
domain
Ω
-method,
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
-emphasize
and
such
that
itO(h
consist
3dea
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
Ω
Ω
∆t
Ωthe
pto
Our
goal
at
each
time
level
ti+1
is
to
an
to
u
in
domain
Ω:
Below
develop
and
numerically
test
Algorithm
Composition
Approach
for
adomain
parabolic
model
p
domains
are
considered
here
only
the
simplicity
of
the
presentation,
and
that
the
discuss
p
i+1
domain,
to
interface
conditions
on
interface
boundary
Γ,
which
we
will
select
to
be
the
Below
we
develop
and
test
the
Algorithm
Composition
Approach
for
athe
parabolic
model
u=
,approximations
(x,
y)
∈
,in=
two
disjoint
Ω
:,find
Ω
Ω
∪
Ω
,,form
Γ
Ω
Ω
with
piecewise
smooth
boundaries
Our
goal
at
each
time
level
tΩ
is
to
find
an
to
u
in
Ω:
p
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
that
the
two
ystem
(2.1)
-and
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
domain
Ω)
in
compo
the
(DPM)
method,
the
reader
can
consult
[7,
4].
2=
i+1
i+1
the
(DPM)
method,
the
reader
consult
[7,
4].
1ti+1
2the
1
2
11to
2(2.26)
i+1
m
0 in
i+1
1can
1Ωextended
2numerically
2i+1
two
disjoint
subdomains
Ω
and
Ω
:find
Ω
∪
ΓΩ∈
=
Ω
∩
Ω
with
piecewise
smooth
boundarie
assume
that
the
boundary
condition
(2.2)
is
approximated
by
with
the
same
or
higher
or
L
[u
]and
=
G
(x,
y)
∈
Ω
,multiple
(p
1,
2)
(3.2)
Ω
be
in
a
straightforward
way
to
subdomains
or
composite
domains.
i+1
i+1
�
2Ω
1tthe
2
1approximations
2and
2Ω
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
pΩ
∆t
2=
(hence,
h
:=
4/2
)
for
approximation
of
the
particular
solution
ū
in
Ω
,
as
well
as
the
Cartesian
Our
goal
at
each
time
level
an
approximations
u
in
domain
Ω:
e
(DPM)
method,
the
reader
can
consult
[7,
4].
Our
goal
at
each
time
level
is
to
an
to
u
in
domain
Ω:
i+1
i+1
Ω
∆t
where
u
(here
p
=
1,
2)
are
the
solutions
to
(2.8)
in
each
subdomain
:
1
N
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
p
ū
u
,
(x,
y)
Ω
,
bdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discusse
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
i+1
p
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
Our
goal
at
each
time
level
t
is
to
find
an
approximations
to
u
in
domain
Ω:
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a parabolic
model
1 ∩
Ωthe
i+1
rssume
goal
at
each
time
level
t
is
to
find
an
approximations
to
u
in
domain
Ω:
under
consideration
(2.1)
-are
(2.3).
�
u
following:
p
Ω
than
p).
Hence,
for
the
central
finite
scheme
that
is
discussed
here,
we
will
expect
O(h
rate
in
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
that
the
boundary
condition
(2.2)
is
by
(2.26)
with
the
same
or
higher
o
Ωdifference
i+1
2
1approximated
where
u
(here
p
=
1,
2)
the
solutions
to
(2.8)
in
each
subdomain
Ω
:
p =
two
disjoint
subdomains
Ω
and
Ω
:
Ω
Ω
∪
Ω
,
Γ
=
Ω
Ω
with
piecewise
smooth
bounda
i+1
under
consideration
(2.1)
(2.3).
�
u
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
i+1
p
i+1
i+1
Ω
1
2
1
2
1
2
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
�
u
,
(x,
y)
∈
Ω
,
mesh
p
subdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
Ω
i+1
1
1∪
22)
2(2.8)
1boundary
u
:=
(3.1)
p an
omain
Ω
arbitrary
bounded
domain
in
R
with
the
∂Ω
and
such
that
it
consis
L-central
[ui+1
]Ω
=
G
, utime-discrete
(x,
y)
∈Ω
Ω
,,each
and
(p
=
1,
2)
(3.2)
2)
�
Γ
:=
∂Ω
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
tw
under
consideration
(2.1)
(2.3).
i+1
i+1
i+1
�
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(
Ω
where
u
(here
p
=
1,
are
the
solutions
to
(2.8)
in
subdomain
Ω
:
p
∆t
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
Our
goal
at
each
time
level
t
is
to
find
an
approximations
to
u
in
domain
Ω:
i+1
i+1
1
1
2
2
1
two
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
∩
Ω
with
piecewise
smooth
boundaries
L
[u
]
=
G
,
(x,
y)
∈
Ω
and
(p
=
1,
2)
(3.2)
p
Ω
∆t
system
(2.1)
(2.3)
(or
formulation
in
its
form
in
domain
Ω)
in
some
composite
under
consideration
(2.1)
�
1
2
1
2
1
2
p
u
,
(x,
y)
∈
Ω
,
i+1
p
than
p).
Hence,
for
the
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
∆t
u
=
,
(x,
y)
∈
Γ,
i+1
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
Ω
i+1
under
consideration
(2.1)
(2.3).
3
Algorithms
Composition
Approach
based
on
the
i+1
pthe the
uand
,in
(x,
y)
∈
Ω
,subdomains
We
start
of
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
Ω
∆t
we
develop
and
numerically
test
Algorithm
Composition
Approach
for
a(3.3)
parabolic
model
ea
can
be
extended
in
ai+1
straightforward
way
to
multiple
or
composite
domains.
i+1
i+1
pthe
�
u
:=
(3.1)
u
uu
,multiple
(x,
y)
∈
Ω
Ωthe
Ω
2Ω
Ω
i+1
Ω
subject
to
appropriate
boundary
conditions
l(u
)simplicity
on∈
the
∂Ω
of
the
original
i+1
1,,Difference
the
(DPM)
method,
the
reader
can
consult
[7,
4].
rate
in
the
norm
in
space.
For
the
detailed
discussion
the
proof
ofMethod
the
accuracy
i+1
i+1
1=
p to
i+1
u
(here
pfor
=introduction
1,
2)
are
the
solutions
(2.8)
each
subdomain
Ω
i+1
Ω
We
start
the
introduction
of
the
scheme,
the
illustration
of
the
ideas,
by
considering
the
under
consideration
(2.1)
-central
(2.3).
,Ω
(x,
y)
Ω
,11boundary
Ω
lΓ
(u
,∂Ω
ustraightforward
)the
:=
han
p).
Hence,
the
finite
difference
scheme
that
is
discussed
here,
we
will
expect
Ω
2u
p :and
Γwhere
:=
∂Ω
∪
Γmaximum
and
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
theO
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
i+1
Ω
,2ψ
(x,
y)
∈
m
m
1also
1
Γ
subdomains
are
considered
here
only
for
the
of
the
presentation,
and
the
discussed
2an
i+1
i+1
Ω
can
be
aΩ
way
to
subdomains
or
composite
domains.
utime-discrete
,detailed
(x,
y)
∈
Ω
,Ω
3Below
Algorithms
Composition
Approach
on
the
Potentials
Method
i+1
1idea
1Ω
2
2
Ω
Ω
u
:=
(3.1)
We
start
the
introduction
of
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
uΩ
goal
at
each
time
level
t
is
to
find
approximations
to
u
in
domain
Ω:
Ω
L
[u
]
=
G
,
(x,
y)
∈
,
and
(p
=
1,
2)
(3.2)
p
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∪
Γ
(see
Figure
3).
We
would
like
to
emphasize
that
the
twothat
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
i+1
u
,
(x,
y)
∈
,
2
i+1
1
�
subdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discusse
Ω
2
×
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
1we
1extended
2in
2and
p
∆t
domain
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
p
wo
disjoint
subdomains
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
bounda
1
1
i+1
u
:=
(3.1)
Below
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
Ω
∇u
·
n
=
β∇u
·
n
,
(x,
y)
∈
Γ,
rate
in
the
maximum
norm
in
space.
For
the
discussion
and
the
proof
of
the
accuracy
Ω
∆t
Ω
i+1
i+1
i+1
2
1
2
1
2
1
2
i+1
Γ
Γ
Ω
p
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
form
in
domain
Ω)
in
some
composite
u
,
(x,
y)
∈
Ω
,
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
i+1
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
i+1
:=
(3.1)
1
u
Ω
Ω
u
,
(x,
∈
Ω
,
i+1
1
Ω
u
(3.1)
domain,
and
to
interface
conditions
on
the
interface
boundary
Γ,
which
we
will
select
to
be
the
Ω
L
[u
]
=
G
,
(x,
y)
∈
Ω
,
and
(p
=
1,
2)
(3.2)
i+1
under
consideration
(2.1)
(2.3).
i+1
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
2
p
i+1
i+1
i+1
subject
to
the
appropriate
boundary
conditions
l(u
=
ψ
on
the
boundary
∂Ω
of
the
original
ur
goal
at
each
time
level
tboundary
is
find
an
approximations
to
u
inin
domain
Ω:composite
Ω
Ωthe
py)
∆t
uuthe
(x,
y)
∈
Ω
,domain
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
i+1
system
(2.1)
-each
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
i+1
i+1
Ω
1detailed
We
start
the
introduction
ofthere
the
scheme,
and
illustration
of
the
ideas,
by
considering
the
the
(DPM)
method,
the
reader
can
consult
[7,
4].
u
:=
(3.1)
Ω
Ω
∆t
2of
(x,
∈∈
ΩΩ
,221subdomains
i+1
Ω
subject
to
the
appropriate
conditions
l(u
),),∈
=
ψ
on
the
boundary
∂Ω
of
the
original
p
subdomains
are
considered
here
only
for
simplicity
the
and
that
the
discussed
u
(x,
y)
∈
Ω
,
2in
Ω
Land
[u
]to
=
G
,Ω
(x,
y)
Ω
,form
and
(p
=
1,
2)
(3.2)
idea
can
be
extended
in
a
straightforward
way
to
multiple
or
composite
domains.
system
(2.1)
-are
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
in
domain
Ω)
in
some
composite
u
,
(x,
,
ate
in
the
maximum
norm
in
space.
For
discussion
and
the
proof
of
the
accurac
subdomains
considered
only
for
the
simplicity
of
the
presentation,
and
that
the
discus
22the
18
�
uΩ
Our
goal
at
time
level
is
to
find
an
approximations
to
u
in
domain
Ω:
2
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
Ω)
some
composite
two
disjoint
subdomains
Ω
Ω
:
Ω
=
Ω
∪
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
ppresentation,
Ω
∆t
u
:=
(3.
Ω
Ω
2
1
2
1
1
2
i+1
ow
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
mod
p
idea
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
Ω
∆t
u
,
(x,
y)
∈
Ω
,
i+1
Ω
i+1
Ω
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
i+1
p
under
consideration
(2.1)
(2.3).
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
1
Below
we
develop
and
numerically
test
the
Algorithm
Composition
2
2
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
the
(DPM)
method,
the
reader
can
consult
[7,
Ω
2(2.8)
following:
i+1
the
(DPM)
method,
reader
can
consult
[7,
Ω
i+1
i+1
1system
1Ω
2formulation
2the
domain,
and
to=
interface
conditions
on
the
boundary
Γ,
which
we
will
select
to
be
the
Lthe
[u
]i+1
=
G
,:=
(x,
∈
Ω
,illustration
and
(p
=
2)
(3.2)
24].
where
u
(here
pand
1,
2)
are
the
solutions
to
in
each
subdomain
Ω
:such
uy)
(x,
y)
∈
Ω
,,1,
We
start
introduction
of
scheme,
and
the
of
the
ideas,
by
considering
�
2interface
under
consideration
(2.1)
-conditions
(2.3).
u
(3.1)
pboundary
∆t
pdomains.
2with
2or
domain
-1the
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
-Approach
and
that
it
consists
ofthe
idea
can
be
extended
in
anumerically
straightforward
way
to
multiple
subdomains
composite
i+1
(2.1)
-develop
(2.3)
(or
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
i+1
�
subject
to
boundary
conditions
l(u
),,4].
=
ψComposition
on
the
∂Ω
of
the
original
domain,
and
to
interface
on
the
boundary
Γ,
which
we
will
select
to
be
the
Ω
∆t
Ω
Below
we
develop
and
test
the
Approach
for
aconsists
parabolic
Ω
pΓ
u
(x,
y)
∈
Ωon
pi+1
domain
Ω
-the
an
arbitrary
bounded
domain
in
R1uinterface
the
∂Ω
-boundary
and
such
that
it
consists
of model
Ω
�
2
ow
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
mo
domain
Ω
-Γappropriate
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
-to
and
such
that
it
ofMethod
i+1
i+1
Γ
:=
∂Ω
∪
Γ
:=
∂Ω
∪of
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
mode
i+1
i+1
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Ω
i+1
Below
we
and
numerically
test
the
Algorithm
Composition
for
a
parabolic
model
8Algorithm
Our
goal
at
each
time
level
t
is
to
find
an
approximations
u
in
domain
Ω:
1
2an
where
u
(here
p
=
1,
2)
are
the
solutions
to
(2.8)
in
each
subdomain
Ω
:
i+1
1can
2the
2i+1
i+1
idea
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
u
,
(x,
y)
∈
Ω
,
he
(DPM)
method,
reader
can
consult
[7,
4].
two
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
p
Ω
i+1
i+1
2
1
2
2
1
2
u
=
,
(x,
y)
∈
Γ,
Our
goal
at
each
time
level
t
is
to
find
approximations
to
u
in
domain
Ω:
1
i+1
Ω
der
consideration
(2.1)
(2.3).
We
start
the
introduction
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
under
consideration
(2.1)
(2.3).
subject
to
the
appropriate
boundary
conditions
l(u
)
=
ψ
the
boundary
∂Ω
of
the
original
i+1
under
consideration
(2.1)
(2.3).
p
i+1
Ω
following:
2
i+1
i+1
i+1
i+1
i+1
u
,
(x,
y)
∈
Ω
,
2
Ω
Ω
ubdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discu
Our
goal
at
each
time
level
t
is
to
find
an
approximations
to
u
in
domain
Ω:
where
u
(here
p
=
1,
2)
are
the
solutions
to
(2.8)
in
each
subdomain
Ω
:
1
2
Ω
1
u
:=
(3
i+1
following:
p
ystem
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
two
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
domain,
and
to
interface
conditions
on
the
interface
boundary
Γ,
which
we
will
select
to
be
the
subject
to
appropriate
boundary
conditions
l(u
)
=
ψ
on
the
boundary
∂Ω
of
the
original
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
u
,
(x,
y)
∈
Ω
,
�
i+1
l
(u
,
u
)
:=
(3.3)
elow
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
mod
Ω
where
u
(here
p
=
1,
2)
are
the
solutions
to
(2.8)
in
each
subdomain
Ω
:
1
2
1
2
1
2
where
u
(here
p
=
1,
2)
are
the
solutions
to
(2.8)
in
each
subdomain
Ω
:
Ω
i+1
under
consideration
(2.1)
- and
(2.3).
two
disjoint
Ωand
Ω
: solutions
Ω
=
Ω
Ω
,i+1
ΓWe
=
Ω
∩y)
Ωon
smooth
boundaries
1to,piecewise
subdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
11=
two
disjoint
Ω
Ω
: is
Ω
=
Ω
∪
Ωan
,2β∇u
ΓΩapproximations
Ω
∩n
Ω
with
piecewise
boundaries
p
psmooth
i+1
i+1
i+1
i+1
�
Ω
i+1
p domain
12
1∪
1in
i+1
i+1
Ω
Ω
Ω
1-(2.3).
2Γ
1
2
1y)
22∈
u
�
where
u
(here
pΓ(2.1)
=
2)
the
to
(2.8)
each
subdomain
Ω
:will
Ω1subdomains
i+1
under
consideration
(2.1)
(2.3).
i+1
11,
2are
ΓWe
∂Ω
∪
Γsubdomains
and
Γ
:=
∂Ω
∪
(see
Figure
3).
would
like
also
emphasize
that
the
two
nder
consideration
(2.3).
Ω
Ω
3
Algorithms
Composition
Approach
based
the
Difference
Potentials
Method
p
er
consideration
(2.1)
u
,
(x,
Ω
pideas,
p
p
1 :=
21,
2i+1
�
u
:=
(3.1
∇u
·
n
=
·
,
(x,
y)
∈
Γ,
i+1
i+1
Our
goal
at
each
time
level
t
to
find
to
u
in
Ω:
�
u
=
u
,
(x,
∈
Γ,
2
Ω
two
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
where
u
(here
p
=
2)
are
the
solutions
to
(2.8)
in
each
subdomain
Ω
:
i+1
domain,
and
to
interface
conditions
on
the
interface
boundary
Γ,
which
we
select
to
be
the
subject
to
the
appropriate
boundary
conditions
l(u
)
=
ψ
on
the
boundary
∂Ω
of
the
original
Γ
Γ
i+1
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
p
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
by
considering
the
u
:=
(3.1)
1
2
1
2
1
2
Ω
p
start
the
introduction
of
the
scheme,
and
the
illustration
i+1
i+1
i+1
2
u
Ω
Ω
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
th
Ω
Ω
i+1
Ω
2
1
2
u
=
u
,
(x,
y)
∈
Γ,
Ω
1
2
L
[u
]
=
G
,
(x,
y)
∈
Ω
,
and
(p
=
1,
2)
(3.2)
Ω1an
Ω
i+1
p∪
i+1
pbe
following:
Ω
two
disjoint
subdomains
and
Ω
:Γ
Ω
=
Ω
Ω
ΓWe
=
Ω
Ω
piecewise
smooth
boundaries
Γ
:=
∂Ω
∪
Γ(2.3)
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
pillustration
and
interface
conditions
on
interface
boundary
we
will
select
to
be Method
the
∆t
dea
can
be
extended
in
a∂Ω
straightforward
way
to
multiple
subdomains
or
composite
domains.
u
,,the
(x,
y)
Ω
,1also
u
,,in
y)
∈
Ω
,,which
i+1
ΓWe
:=
∂Ω
∪
Γ=
and
Γ
:=
(see
Figure
3).
We
would
like
emphasize
the
two
(u
,u∂Ω
ui+1
)of
:=
(3.3)
domain
-(here
arbitrary
bounded
in
R
with
the
boundary
∂Ω
-and
and
such
that
it
consists
idea
can
aformulation
straightforward
way
to
multiple
subdomains
or
composite
domains.
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Ω
system
(2.1)
-Γ1extended
(or
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
1L
2domain
1i+1
2
11(x,
2∈with
2Γ(u
2
ere
u
pto
1,
2)
are
the
solutions
to
(2.8)
each
subdomain
Ω
:that
Γ
:=
∂Ω
∪
and
Γ
:=
∪
Γ∪
(see
Figure
3).
would
to
also
emphasize
that
the
two
We
the
introduction
of
the
scheme,
and
of1,
the
ideas,
by
considering
theof
Ω
∆t
2,Γ,
1Ω
2Ω
22i+1
1the
2,
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
u
(x,
y)
∈
Ω
,Ω
(x,
y)
∈
Ω
,∩
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
i+1
nder
consideration
(2.1)
(2.3).
p
11domain,
1 start
2Γlin
2
i+1
pideas,
Ω
Ω
uuu
subdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
that
the
discussed
Ω
Ω
2to
u
(x,
y)
∈
Ω
[u
]
=
G
,
(x,
y)
∈
Ω
,
and
(p
=
2)
(3.2)
�
l
)
:=
(3.3)
i+1
1 ,∆t
1the
Ω
Ω
2
start
the
introduction
the
scheme,
and
illustration
of
the
ideas,
by
considering
the
2
Ω
i+1
p
1
Ω
p
i+1
p
i+1
i+1
We
start
the
introduction
of
the
scheme,
and
illustration
of
the
ideas,
by
considering
the
∇u
·
n
=
β∇u
·
n
,
(x,
y)
∈
Γ,
Ω
2
�
Ω
Ω
i+1
i+1
u
Ω
∆t
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
Ω
p
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
by
considering
i+1
domain,
and
to--arbitrary
interface
conditions
on
the
interface
boundary
which
will
the
Γwith
Γ ,Γ,
following:
i+1
where
p = (or
1,
the
solutions
(2.8)
in
subdomain
Ωsuch
: Ω)
1 are
pu
1the
Ω
1u Ω
1an
22)
2[u
uG
:=
(3.1)t
:=
(3.1)
domain
-(here
bounded
domain
in
∂Ω=
and
that
itbe
consists
of(3.2)
u
Ω
Ω
system
(2.1)
(2.3)
formulation
(2.8)
its
time-discrete
form
in
domain
intosome
composite
p
pselect
i+1
i+1
i+1
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
1in
∇u
·R
=
β∇u
· n,i+1
(x,
y)
∈-we
Γ,
Ω
i+1
Li+1 2i+1
]=
,nto
(x,
y)
∈2 boundary
Ωeach
and
(p
1,
2)
b
bb
"
"
""
2
2 22
1.5
2 2
1.5
1.51.5 2
""
2
1
1.511.51.5
1 1
!
!
!
!
0.5
!
1 !
0
0 0.5
0.5
0
0.5
−0.5
−0.5
−1
"
"
!
!
−0.5 0
0
"
"
!!
!
−0.5 0
"
! !!
!
0.5
0
""
""
""
! !
1.5
0.5
0.5
1
1 !
0.5
1
"
"
"
"
"
"
"
0
−0.5
−0.5
−1 −1
−0.5−0.5
−1.5
−1
!
−1.5 −1
−1.5−1
−1
−0.5
−1.5
−2
−2 −2
−1.5
0
0.5
1
!
−1 −0.5 −0.5 0
0 0.5
0.51
1.5
1
0
0.5
1
1.5
1
1.5
!
−1.5
−2 −2
−1.5
−2 −2 −1.5 −1.5 −1
−1.5
−2
−2
−0.5
−1
−0.5
−0.5
−1
−1
−1.5
0
−0.5
−0.5
0
2
""
1.5
1 1.5
1 1.5
1
2
1
0.5
1!
!
0.51 1.5
!
!
0.5
!
!
−0.50
−0.5
""
"
"
"
−0.5
−1.5
2
2
1.5
1
2
1.5
2
"
"
!
−1
−0.5
−1.5
−2
−1.5
−1
−2
1.5
1
""
!
−0.5
−1 −0.5
0
−1
−1
−1.5−1
2
2
"
!!
!
−1
2
1.5
!!
!
0
0.5
1
0.5
0.5
"
−0.5
−0.5 0
0.5
"
""
""
""
!! !!
0
0.5
0.5
0
0
1
0.5
2
""
2
2 22
2
1.51.5
1.52
0.5
0
0
""
γ
1.5
!
−1
−2
−1.5
−1
−2 −1.5
−1.5
−2 −2
−2
!
!
−2
−1 −1 −1.5
−2
−1.5
!
−1.5
−0.5!!
−1
0
0.5
−2 −1.5
−1.5
−2
−1.5
−1
−0.5 ! 0
−1
−2
−2
−2
−1.5
−1
−0.5
−1.5
−2 −2 −1.5
−1
−0.5
−2 −1.5
−2 −2 −1.5 −1.5 −1 −1 −0.5!−0.5
−2
−2 −2
−2
−1.5
−1.5
−1
−1
1
0.5
0
0
0
−0.5
−0.5
0
1.5
1
0.5
0.5
0.5 0.5
0
0
0.5
0.5
2
1.5
1
1
2
1.5
1.5
1.5 1.5
1
1
1
1
2
2
1.5
1.5
2
2
2
2
γ
1
1
N1
1
2
2
1
N2
2
2
p
1
1
2
2
Ωp
p
1
2
1
1
1
1
1
1
2
2
2
1
2
2
1
N1
1
bb1
1
1
b1
1
b010
b
1
11b1
b
1
b
1
0
bb1
b1
0 0
1 b1
1
δΩb 1
0
b
1
1
1
11111
1
1
b 10
b1
11 1 1
1
1
1
1 1
111
b 1b 1
1
b1
1
1
b1
1
11
1
b1
1 111
1 1
b1
11
KK
K1
K
K
KK
1
1
K
K
K
K
K
1
b
b
"
"
" "
2 2
22
2 2
1.51.5
"
2
" "
1.5
1.5
1.5 1.5
1.5 2
11 1
!!
!
1
0.5
1
0.5
1
0.5
0.5
00.5
0.5 0
1.5
1
!!
!
0.5
−0.5
−0.5 0
0
1
!
!
0.5
0
−0.5
−1 −0.5
"
"
"
"
"
"
!
"
"
!!
!
!
" "
"
"
!
−0.5 −0.5
−1
"
!
!!
!
0
0
"
"
"
0
"
−0.5
−1 −1
−1.5
−1.5−1
−0.5
−1
!
1
!
Figure
Sketch
domain
Ωwith
withboundary
boundary
two
subdomains
Ω1Ω
, the
ΩΩ
,interface
and
theΓthe
interface
Γ
1∂Ω,
Figure
2:
Sketch
domain Ω
Ω
∂Ω,
two
subdomains
KK
Figure
2:2:2:Sketch
ofof
Ω
with
boundary
∂Ω,
two
subdomains
and
K
KΩ1Ω, 1Ω, 2Ω, 2and
Figure
Sketch
ofofdomain
domain
with
boundary
∂Ω,
two
subdomains
, and
Γ interface Γ
1 , 2the
2 ,interface
1
igure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
gure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
Figure
2:Sketch
Sketch
ofdomain
domain
Ω
with
boundary
∂Ω,
two
subdomains
,,2Ω
, 1and
interface
K ΩΩ
Figure
Sketch
with
boundary
∂Ω,
two
subdomains
Ω
,and
the
interface
ΓΓ Γ
Figure
2:2:
ofofof
domain
ΩΩwith
boundary
∂Ω,
two
subdomains
,1Ω
interface
ΓΓ interface
1,the
2 and
1and
22the
2Ω
Figure
2:
Sketch
of
domain
Ωwith
with
boundary
∂Ω,
two
subdomains
,the
Ω
,interface
the
Γ
1Ω
2
1Ω
2Ω
Figure
2:
Sketch
ofdomain
domain
Ω
with
boundary
∂Ω,
two
subdomains
, ,1Ω
,and
Figure
2:
Sketch
Ω
boundary
∂Ω,
two
subdomains
,
Ω
and
the
interface
1
2
Figure
2:
Sketch
of
Ωwith
with
boundary
∂Ω,
two
subdomains
Ω
, the
ΩΩ
,and
the
interface
Γ Γ
Figure
2:
Sketch
ofdomain
domain
Ω
with
boundary
∂Ω,
twotwo
subdomains
Ω1level.
, Ω2 ,Ωand
Γ interface
where
uSketch
denote
the
solution
to
the
difference
scheme
(2.15)
at subdomains
each
Hence,
to the
Figure
2:2:
Sketch
of
Ω
with
boundary
∂Ω,
,1level.
Γ
Figure
ofdomain
domain
boundary
∂Ω,
two
subdomains
,Ω
,and
and
the
interface
1Ω
222,due
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
subdomains
Ω
,1Ω
,interface
the
interface
2 and
where
u udenote
the
solution
toΩthe
the
difference
scheme
(2.15)
attime
each
time
Hence,
due
to
1
K
where
denote
the
solution
to
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
whereFigure
denote
the
solution
difference
(2.15)
time
level.
due to
auuniqueness
argument,
fΩ
coincides
with scheme
the ∂Ω,
solution
uK
theeach
scheme
N : Hence,
2: Sketch
of P
domain
boundary
two
subdomains
Ω1(2.15)
, ΩHence,
and
the due
interface
Γ
γ +the
N γ vto
N with
N ofat
2 ,on
0 atat
udenote
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
to
where
uu
denote
the
solution
to
the
scheme
(2.15)
each
time
level.
due
to
a where
uniqueness
argument,
PNN
vto
+the
fdifference
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
where
u
the
solution
to
the
difference
scheme
(2.15)
each
time
level.
Hence,
due
to
u
denote
the
solution
to
difference
scheme
(2.15)
each
time
level.
Hence,
due
to
awhere
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
γ
γ
N
N
Fig.
5.4:
Test
Problem
4:
Example
of
the
auxiliary
domain
Ω
,
domains
Ω
and
Ω
and
the
boundary
Γ
of
γ
γ
N
N
1
2
re
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
t
ere
u
denote
the
solution
difference
scheme
(2.15)
at
each
time
level.
Hence,
where
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
KuKK
uwhere
= denote
PNuγargument,
vdenote
fN .the
�Hence,
solution
toto
the
difference
scheme
(2.15)
at each
time
level.
due
to
γ +the
Nu
aawhere
uniqueness
P
+
fwith
coincides
with
the
solution
utime
of
the
scheme
(2.15)
on
N :due
solution
the
difference
scheme
(2.15)
at
each
level.
Hence,
due
to
γ
N
γ vv
N
N
Figure
2:
Sketch
of
domain
Ω
boundary
∂Ω,
two
subdomains
Ω
,
Ω
,
and
the
interface
Γ
the
domain
Ω
;
k
=
5
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
of
the
scheme
(2.15)
on
N
:
a
uniqueness
argument,
P
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
2
θ
a
uniqueness
argument,
P
v
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
1
2
u
=
P
v
+
f
.
�
u
=
P
v
+
f
.
�
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:
γ
N
γ
N
N
K
γ
N
γ
N
γ
N
γ
N
N
γ
N
N
γ
N
γ the
N u denote
N
N the
γ +to
N γv
Nthe
Nateach
here
solution
the
scheme
(2.15)
time
level.
Hence,
due
to
uγargument,
denote
solution
difference
scheme
(2.15)
each
time
level.
Hence,
to
niqueness
f+
coincides
with
the
solution
of
the
scheme
(2.15)
N
Finally,
let
about the
space
accuracy
of
the
u
= the
aNawhere
uniqueness
argument,
P
vvγγ+
+
fNdifference
coincides
with
solution
u
scheme
on
N
Figure
2:
Sketch
ofP
domain
with
boundary
∂Ω,
two
subdomains
,of
Ωthe
,(2.15)
and
interface
Γ on
here
uP
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
argument,
P
ffbriefly
coincides
with
the
solution
uNΩ
scheme
(2.15)
N
aNRemark:
uniqueness
P
fN
coincides
with
thethe
solution
uNat
of
the
scheme
on
N
:(2.15)
uniqueness
argument,
P
+
coincides
with
solution
uapproximation
of
the
scheme
(2.15)
on
N�
:due
γto
Nus
γN
N
N
Nv
γv
N
1of
2the
uniqueness
=
P
+
f+
.argument,
γΩ
N
γcomment
N
N
N
γγ
γ
Nu
γγNdenote
γγ v
N
u
=
P
v
+
f
.
�
uuN
=
v
+
f
.
�
u
=
P
v
f
.
�=
where
u
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
γ
N
γ
N
=
P
v
+
f
.
�
γ
N
N
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
:Γ
γ
N
N
N
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
γ
N
N
γ
N
γ
N
γ
N
N
N
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
N
N
N
Figure
2:
of
domain
Ω
with
boundary
∂Ω,
subdomains
Ω
,1Ω,Ωscheme
and
the
interface
Figure
2:
Sketch
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
and
the
interface
Γ
γ
uP
=γγ+
P
v.Sketch
+.:=
fof
.3.4/2
�
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
of
the
scheme
(2.15)
N
:
1, the
2,,scheme
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
(2.15)
on
N
u
=
v
+
�
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
Ω
,
and
the
interfac
1
2
m
0u
γ.fN
N
N
γ
N
=
v
f
�
γ
N
γ
N
N
uNaP
=
P
v
�
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
(2.15)
on
γ
N
γ
N
N
γ
N
2
=
P
v
+
f
γ
N
γ
N
γ
N
N
γ
(hence,
h
)
for
the
computation
of
the
particular
solution
ū
in
Ω
,
see
Tables
5.1
5.9.
γ
N
γ
N
N
γ
N
γ
N
2 let
N ofexpect
Finally,
let
uscontinuous
comment
briefly
about
the
space
accuracy
of
the
uon
=N�:Γ:N
ū
Figure
2:
Sketch
of
domain
Ω
with
boundary
∂Ω,
two
subdomains
Ω
, approximation
Ωthat
, and
the
interface
P
vP
+
fconverge
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
-the
(2.3).
One
would
that
the
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
uNN
=
aRemark:
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
N
1+
2solution
Remark:
Finally,
us
comment
briefly
about
the
space
accuracy
the
approximation
u
=
uRemark:
=
v
+
f
.
γN
N
N
u
will
to
the
solution
u
in
the
discrete
Holder
norm
of
order
q
�
(with
Remark:
Finally,
let
us
comment
briefly
about
space
accuracy
of
the
approximation
u
=
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
to
γ
γ
N
γ
N
N
N
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
the
solution
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
=
γ
N
γ
N
N
γ
N
N
N
γ = P
0
u
0
Remark:
let
uscomment
comment
briefly
about
the
space
accuracy
of theoforder
approximation
= u =u �
uwhere
v
+
f(2.25)-(2.26)
. the
compute
particular
solution
ūdifference
Ω
, we
make
the
smooth
extension
g̃of
∈the
\Ω
ofHence,
=
P
v
+
fFinally,
.Finally,
�=
Nthe
γ
N
N
γTo
N
p−�in
N
1 u
γ
N
N
γγγv
N
ūof
ū the
=
P
v
+
fγ(2.25)-(2.26)
.<
�
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
u
will
converge
to
the
continuous
solution
u)of
in
the
discrete
Holder
norm
qΩthe
+
�approximation
(with
Remark:
let
us
briefly
about
space
accuracy
the
approximation
P
+
fv
of
to
the
solution
uu
of
(2.1)
-Here,
(2.3).
One
would
expect
that
solution
u
denote
the
solution
to
the
scheme
at
each
time
level.
due
to
the
arbitrary
0+
�let
<
1)
with
the
rate
O(h
as
h(2.1)
→
0.the
p(2.15)
is
the
order
of
the
accuracy
of
γ
NP
N
N
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
=
N
N
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
(2.1)
(2.3).
One
would
expect
that
the
solution
γN
N
N
v
+
of
(2.25)-(2.26)
to
the
solution
u
(2.1)
(2.3).
One
would
expect
that
the
solution
u
=
P
f
.
Nu
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
of
(2.1)
(2.3).
One
would
expect
that
the
solution
γ
N
Remark:
Finally,
us
comment
briefly
about
the
space
accuracy
of
approximation
γ
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on N
Nu
: N�
P
v
+
f
of
to
the
solution
u
of
(2.3).
One
would
expect
that
the
solution
γ
uN
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
order
q
+
�
(with
N
N
γ
N
γ
γ
N
N
γ
N
γ
N
N
N
u
i+1
i
γ
γ
N
N
N γRemark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
=
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
(see
Def.
2.8)
outside
of
the
domain
Ω
(in
other
words
we
make
a
right-hand
side
g
=
−q
−
u
p−�
γ
N
N
1 operator
γ
Ω solution
the
arbitrary
0Finally,
<
<
1)
with
the
rate
O(h
)uin
habout
→
0.
Here,
psolution
isdiscrete
the
order
of
the
accuracy
of
the
approximation
of
the
continuous
differential
operator
L
by
the
L
(we
uRemark:
will
converge
to
the
continuous
u
in
the
discrete
Holder
norm
of
order
qorder
+
�that
(with
let
us
comment
briefly
the
space
accuracy
of
the
approximation
uN
=
where
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
∆t
∆t,h
N
v
+
fuof
of
(2.25)-(2.26)
to
solution
uabout
of
(2.1)
- Figures
(2.3).
One
would
expect
that
the
solution
P
v
+
fconverge
(2.25)-(2.26)
solution
uas
of
(2.1)
(2.3).
One
would
expect
that
the
solution
uN
converge
to
the
continuous
solution
up−�
the
discrete
Holder
norm
of
order
qapproximation
+
(with
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
aNu
uniqueness
P
vthe
+
fsolution
coincides
with
the
unorm
of
the
scheme
(2.15)
on
Nu
0
where
denote
the
to
the
difference
scheme
(2.15)
each
time
level.
Hence,
due
uP
will
converge
to
the
continuous
solution
u
in
the
Holder
norm
ofto
q�the
+
�Hence,
(with
N
γ
N
N
will
to
the
continuous
u
in
the
discrete
Holder
order
q��+
+
(with
N
v
+
fγu
of
(2.25)-(2.26)
to
the
solution
of
(2.1)
-discrete
(2.3).
One
would
expect
the
solutio
γthe
Nto
γthe
N
N
N
uwill
=
P
v
+
f�N
.solution
�
γwhere
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
� that
(with
N
Remark:
Finally,
let
us
comment
briefly
the
space
accuracy
of
the
approximation
the
arbitrary
0
<
�
<
1)
with
rate
O(h
)
as
h
→
Here,
p
is
the
order
of
the
accuracy
of
γN
where
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
du
N
smooth
extension
to
the
entire
auxiliary
domain/square
Ω
,-0.
see
5.1
-at
5.4).
Similarly,
compute
γargument,
Nv
N
γ(2.25)-(2.26)
+
f
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
the
solut
γ
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
(with
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
uuN
=:=
N
u
denote
the
solution
to
the
difference
scheme
(2.15)
at
each
time
level.
Hence,
due
t
N
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
γ γN
N
p−�
γ vP
γarbitrary
N the
that
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
γassume
p−�
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
the
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
0
u
i+1
i
p−�
∆t
∆t,h
p−�
p−�
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
N
p−�
particular
solution
ū
in
Ω
,
we
make
smooth
extension
of
the
right-hand
side
g
=
−q
−
u
outside
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(with
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(with
γ
N
N
the
arbitrary
0
<
�
<
1)
with
the
rate
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
u
=
P
v
+
f
.
�
N with
γ
N
γ
N
N
γ
v
+
f
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
solution
N
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
ū
N
Ω
the
arbitrary
0
<
�
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
a
uniqueness
argument,
P
v
+
f
coincides
with
the
solution
u
of
the
scheme
(2.15)
on
γ
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
N
N
γ
N
2
the
arbitrary
0
<
�
<
1)
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(wit
γ
N
N
γ
N
γ
N
Nu
∆t with
∆t,h
γuniqueness
Remark:
Finally,
let
us
comment
briefly
the
space
accuracy
of
the
approximation
u+solution
=soluti
PaNwill
vNP
+uniqueness
fγNp).
(2.25)-(2.26)
to
the
solution
uabout
of
(2.1)
-∆t
(2.3).
One
would
expect
that
the
+of
fNargument,
of
(2.25)-(2.26)
to
the
solution
u
of
(2.1)
(2.3).
One
would
expect
that
the
P
v
+
f
coincides
with
the
solution
of
the
scheme
(2.15)
o
argument,
P
v
+
f
coincides
with
solution
u
of
the
scheme
(2.15)
on
N
than
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
wesame
will
expect
O(h
)∆t,h
N
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(w
γa
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
�
(with
N
γ
N
γ
N
N
γ
N
γ
N
N
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
the
or
higher
order
γ
γv
0
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
p−�
p−�
of the
domain
to
the
auxiliary
domain
Ωp−�
We
compute
the
L
(L
)discrete
5.2
- 5.10)
and
L
(L
the
approximation
ofof
the
differential
L∆t
by
operator
L
(we
2 continuous
2the
∞the
∞
∞ ) q(we
u=
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
+ ��u(we
(with
ū operator
∆t,h
uN
=
P
v0maximum
+
.Ωofnorm
the
approximation
of
the
continuous
differential
operator
L
by
discrete
operator
L
the
arbitrary
�fthe
<
1)
with
the
rate
O(h
as
hthe
→
Here,
p(Tables
the
order
of
accuracy
of
the
arbitrary
<
<
1)
with
the
rate
O(h
)operator
as
h
→
0.
Here,
pwe
is
the
order
of
the
accuracy
oo
the
approximation
the
continuous
differential
operator
L0.
by
the
discrete
operator
Lthe
(we
∆t
the
approximation
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
γ�
N
N
γf+
N
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
the
approximation
the
continuous
differential
L
by
the
discrete
operator
L
2the
∆t
∆t,h
will
converge
to
continuous
solution
u.))about
in
discrete
Holder
norm
of
order
qaccuracy
+
(with
∆t
rate
the
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
∆t,h
uNu
P
fthe
.(2.25)-(2.26)
p−�
∆t
P
v=
+
of
to
the
solution
up−�
of
(2.1)
(2.3).
One
would
expect
that
∆t,h
Remark:
Finally,
let
us
comment
briefly
the
space
accuracy
of
the
approximation
arbitrary
0
<
<
1)
with
the
rate
O(h
as
h the
→
0.
Here,
pis
is
the
order
of
the
accuracy
uthan
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
qsolution
+
�+(with
γthat
N
γin
N
γγv
N
N
p).
Hence,
for
the
central
finite
difference
scheme
that
is-→
discussed
here,
will
expect
O(h
)∆t,h
N
u
P
v
+
f
.
will
converge
to
the
continuous
solution
u
in
discrete
Holder
norm
of
order
q
� =
(wi
γN
assume
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
N
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
Here,
p
is
the
order
of
the
of�
=
P
v
+
f
.
γ
N
N
γ
N
N
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
0.
Here,
p
is
the
order
of
the
accuracy
(Tables
5.1
5.10)
errors
respectively:
p−�
γ
N
N
N
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
2
assume
that
the
boundary
condition
(2.2)
approximated
by
(2.26)
with
the
same
or
higher
order
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
of
p−�
the
approximation
of
the
continuous
differential
L
by
the
discrete
operator
Lthe
(we
the
(DPM)
the
reader
can
consult
[7,briefly
4].
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
same
or
higher
order
p−�
the
approximation
of
the
continuous
differential
operator
LHere,
by
the
discrete
operator
Lorder
(we
2∆t,h
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
∆t
assume
boundary
condition
(2.2)
isO(h
approximated
by
(2.26)
with
the
same
orof
∆t
∆t,h
p−�
u
will
to
the
continuous
solution
u )of
in
discrete
Holder
norm
of
order
qhigher
+
Remark:
let
usspace.
comment
about
the
space
accuracy
of
the
approximation
u
eapproximation
arbitrary
0the
<method,
�Finally,
<
1)
with
the
rate
O(h
as
hthe
→
0.(2.3).
Here,
phere,
is
the
order
the
accuracy
of
rate
in
the
maximum
norm
in
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
P
vthat
+
fconverge
of
(2.25)-(2.26)
to
solution
u)about
(2.1)
-isdiscussed
One
would
expect
that
p).
Hence,
for
the
central
finite
difference
scheme
that
discussed
we
will
expect
O(h
)� (with
N
the
arbitrary
0of
<
�let
<
1)
with
the
rate
)operator
as
h
→
0.
pthe
is
the
order
of
the
of
N=
of
the
continuous
differential
operator
L
by
discrete
operator
L
(w
γHence,
Nthan
N
the
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
22)accuracy
γapproximation
∆t
he
arbitrary
0
<
�
<
1)
with
the
rate
O(h
as
h
→
0.
Here,
p
is
the
order
the
accuracy
∆t,h
Remark:
Finally,
us
comment
briefly
the
space
accuracy
of
the
approximation
u
2solution
∆t
∆t,h
N scheme
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
u
than
p).
for
the
central
finite
difference
scheme
that
is
here,
we
will
expect
O(h
)
Nu
approximation
the
continuous
differential
operator
L
by
the
discrete
operator
L
2
than
p).
Hence,
for
the
central
finite
difference
that
is
discussed
here,
we
will
expect
O(h
N
X
∆t
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
∆t,h
Remark:
Finally,
let
us
comment
briefly
about
the
space
accuracy
of
the
approximation
2
p−�
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
(we
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
∆t
∆t,h
assume
that
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
orqof
higher
orde
the
(DPM)
method,
reader
consult
[7,
4].
i(2.1)
i-L
2 1/2
the
arbitrary
0boundary
<
�the
<
1)
with
the
rate
O(h
)uuas
hdiscussion
→
0.
Here,
pby
isthe
the
order
of
the
accuracy
of
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
)
rate
in
the
maximum
norm
incan
space.
For
the
detailed
and
the
proof
ofthe
the
accuracy
P
v
+
f
of
(2.25)-(2.26)
to
the
solution
of
(2.3).
One
would
expect
that
the
soluti
e
approximation
of
the
continuous
differential
operator
by
discrete
operator
L
(we
u
will
converge
to
the
continuous
solution
in
the
discrete
Holder
norm
of
order
+
�
(with
the
approximation
of
the
continuous
differential
operator
L
the
discrete
operator
L
(we
∆t(max
|u
−
u
|)
)
,
L
(L
)
:=
(
γ
N
N
∆t
∆t,h
N
2
∞
γ
∆t
∆t,h
ume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
same
or
higher
orde
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
calc
exact
rate
in
the
norm
in
space.
For
the
detailed
discussion
the
proof
of
the
accuracy
of
rate
in
the
maximum
norm
inspace.
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
he
of(2.25)-(2.26)
the
continuous
differential
operator
L
by
the
discrete
operator
Lthe
(w
P
vvP
+
fAlgorithms
ofof
(2.25)-(2.26)
to
the
solution
uapproximated
of
(2.1)
- (2.3).
One
would
expect
that
the
soluti
v
+
fboundary
ofboundary
to
the
solution
uof
of
(2.1)
--isand
(2.3).
One
would
expect
that
the
solutio
2order
3
Composition
Approach
based
the
Difference
Potentials
Method
∆t
rate
in
the
maximum
norm
in
For
the
detailed
discussion
and
the
proof
ofthe
the
accuracy
of(we
(x,y)
γ
∆t,h
N
N
γmaximum
N
N
ume
that
the
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
orof
higher
or
2
γ
γapproximation
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
accuracy
that
the
condition
(2.2)
is
by
(2.26)
with
the
same
or
higher
the
(DPM)
method,
the
reader
can
consult
[7,
4].
P
+
fHence,
(2.25)-(2.26)
to
the
solution
uon
(2.1)
(2.3).
One
would
expect
that
sol
than
p).
for
the
central
finite
difference
scheme
that
discussed
here,
we
will
expect
O(h
)�ofof
p−�
i=0
the
(DPM)
method,
reader
can
consult
[7,
4].
γp).
the
approximation
ofthe
the
continuous
differential
operator
Ldiscrete
by
the
discrete
operator
Lhigher
Nassume
N
than
Hence,
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
γu
2
∆t
∆t,h
2
will
converge
to
the
continuous
solution
u
in
the
Holder
norm
of
order
q
+
(wi
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
or
order
the
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
sume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
order
N
the
(DPM)
method,
the
reader
can
consult
[7,
4].
the
(DPM)
method,
the
reader
can
consult
[7,
4].
than
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
will
expect
O(h
)
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
2
u
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
of
order
q
+
�
(wit
the
(DPM)
method,
the
reader
can
consult
[7,
4].
ssume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
ord
un
will
converge
to
the
continuous
solution
u
in
the
discrete
Holder
norm
order
q
+
�
(w
N
the
(DPM)
method,
the
reader
can
consult
[7,
4].
N
n
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(
than
p).
Hence,
for
the
central
finite
difference
scheme
is
discussed
here,
we
will
expect
O(h
)
3
Algorithms
Composition
Approach
based
on
thediscussion
Difference
Potentials
Method
rate
in
the
maximum
norm
in
space.
the
detailed
and
proof
of
the
accuracy
of+
p−�
2 )�
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
same
or
higher
order
and
u
will
converge
to
continuous
solution
uscheme
in
the
discrete
Holder
norm
order
qO(h
(
2)
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
ofof
the
accuracy
Below
we
develop
and
numerically
test
theFor
Algorithm
Composition
Approach
forwith
athe
model
the
(DPM)
method,
the
reader
can
consult
[7,
4].
N
the
arbitrary
0the
<
�the
<
1)
with
the
rate
O(h
))operator
as
h
→
0.∆t
Here,
pparabolic
is
the
order
of
the
accuracy
than
p).
Hence,
for
the
central
finite
difference
that
is
discussed
here,
we
will
expect
O(h
the
approximation
of
the
continuous
differential
L
by
the
discrete
operator
L
(we
p−�
an
p).
Hence,
for
central
finite
difference
scheme
that
is
discussed
here,
will
expect
p−�
∆t,h
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
ofooo
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
the
accuracy
the
arbitrary
0
�
<
1)
with
the
rate
O(h
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
2
he
arbitrary
0
<
�
<
1)
with
the
rate
O(h
)
as
h
→
0.
Here,
p
is
the
order
of
the
accuracy
han
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
p−�
3(DPM)
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
rate
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
ofMethod
the
accuracy
under
consideration
(2.1)
-central
(2.3).
the
method,
the
reader
can
consult
[7,
4].
than
p).
Hence,
for
the
finite
difference
scheme
that
is
here,
weproof
will
expect
O(h
) ofof(w
i discussion
i discussed
3
Algorithms
Composition
Approach
based
on
the
Potentials
ethe
in
the
maximum
incondition
space.
For
the
detailed
and
of
the
accuracy
(DPM)
method,
the
reader
can
consult
[7,
4].
the
arbitrary
0maximum
<Composition
� norm
<
with
the
rate
O(h
)(max
as
h
→
0.
Here,
pthe
isthe
the
order
of
the
LApproach
(L
)For
:=
max
|u
− Difference
uby
|).
3in
Algorithms
Composition
based
on
the
Difference
Potentials
Method
rate
in
the
norm
in
space.
detailed
discussion
and
the
proof
of
the
accuracy
the
approximation
of1)
the
continuous
differential
operator
L(2.26)
by
discrete
operator
Laccura
∞the
∞Algorithm
assume
that
the
boundary
(2.2)
isthe
approximated
with
the
same
or
higher
order
calc
exact
∆tand
3
Algorithms
Approach
based
on
the
Difference
Potentials
Method
∆t,h(w
te
in
the
maximum
norm
in
space.
For
the
detailed
discussion
proof
of
the
accuracy
of(
Below
we
develop
and
numerically
test
Composition
Approach
for
athe
parabolic
model
the
(DPM)
method,
the
reader
can
consult
[7,
4].
(i=0,...,N
) (x,y)
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
(DPM)
method,
the
reader
can
consult
[7,
4].
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
∆t
∆t,h
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
he
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
L
ate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
rate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
of
the
(DPM)
method,
the
reader
can
consult
[7,
4].
∆t
∆t,h
2
(DPM)
method,
the
reader
can
consult
[7,
4].
the
(DPM)
method,
the
reader
can
consult
[7,
4].Composition
the
approximation
of
the
continuous
differential
operator
L
by
the
discrete
operator
Lorde
under
consideration
(2.1)
-central
(2.3).
assume
that
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
the
same
or
higher
ord
than
p).
Hence,
for
the
finite
difference
scheme
isApproach
discussed
here,
we
will
expect
O(h
)
3e Below
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
∆t
∆t,h
Below
we
develop
and
numerically
test
the
Algorithm
for
acomposite
parabolic
model
system
(2.1)
-the
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
formthat
inofdomain
Ω)
in some
(DPM)
method,
the
reader
can
consult
[7,
assume
that
the
boundary
condition
(2.2)
is4].
approximated
by
(2.26)
with
the
same
or
higher
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
athe
parabolic
model
Let
us
first
make
areader
few important
remarks
about
theComposition
accuracy
the
Algorithms
Composition
Framework
Below
we
develop
and
numerically
test
the
Algorithm
Approach
for
afor
parabolic
model
the
(DPM)
method,
the
can
consult
[7,
4].
assume
that
the
boundary
condition
(2.2)
is
approximated
by
(2.26)
with
same
or
higher
ord
he
(DPM)
method,
the
reader
can
consult
[7,
4].
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
2
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
domain
Ω
- an
arbitrary
bounded
domain
in
R
with
boundary
∂Ω
- and
such
that
it
consists
of
under
consideration
(2.1)
-central
(2.3).
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
aPotentials
parabolic
model
than
p).
Hence,
for
the
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
rate
in
the
maximum
norm
inapproach
space.
For
the
detailed
discussion
and
the
of
the
accuracy
of 2
3Below
Algorithms
Composition
Approach
based
on
the
Difference
Method
assume
that
the
boundary
condition
(2.2)
isthe
approximated
by
(2.26)
with
the
same
or higher
and
validate
the
developed
for
problems
with
complex
interfaces.
We
will
select
ahere,
suitable
problem
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
aproof
parabolic
model
under
consideration
(2.1)
-central
(2.3).
under
consideration
(2.1)
- (2.3).
than
p).
Hence,
for
the
finite
difference
scheme
that
is
discussed
we
will
expect
O(h
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
under
consideration
(2.1)
(2.3).
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
han
p).
Hence,
for
the
central
finite
difference
scheme
that
is
discussed
here,
we
will
expect
O(h
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
two
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
andmethod,
test
the
accuracy
the
developed
approach
against
the
of
numerical
method
on
the single
We
start
the
introduction
ofof
scheme,
the
ofthe
the
ideas,
by
considering
the
1reader
2the
1 For
2and
1 illustration
2accuracy
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a parabolic
model
under
consideration
(2.1)
- of(2.3).
the
(DPM)
the-central
can
consult
[7,
4].detailed
rate
in
the
maximum
norm
in
space.
the
and
the
proof
ofthe
the
accuracy
3domain
Algorithms
Composition
Approach
based
on discussion
the
Difference
Potentials
Method
We
start
the
introduction
the
scheme,
the
illustration
of
the
ideas,
by
considering
under
consideration
(2.1)
(2.3).
than
p).
Hence,
for
the
finite
difference
scheme
that
isApproach
discussed
here,
we
will
expect O
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
2For
rate
inΩ
the
maximum
norm
in
space.
the
detailed
discussion
and
the
proof
of
the
accuracy
o
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
and
geometrically
simple
domain.
In
the
current
paper
we
combined
the
proposed
Algorithms
Composition
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
3
Algorithms
Composition
Approach
based
the
Difference
Potentials
Method
Below
we
develop
and
numerically
test
the
Algorithm
Composition
for
a
parabolic
model
1
1
2
2
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
ate
in
the
maximum
norm
in
space.
For
the
detailed
discussion
and
the
proof
of
the
accuracy
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
under
consideration
(2.1)
(2.3).
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
mode
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
the
(DPM)
method,
the
reader
can
consult
[7,
4].
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
Framework
with
second-order
central
finite
difference
scheme
space
first
order
Backward
Euler
scheme
rate
in
the
maximum
norm
inonly
space.
For
the
detailed
discussion
and
the
proof
ofofthe accura
the
(DPM)
method,
reader
can
subdomains
are
considered
for
the
simplicity
the
presentation,
and
that
thethat
discussed
two
disjoint
Ωthe
and
Ω2(2.8)
:domain
Ω
=
Ωits
∪Algorithm
Ω222Algorithm
,with
Γ [7,
=of
Ω4].
Ωin
piecewise
smooth
boundaries
system
(2.1)
- (2.3)
(or
formulation
in
time-discrete
form
inand
domain
Ω)
in
some
composite
1- here
1consult
1∩
2 with
domain
Ω
-subdomains
an
arbitrary
bounded
in
R
the
boundary
∂Ω
-of
and
such
it
consists
under
consideration
(2.1)
(2.3).
Below
we
develop
and
numerically
test
the
Composition
Approach
for
acomposite
parabolic
model
ow
we
develop
and
numerically
test
the
Composition
Approach
ageneral.
parabolic
mode
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
he
(DPM)
method,
the
reader
can
consult
[7,
4].
2
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
We
start
the
introduction
of
the
scheme,
and
the
illustration
the
ideas,
by
considering
the
in
time:
the
backward-centered
scheme.
(However,
the
developed
algorithms
composition
approach
is
under
consideration
(2.1)
(2.3).
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
2
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
idea
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
under
consideration
(2.1)
(2.3).
ow
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
mo
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
1
1
2
2
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
the
(DPM)
method,
the
reader
can
consult
[7,
4].
elow
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
model
two
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
2
1
2
1
2
1
2
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
Future
research
will
include
the
development
ofΩwith
the
higher-order
and
time
discretization
within
the
under
consideration
(2.1)
-i+1
2R
two
disjoint
subdomains
Ω
and
Ω
: Ωscheme,
=
ΩR1its
∪
, Γthe
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
i+1
domain
Ω
an
arbitrary
bounded
domain
in
the
∂Ω
and
such
that
it
consists
of
er
consideration
(2.1)
(2.3).
1-(2.3).
2test
2and
1boundary
2space
under
consideration
(2.1)
(2.3).
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
time-discrete
form
in
domain
Ω)
in
some
composite
Our
goal
at
each
time
level
t
is
to
find
an
approximations
to
u
in
domain
Ω:
domain
Ω
an
arbitrary
bounded
domain
in
with
boundary
∂Ω
and
such
that
it
consists
of
We
start
the
introduction
of
the
the
illustration
of
the
ideas,
by
considering
the
elow
we
develop
and
numerically
the
Algorithm
Composition
Approach
for
a
parabolic
mod
two
disjoint
subdomains
Ωand
and
Ω
:for
ΩFramework,
=
Ω
,=Conclusion
Γof
Ω1Ω
∩
Ω
with
piecewise
smooth
boundaries
subdomains
are
here
only
the
simplicity
the
presentation,
and
that
the
discussed
We
start
the
introduction
of
the
scheme,
illustration
the
ideas,
by
considering
the
1
1Ω∪2 ,Ω
2and
2 the
under
consideration
(2.1)
-∂Ω
(2.3).
two
disjoint
subdomains
Ω
Ω
: Γ2Ω
=
Ω
∪
Γ
Ω=the
∩
with
piecewise
smooth
boundaries
ΓWe
∂Ω
ΓΓconsidered
and
Γ
∂Ω
∪
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
proposed
Algorithms
Composition
see
Section
6 also
for
aof
more
detailed
discussion).
3
Algorithms
Composition
Approach
based
Difference
Potentials
Method
der
consideration
(2.1)
(2.3).
1-:=
1Figure
1the
2 on
1Γ1:=
1-1∪
2formulation
222
nder
consideration
(2.1)
(2.3).
:=
∂Ω
∪
and
Γ
:=
∪
Γ
(see
3).
We
would
like
to
emphasize
that
the
two
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Method
system
(2.1)
(2.3)
(or
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
�
2
start
the
introduction
of
the
scheme,
and
illustration
of
the
ideas,
by
considering
the
2
two
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
1
2
1
2
1
2
We
start
the
introduction
of
scheme,
and
the
illustration
of
the
ideas,
by
considering
We
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
theth
two
disjoint
subdomains
Ω
and
Ωof
: the
Ω
=
Ω
∪
Ω
,with
Γ
=
Ω
∩
Ωof
with
piecewise
smooth
domain
Ωcan
-1start
an
arbitrary
bounded
in
R
the
boundary
∂Ω
-and
and
such
that
it
consists
of
idea
be
extended
in
aformulation
way
to
multiple
subdomains
or
composite
domains.
system
(2.1)
-Γ
(2.3)
(or
(2.8)
in
time-discrete
form
in
domain
Ω)
in
some
composite
1-straightforward
2domain
1
23).
1would
2 the
Therefore,
the
obtained
space
accuracy
isthe
limited
by
the
accuracy
the
order
finite
difference
scheme.
Γsystem
:=
∂Ω
∪
and
Γ
:=
∂Ω
∪the
Γ
(see
Figure
We
like
to
also
emphasize
that
the
two
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
aboundaries
parabolic
model
nder
consideration
(2.1)
1:=
2
2here
subdomains
are
considered
here
only
for
simplicity
the
presentation,
that
the
discussed
We
start
the
introduction
scheme,
and
of
the
ideas,
by
considering
the
ΓWe
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
tosecond
also
emphasize
that
the
two
-Γ
(2.3)
(or
formulation
(2.8)
time-discrete
form
in
domain
Ω)
in
some
composite
ui+1
, its
(x,
y)
∈the
Ωof
3
Algorithms
Composition
Approach
based
on
Difference
Potentials
Method
2its
1Below
1(2.1)
2considered
2(2.3).
1 ,illustration
subdomains
are
only
for
the
of
the
presentation,
and
that
the
discussed
Ωin
i+1
We
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
1simplicity
start
the
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
the
i+1
i+1
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
Γ
:=
∂Ω
∪
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
1
1
2
2
Let
us
illustrate
this
point,
first
using
theoretical
and
then
numerically.
complex
interface
u
:=Figure
(3.1)
22estimates
i+1
Our
goal
at
each
time
level
there
isΓ
to(see
find
an
approximations
to
domain
Ω:The
Γ
:=
∂Ω
∪the
Γ
and
Γ
:=
∂Ω
∪
We
like
to
also
emphasize
that
the
two
3
Algorithms
Composition
Approach
based
on
the
Difference
Potentials
Meth
system
-extended
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
Ω(2.8)
em
(2.1)
-Ω
(2.3)
(or
insimplicity
time-discrete
form
in
domain
Ω)
in
some
composit
1Below
1we
2formulation
2 (2.3).
two
disjoint
subdomains
Ω
and
Ω
:domain
Ω
=
Ω
∪its
Ω
,with
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
subdomains
are
considered
only
for
the
simplicity
the
presentation,
and
that
the
discussed
domain
Ω
-(2.1)
an
arbitrary
bounded
in
R
boundary
∂Ω
and
such
that
it
consists
idea
can
be
extended
in
a1numerically
straightforward
way
to
multiple
subdomains
or
composite
domains.
u
,3).
(x,
∈
Ωwould
,the
2
1its
2y)
1u
2in
subdomains
are
considered
here
only
for
the
of
the
presentation,
and
that
the
discussed
under
consideration
(2.1)
system
(2.1)
(2.3)
(or
formulation
(2.8)
in
time-discrete
form
in
domain
Ω)
in
some
composite
2of
develop
and
test
the
Algorithm
Composition
Approach
for
a
parabolic
mod
We
start
introduction
of
the
scheme,
and
the
illustration
of
the
ideas,
by
considering
domain
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
ofot
idea
can
be
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
Ω
2
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
a
parabolic
mode
2
test
problem
(5.6)
which
we
consider
in=
this
Section
be=of
viewed
as
the
classical
solution
(without
complex
subdomains
are
considered
here
only
for
the
simplicity
the
presentation,
and
that
the
discussed
�
22can
two
disjoint
subdomains
Ω
and
Ω
:
Ω
Ω
∪
Ω
,
Γ
Ω
∩
Ω
with
piecewise
smooth
boundaries
tem
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
compo
stem
(2.1)
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
composite
i+1
i+1
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
2
1
2
1
1
2
subdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
2
i+1
i+1
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
i+1
Our
goal
at
each
level
isis(2.5)
to
an
approximations
to
u∩
domain
Ω:
can
be
extended
in
alevel
straightforward
way
to
multiple
subdomains
or
composite
domains.
Γtwo
:=
∂Ω
∪consideration
Γ
and
Γtime
∂Ω
∪(2.3).
Γ
(see
3).
would
to
also
emphasize
the
two
main
Ω
- an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
-piecewise
and
such
that
consists
o
idea
can
be
extended
in
a:=
straightforward
way
to
multiple
subdomains
or∂Ω
composite
domains.
disjoint
subdomains
Ω
Ωthe
:find
=
Ω
∪
Ω
,,the
Γ
Ω
∩×like
Ω
with
smooth
boundarie
domain
Ω
-(2.3)
an
arbitrary
bounded
domain
in
with
the
boundary
-Approach
and
such
that
it
consists
of
Our
goal
at
each
to
an
approximations
to11 2]
u
domain
Ω:
1idea
1
2time
interface)
to
the
heat
equation
inΩ
the
square
domain
Ω
≡
[−2,
[−2,
2].
Hence,
the
solution
(5.6)
toitboundaries
We
start
the
introduction
of
scheme,
and
of
the
ideas,
by
considering
the
Below
we
develop
and
numerically
the
Algorithm
Composition
for
athat
parabolic
mod
uin
(x,
y)22We
∈
Ω
,illustration
11t2and
2test
1R
2in
under
consideration
(2.1)
-tand
two
disjoint
subdomains
Ω
Ω
:find
ΩFigure
=
∪
Ω
Γ
=
Ω
Ω
with
piecewise
smooth
1=
ystem
(2.1)
-extended
(or
formulation
(2.8)
time-discrete
form
in
domain
Ω)
in
some
composi
2
1,2to
2in
i+1
2
under
(2.1)
-and
(2.3).
ΩΩ
i+1
idea
can
be
extended
in
astraightforward
straightforward
way
multiple
subdomains
or
composite
domains.
1 its
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
i+1
i+1
where
u
(here
p
=
1,
2)
are
the
solutions
to
(2.8)
in
each
subdomain
Ω
:
�
i+1
i+1
two
disjoint
subdomains
Ω
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundaries
omain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consists
of
main
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consist
1
1
2
2
idea
can
be
in
a
way
to
multiple
subdomains
or
composite
domains.
u
:=
(3.1)
p
i+1
�
1
2
1
2
1
2
(2.5)
can
be
computed
approximately
using
standard
backward-centered
finite
difference
scheme
for
the
heat
Ω
two
disjoint
subdomains
and
:=
Ωthe
=
Ω
Ω
Γ
=
Ω
∩Ω
Ω
with
piecewise
smooth
boundaries
Below
we
develop
and
numerically
test
the
Algorithm
Composition
Approach
for
aconsidering
parabolic
m
Our
goal
at∪
each
time
level
tΩ
is2∪of
to
find
an
approximations
u
inpiecewise
domain
Ω:
p
Our
goal
at
each
time
level
tand
to
find
an
approximations
to1the
u
in
domain
Ω:
Ω
1isand
1
2∈
1toΩ
2like
subdomains
are
considered
only
for
simplicity
presentation,
and
that
the
discussed
two
disjoint
Ω
Ω
:2the
Ω
=
Ω
∪
,Γ
Γ, =
=
Ω
∩∩
with
smooth
boundaries
2∪
u
,i+1
(x,
y)
Ω
,Ω
Γ
∂Ω
Γ
and
Γ
:=
∂Ω
Γ
(see
Figure
3).
We
like
to
also
emphasize
that
thetwo
two
disjoint
subdomains
Ω
:domain
Ω
Ω
∪
Ω
,and
with
piecewise
smooth
boundarie
i+1
1i+1
2the
1i+1
2
22
(2.1)
-subdomains
(2.3)
(or
formulation
(2.8)
in
its
time-discrete
domain
Ω)such
in
some
composite
2of
1
1start
2Ω
2simple
1:=
1
1would
We
the
introduction
scheme,
the
of
the
ideas,
by
t
Γsystem
:=
∂Ω
∪
Γ
and
Γ
∂Ω
ΓΩ
(see
Figure
3).
to
also
emphasize
that
the
under
consideration
(2.1)
-there
(2.3).
,Ω
(x,
y)
∈
,,illustration
Ωu
2u
1:=
1start
2single
2∪
i+1
i+1
11the
Our
goal
at
each
time
level
t∂Ω
is
to
find
an
approximations
to
uform
inin
domain
Ω:
We
the
introduction
of
scheme,
and
illustration
of
the
ideas,
by
considering
th
omain
Ω
an
arbitrary
bounded
in
R
with
the
boundary
∂Ω
and
that
it
consists
,2≡
(x,
y)=
∈
Ω
Ω
equation
on
the
and
square
domain
Ω
[−2,
2]
×
[−2,
2].
Thus
we
can
test
the
accuracy
of
our
i+1
11Ω
subdomains
are
considered
here
only
for
the
simplicity
of
presentation,
and
that
the
discussed
Our
goal
at
each
time
level
is
to
find
an
approximations
to
u
in
domain
Ω:
Ω
i+1
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
�
u
�
wo
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
,
Γ
Ω
∩
Ω
with
piecewise
smooth
boundaries
Ω(see
1
1
2
2
ounder
disjoint
subdomains
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
bounda
u
(3.1)
p:=
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
Figure
3).
to
also
emphasize
that
the
two
i+1
2
1
2
1
2
1
2
i+1
2
1
2
1
2
1
1
2
2
Γ
:=
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
u
:=
(3.1)
consideration
(2.1)
(2.3).
idea
can
extended
in (or
a2(or
straightforward
to
multiple
subdomains
or
composite
domains.
ΩΩ against
i+1
i+1
11start
2Γ] =
subdomains
considered
here
only
for
the
simplicity
presentation,
and
that
the
discussed
L
[u
G
,(2.8)
(x,
∈
Ω
and
(p
=
1,the
2) like
(3.2)
domain
Ω
-1are
an
arbitrary
bounded
domain
in
R
with
the
∂Ω
-piecewise
and
such
that
it
consists
=
∂Ω
∪beΓ
and
Γ
:=
∂Ω
∪
(see
Figure
3).
We
would
to
also
emphasize
that
the oftwt
�
uuy)
,its
(x,
∈
Ω
,boundary
Algorithms
Composition
Framework
the
accuracy
of
the
“single
domain
computations”.
Moreover,
p ,time-discrete
∆t
subdomains
are-introduction
considered
here
only
for
the
simplicity
presentation,
and
the
discussed
system
(2.1)
formulation
in
its
form
in
domain
Ω)
in
some
compos
22of
2
2formulation
Ωof
,,Ω
(x,
y)
∈
,subdomains
We
the
the
scheme,
and
the
of
the
ideas,
by
considering
uway
,i+1
(x,
y)
∈
Ω
,illustration
p Ω
(x,
y)
∈
Ω
system
(2.1)
-(2.3)
(2.3)
(2.8)
in
form
in
domain
Ω)
in
some
composit
Ω
�
i+1
2
1Ω
wo
disjoint
subdomains
Ω
and
: ∆tΩ
=
Ω
∪
,y)
Γwith
=
Ω
with
smooth
boundari
11, ∩ Ω
i+1
idea
can
be
extended
in
a
straightforward
way
multiple
or
composite
domains.
Ω
2
1
2
1
2
2
Ω
i+1
i+1
1i+1
subdomains
are
considered
here
only
for
the
simplicity
of
presentation,
and
that
the
discussed
1to
i+1
i+1
where
u
(here
p
=
1,
2)
are
the
solutions
to
(2.8)
in
each
subdomain
Ω
:
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
two
subdomains
are
considered
here
only
for
the
simplicity
the
presentation,
and
that
the
discussed
u
,
(x,
y)
∈
Ω
,
p
subdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
one
would
expect
that
the
accuracy
of
the
problem
(5.6)
the
complex
interface
will
be
limited
by
the
:=
∂Ω
Γ
and
Γ22 level
∪
(see
Figure
3).
would
like
to
also
emphasize
that
the
1Our
1∪
2 and
u
:=
(3.1)
1 ∩to
Ω
22y)
:=
(3.1)
i+1
goal
at
each
time
tstraightforward
is
to
find
an
approximations
u
in
domain
Ω:
p
1start
i+1
We
the
introduction
ofΓ
the
scheme,
and
the
illustration
of
the
ideas,
by
considerin
Ω(x,
uΩ
∈
Ω
idea
can
be
extended
inin
aaboundary
way
to
multiple
subdomains
or
composite
domains.
two
disjoint
subdomains
Ω
Ω
:domain
Ω
=
∪1, (x,
Ω
,with
ΓWe
=
Ωpresentation,
with
piecewise
smooth
boundaries
Ωu
Ωi+1
1Ω,the
idea
can
be
extended
straightforward
way
to
subdomains
or
composite
domains.
domains
are
considered
here
only
for
the
simplicity
of
and
that
the
discusse
i+1
i+1
domain
Ω
-the
an
arbitrary
bounded
in
R
the
boundary
∂Ω
and
such
that
it
consists
12
2(2.8)
11its
2ψ
1, boundary
2 i+1
i+1
u
,
y)
∈
Ω
,
Ω
u
,
(x,
y)
∈
Ω
domain
Ω
an
arbitrary
bounded
domain
in
R
∂Ω
and
such
that
it
consists
o
ystem
(2.1)
(2.3)
(or
formulation
in
time-discrete
form
in
domain
Ω)
in
some
compos
i+1
2
u
:=
(3.1)
2
i+1
subject
to
appropriate
conditions
l(u
)
=
on
the
boundary
∂Ω
of
the
original
Ω
Our
goal
at
each
time
level
t
is
to
find
an
approximations
to
u
in
domain
Ω:
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
like
to
also
emphasize
that
the
tw
accuracy
of
“single
domain
computations”.
Let
us
recall
as
well,
that
the
theoretical
error
L
(L
)
of
the
i+1
Ω
2i+1
Ωfor
∞ that
∞ domains.
idea
can
be
extended
in
aaaare
straightforward
way
to
multiple
subdomains
or composite
composite
domains.
1bdomains
1 u
2==1,
2i+1
i+1
uΩsolutions
:=
(3.1)
Ω2 to
idea
can
extended
in
straightforward
way
to
subdomains
or
domains.
idea
be
extended
in
straightforward
way
multiple
composite
u
, 3).
(x,
y)
∈Ω
Ω
, the
i+1
are
considered
here
only
the
simplicity
ofsubdomain
the
presentation,
and
the
discussed
where
(here
p(or
2)2)
the
to
(2.8)
in
each
Ωi+1
: or
�
ufor
2subdomains
i+1
Ωp
where
u
(here
p
1,
are
the
solutions
to
(2.8)
in
Ω
:
pto
domains
are
considered
here
only
the
simplicity
of
presentation,
and
that
the
discus
i+1
Ω
p
Ω
u
,
(x,
y)
∈
,
2
Our
goal
at
each
time
level
t
is
to
find
an
approximations
in
domain
Ω:
2
p
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
We
would
like
also
emphasize
that
the
two
system
(2.1)
(2.3)
formulation
(2.8)
in
its
time-discrete
form
in
domain
Ω)
in
some
comp
Ω
2
Our
goal
at
each
time
level
t
is
to
find
an
approximations
to
u
in
domain
Ω:
p
backward-centered
finite
difference
scheme
in
the
square
domain
with
Dirichlet
boundary
conditions
is
given
i+1
1
1
2
2
two
disjoint
subdomains
Ω
and
Ω
:
Ω
=
Ω
∪
Ω
,
Γ
=
Ω
∩
Ω
with
piecewise
smooth
boundar
aubdomains
can
be
extended
in
a straightforward
way
to
multiple
subdomains
or
composite
domains.
domain,
and
tosubdomains
interface
on
the
interface
Γ,
select
to
besuch
the
Lconditions
[uΩ
]and
=domain
G∆t
(x,
(p
=
2)we
(3.2)
Ωy)
two
disjoint
Ω
:,find
Ω
=
Ω
∪
with
piecewise
smooth
i+1
i+1
�
2
1∈
p , 2and
∆t
domain
Ω
- an
arbitrary
bounded
in
with
boundary
-domain
and
that
itboundarie
consists
i+1
i+1
1pi+1
2find
1boundary
1
22will
u2R
,2Ω
(x,
y)the
∈
,1,
are
considered
here
only
for
the
simplicity
ofΩwhich
and
that
the
discuss
Our
goal
at
time
level
is
to
find
an
approximations
u
inor
Ω:
1the
Our
goal
ateach
each
time
level
tΩ1ti+1
is
to
an
approximations
to to
upresentation,
in∂Ω
domain
Ω:
Our
each
time
level
is
to
an
in
domain
Ω:
i+1
i+1
Ω
ea
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
composite
domains.
i+1
i+1
1approximations
as
�
u
i+1
i+1
�
u
following:
where
u
(here
p
=
1,
2)
are
the
solutions
to
(2.8)
in
each
subdomain
Ω
:
Ω
a
can
be
extended
in
a
straightforward
way
to
multiple
subdomains
or
composite
domains.
u
,
(x,
y)
∈
Ω
,
p
subdomains
are
considered
here
only
for
the
simplicity
of
the
presentation,
and
that
the
discussed
Ω
i+1
p
where
u
(here
p
=
1,
2)
are
the
solutions
(2.8)
in
each
subdomain
Ω
:
p to
1
i+1
u
:=
(3.1)
domain
Ω
an
arbitrary
bounded
domain
in
R
with
the
boundary
∂Ω
and
such
that
it
consi
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
3).
We
like
to
also
emphasize
that
thetw
tw
Ω
i+1
p
i+1
Ω
Γ
:=
∂Ω
∪
Γ
and
Γ
:=
∂Ω
∪
Γ
(see
Figure
would
like
to
also
emphasize
that
the
�
goal
at
each
time
level
t
is
to
find
an
approximations
to
u
in
domain
Ω:
p
i+1
i+1
1
1
2
2
1 i+1
Ω
L
[uthe
=
G
,, i+1
(x,
y)
∈
, =
and
2)
(3.2)domains.
1 be
1
2straightforward
2]solutions
wo
disjoint
subdomains
Ω
and
:find
Ω∆t
=to
Ω
∪uu
Ω
,Ω
Γ
Ω
∩
Ω
with
smooth
boundar
�
L∆t
[uΩΩppΩ
]Ω
G
(x,
y)
∈
(p
=boundary
1, 22)
(3.2)
�
i+1
p
i+1
�
i+1
i+1
where
upΩto
(here
p=
to
each
subdomain
Ω:pin
:∂Ωpiecewise
i+1
i+1
py)
∆t
1are
2=
1,(2.8)
2in
12 Ω
u
,in
(x,
∈
Ωthe
,subdomains
dea
can
extended
in1,
a2)
way
to
multiple
or
composite
∆t
u
:=
(3.1)
,
(x,
y)
,
i+1
u
=
u
(x,
y)
∈
Γ,
subject
the
appropriate
boundary
conditions
l(u
)
=
ψ
on
of
the
original
1
i+1
i ∈
2i+1
i+1
where
u
p
=
1,
2)
are
the
solutions
(2.8)
each
subdomain
Ω
i+1
p(here
Ω
ur
goal
at
each
time
level
t
is
to
an
approximations
to
u
domain
Ω:
i+1
i+1
i+1
Ω
Ω
2
p
Ω
Ω
i+1
Ω
Ω
1
1
2
i+1
max
max
|u((jh,
kh),
−y)
Uy)
|Ω
≤of
(∆t
+presentation,
h ),
(5.7)
Ωpeach
idea
can
besubdomains
extended
atuΩ1straightforward
to
multiple
orpiecewise
composite
domains.
uapproximations
,1i∆t)
y)
∈jk∈
,Ω
r goal
at
time
level
is Γto
find
anway
u
in
domain
Ω:
uΩ
,(x,
Ω
,1the
(uin
)0≤i≤N
:=
(3.3)
u
,2(x,
∈
2subdomains
�
u
subdomains
are
considered
here
only
for
the
simplicity
presentation,
and
that
the
discuss
u∪
,∈toΓ,
1cT
Γ
subdomains
are
here
only
for
the
simplicity
and
that
the
discusse
disjoint
Ω
:i+1
=
Ω
Ω(x,
=
Ω
Ω
with
smooth
bound
Ω
i+1
u∇u
(3.1
1∩
pi+1
Ω1 ,Ω
u
:=
(3.1)
uΩ
i+1
Ω
i+1
i+1
2
2Ω
1β∇u
2 i+1
0≤j,k≤N
i+1
Ω
2 and
i+1
1
Ωp:=
Ω
Γtwo
∂Ω
Γthe
and
Γlconsidered
:=
∂Ω
∪
(see
Figure
3).
We
would
like
to in
also
emphasize
that
the
11i+1
i+1
·:=
nan
·,and
nΓ
, and
(x,
y)1,
domain,
and
to
interface
conditions
on
the
interface
boundary
Γ,
which
we
will
select
to
be original
the
Ω
i+1
i+1
1 :=
1 ∪each
2level
2i+1
i+1
i+1
Γ =
Γy)
L
[u
]
=
G
,
(x,
y)
∈
Ω
,
(p
=
2)
(3.2)
Our
goal
at
time
t
is
to
find
approximations
to
u
domain
Ω:
u
:=
(3.1)
u
,
(x,
∈
�
u
Ω
Ω
i+1
u
Ω
,
p
∆t
u
(3.1)t
i+1
1
2
u
:=
(3.1)
Ω
L
[u
]
=
G
,
(x,
y)
∈
Ω
,
(p
=
1,
2)
(3.2)
i+1
2
Ω
∆t
subject
to
appropriate
boundary
conditions
l(u
)
=
ψ
boundary
∂Ω
of
the
p
i+1
i+1
subject
to
the
appropriate
boundary
conditions
l(u
)
on
the
boundary
∂Ω
of
the
original
pΩ
Ω
p
∆t
u
,
(x,
y)
∈
Ω
,
Ω
Ω
Ω
�
u
Our
goal
at
each
time
level
t
is
to
find
an
approximations
to
u
in
domain
Ω:
2
19
Ω
∆t
1
2
u
,
(x,
y)
∈
Ω
,
Ω
Ω
Ω
i+1
p
p
idea
can
be
extended
in
ahere
straightforward
way
to
multiple
subdomains
or composite
composite
domains.
u
,Ω
(x,
y)
∈Ω
i+1
2Ω
i+1
L∆t
]i+1
=GG
(x,
,and
and
(p=
=
1,2)like
2) Ωto: also
(3.2)
Ω
idea
can
be
extended
in
a[u
straightforward
way
subdomains
or
domains.
uy)
, 1,presentation,
1
p
Ω
Γ
∂Ω
∪are
Γ
and
Γ22)L
:=
∂Ω
∪
Γ
(see
Figure
3).
We
would
emphasize
that
the
following:
2
Ω
∆t
i+1
1 :=
1 (here
2Ω]pu
Ω
u
(x,
∈of∈
,2which
[u
=
, ,the
(x,
y)
∈,22∈
Ω
(p
(3.2)
ubdomains
only
for
simplicity
the
and
that
the
discuss
where
udomain,
pto=
1,
are
the
solutions
to
(2.8)
in
each
subdomain
�
1Ω
:=
(3.1
p ,y)
∆t
p select
�
i+1
domain,
and
to
interface
conditions
on
the
interface
which
we
will
select
to be
be
the
Ω
andconsidered
interface
conditions
on
the
interface
boundary
Γ,
we
will
to
the
Ωthe
∆t
Ω
i+1
p i+1
1i+1,boundary
u
(x,
y)
,
pi+1 (here
�
Ω
i+1
i+1
i+1
i+1
i+1
i+1
1
where
u
p
=
1,
2)
are
solutions
to
(2.8)
in
each
subdomain
Ω
:
i+1
uan
,approximations
(x,
y)Γ,
∈i+1
Ω
,the
:=
i+1
i+1
uisΩto
=find
uΩ8l(u
,an
(x,
y)
∈
Our
goal
at
each
time
level
tt u
is
inor
domain
Ω:that
1
2
Our
goal
atappropriate
each
time
level
tofor
find
uu p∂Ω
in
domain
Ω:
Ω
i+1
subject
to
the
appropriate
boundary
conditions
)approximations
=
ψ=
on∈
the
boundary
of∂Ω
theofand
original
i+1
p
i+1
i+1
i+1
uu2Ω
,,i+1
(x,
y)
Ω
, topresentation,
Ω
subdomains
are
considered
here
the
simplicity
ofΩ
the (3.1)
discu
1 :=
2uΩ
Ω
subject
the
boundary
conditions
l(u
))(x,
ψ
on
boundary
the
original
1, the
uonly
(3
i+1
following:
following:
dea
can
extended
in
a2)
way
to
multiple
subdomains
composite
domains.
,
(x,
y)
∈
Ω
,
i+1
i+1
y)
∈
i+1
l=
(u
,2)
ustraightforward
) :=
(3.3)
Ω
where
ube
(here
p
1,
2)
are
the
solutions
to
(2.8)
in
each
subdomain
Ω
:
i+1
where
utoi+1
(here
p
1,
are
the
solutions
to
(2.8)
subdomain
Ω
:
2
1
1
Ω
Γ=
i+1
Ω
i+1
i+1
p
p
Ω
Ω
Ω
subject
to
the
appropriate
boundary
conditions
l(u
=
ψ
on
the
boundary
∂Ω
of
the
original
u
Ω
�
i+1
�
i+1
i+1
1
2
where
u
(here
p
=
1,
are
the
solutions
to
(2.8)
in
each
subdomain
Ω
:
2
1
Ω
Ω
Ω
u
,
(x,
y)
∈
Ω
,
pΩpto
puthe
p· nΓ l(u
p
i+1
�
�
u
:=
(3.
∇u
=
β∇u
·
n
,
(x,
y)
∈
Γ,
i+1
i+1
i+1
i+1
Ω
2
domain,
and
interface
conditions
on
the
interface
boundary
Γ,
which
we
will
select
to
be
the
where
(here
p
=
1,
2)
are
the
solutions
to
(2.8)
subdomain
Ω
:
where
u
(here
p
=
1,
2)
are
the
solutions
to
(2.8)
in
each
subdomain
Ω
:
i+1
subject
to
appropriate
boundary
conditions
)
=
ψ
on
the
boundary
∂Ω
of
the
original
Γ
u
:=
(3.1)
p
Ω
p
u
Ω
Ω
i+1
i+1
i+1
Ω
2
1
2
u
=
u
,
(x,
u
=
u
,
(x,
y)
∈
Γ,
Ω
L
[u
]
=
G
,
(x,
y)
∈
Ω
,
and
(p
=
1,
2)
(3.2)
Ω
Ω
Ω
domain,
and
to
interface
conditions
on
the
interface
boundary
Γ,
which
we
will
select
to
be
the
p
i+1
i+1
p
peach
Ωfind
py)
ideadomain,
can at
beand
extended
ini+1
a∆t
way
to
multiple
orselect
composite
domains
,, (x,
(x,
y)and
Ω2Γ,
,to
i+1
i+1
Ωu
ΩΩ
Ω
Our
goal
timelΓlΓlevel
is:=
toon
inwill
domain
Ω:
2 Ωy)
Ωi+1
∆t
2Ω
11 , an
2approximations
u
∈∈Ω
,subdomains
p
conditions
the
interface
boundary
which
the
u
∈
, u
,Ω(x,
L
[u
] on
=
G
(x,
(p
=
1, 2)
(3.2)
,utui+1
(3.3)
, straightforward
)):=
(3.3)
2
11
p , y)
Ωu
i+1
2Ω
Ω
Ω
uinterface
following:
Ω∈
Ω
Ω
ui+1
∆t
i+1
domain,
and toptointerface
conditions
the
boundary
Γ,
which
wewewill
toto
bebethe
i+1
1 1∆t
2 2Ωp
1 i+1
Ωto
1
Ω
u
re
ui+1 (here
=interface
1, 2)(u(u
are
the
solutions
in
each
subdomain
Ω : select
p (2.8)
−1
−1.5
−2−1.5
−1
−2 −1.5
−1.5
−2
−1.5
−2
−1
−1.5
−1
!
−0.5
!
−0.5
!
0
0.5
0
1
0.5
1.5
1
2
1.5
2
−1.5 −2
−2
−1.5
! 0
0 0.5 0.5 1
−2 −2 −2
−1.5 −1.5 −1 −1 −0.5 −0.5
−2
−2 −2 −1.5 −1.5 −1 −1 −0.5 −0.5 0 0 0.5 0.5 1
−2
−2
−2
−1.5
−2
−1
−1.5
−0.5
−1
−0.5 0
0 0.5 0.5
1
1 1.5
1.5 2
1
1.5
1
1.5
2
2
1.51.5
2
22
2
2
1
N2
N1
N1
1
2
N2
2
N2
t
t
t
i+1
i+1
i+1
i+1
i+1
s
∇u Ωpp·· nnΓ =
= β∇u
β∇u
∇u
· nΓ ,
(x, y)
y) ∈
∈ Γ,
Γ,
i+1 p
4
4
2
where c = max{(|| ∂∂xu4 ||L∞ (Ω̄×[0,T ]) + || ∂∂yu4 ||L∞ (Ω̄×[0,T ]) )/12, || ∂∂t2u ||L∞ (Ω̄×[0,T ]) /2}, where u((jh, kh), i∆t) is
i
the exact solution at the discrete points of the space and time mesh, and Ujk
is the approximate solution
obtained by the standard backward-centered scheme. If we now take the smallest mesh size h ≈ 0.0033
(which was used as the smallest mesh size in the problems with complex interfaces below), then by performing
rough estimates using formula (5.7) and expression for the analytical solution (5.6), we can show that the
best accuracy that can be achieved by the backward-centered finite difference scheme on the single and
geometrically simple domain Ω ≡ [−2, 2] × [−2, 2] is about O(10−4 ) − O(10−5 ) (note that the second order
continuous piecewise-linear finite element method will result in an error with a similar range). Therefore,
our results in Tables 5.1 - 5.10 (as well as the results in Tables 5.11 - 5.12, and Table 5.13) below will be
limited by the second order space accuracy. We illustrate this point numerically below as well, see Table 5.1.
For the experiment in Table 5.1 we used the settings as described above except in the last two rows of the
table: 2n1 × 2n1 ≡ 2n2 × 2n2 with n1 ≡ n2 = 10 for the auxiliary domains Ω0G1 and Ω0G2 , and the time step
was set to dt = 8.e − 7 (instead of 1.e − 6) to avoid the influence of the time discretization error. Notations
in Table 5.1: “Finite Difference” stands for the “single domain computations” approach when solution to
the problem (5.6) viewed as the classical solution (without complex interface) to the heat equation (2.5)
in the square domain Ω ≡ [−2, 2] × [−2, 2], and backward-centered finite difference scheme on the single
and geometrically simple domain Ω ≡ [−2, 2] × [−2, 2] is used to approximate the solution numerically.
Mesh h0 is used everywhere in the domain Ω ≡ [−2, 2] × [−2, 2] for the computation in this “single domain
computations”. “Algorithms Composition Framework” in Table 5.1 is used when we consider (5.6) as the
complex interface problem. Meshes with h1 and h2 are used for the computations of the particular solutions
ūN1 ∈ Ω1 and ūN2 ∈ Ω2 , respectively, in the Algorithms Composition Framework. We consider test Problem
3 with 2L = 42 in this experiment with complex interface problem. Notation “ —//—” in Table 5.1 is used
to denote the same value as in the above row.
We see that the accuracy of our Algorithms Composition Framework for complex interface problem (5.6)
does not differ from the accuracy of the “single domain computations”. Moreover, Table 5.1 illustrates
an important flexibility of Algorithms Composition Framework - different meshes can be used in different
subdomains, and the same accuracy is achieved using a much coarser mesh (with bigger size of h) in the
regions/subdomains of the problems where solutions exhibit smaller gradients. This feature of our method is
very important for the future development of the adaptive schemes/simulations for heterogenous problems,
as well as for the parallel computations.
Next, let us continue with more accuracy tests of Algorithms Composition Framework alone using different
curves as the interfaces below (see Tables 5.2 - 5.9). We again observe a second order convergence for the
space discretization for all four test problems from Tables 5.2 - 5.9. Also, there is not much difference in the
results when we change the total number 2L of the basis functions for the approximation of the Cauchy data
from 2L = 42, Table 5.1 to 2L = 22 and 2L = 30, Tables 5.2 - 5.9. This is explained by the low dimension
of the spaces of the Cauchy data uΓ (2.6) of the exact solution u.
Finally, in the last Table 5.10 in this Section, we again consider the test Problem 3 with 2L = 22, but we
fix the mesh size m1 = 7 for the construction of the particular solution ūN1 in the domain Ω0ūN1 , and we only
vary the mesh size m2 . The accuracy of the results is not affected in comparison to the results presented in
Tables 5.2-5.9 with m1 = m2 (this is similar to the performance of Algorithms Composition Framework that is
illustrated in Table 5.1). Again, this is expected due to the highly oscillatory behavior of the solution (5.6) in
the subdomain Ω2 , and hence computations of the solution in the subdomain Ω2 requires finer/smaller mesh
sizes. This example again illustrates the advantage of our algorithms composition approach since we have
flexibility to use different meshes in the different parts of the domain, and to solve problems independently
in each domain. This makes the numerical scheme much more computationally efficient and very suitable
for adaptive and parallel computations.
5.2. Second Example
As the second test problem we will again consider the heat equation (2.5), but with the exact solution
given below:
100tuΩ1 (x, y) = 100tu(1) , (x, y) ∈ Ω1 ,
u(x, y, t) =
(5.8)
100tuΩ2 (x, y) = 100t(u(1) + u(2) ), (x, y) ∈ Ω2 .
We consider here c1 = c2 = 4 in (5.3). The function u(2) is much less oscillatory now than the one in the
20
m1
6
8
7
9
8
10
“Algorithms
m2
h1
8
0.0625
8
0.0156
9
0.0312
9
0.0078
10 0.0156
10 0.0039
Composition Framework”
h2
L∞ (L∞ ) error of u(x, y)
0.0133
0.00387
0.0133
0.00387
0.0066
0.00097
0.0066
0.00097
0.0033
0.00024
0.0033
0.00024
“Finite Difference”
h0
0.0133
0.0133
0.0066
0.0066
0.0033
0.0033
L∞ (L∞ ) error of u(x, y)
0.00388
—//—
0.00097
—//—
0.00024
—//—
Table 5.1. Comparison of the L∞ (L∞ ) Errors for the “Algorithms Composition Framework” (with complex interface:
test problem 3, kθ = 3, number of the basis functions 2L = 42;) and “Classical Approach” (without complex interface) as
functions of the mesh size;
7
8
9
L2 (L∞ ) error of u(x, y)
0.00130
0.00032
0.00008
Ratio
4.06
4.00
L∞ (L∞ ) error of u(x, y)
0.01523
0.00382
0.00096
Ratio
3.99
3.98
Table 5.2. Errors as functions of the mesh size m1 = m2 ; number of the basis functions 2L = 22; kθ = 0, Test problem 1
previous Section 5.1.
As before, we set time step dt = 1.e − 6 for all tests in Section 5.2, and we consider the same time interval
[0, 0.01]. Here, for the construction of the G1 (ṽΓ1 ), as well as for the computation of the particular solution
ūN1 , we will choose the same auxiliary domains (as in the Section 5.1)
Ω0G1 ≡ Ω0ūN1 := [−2, 2] × [−2, 2].
However, we will select the auxiliary domain
Ω0G2 = [−1.5, 1.5] × [−1.5, 1.5],
for the construction of the G2 (ṽΓ2 ), and we will consider the auxiliary domain
Ω0ūN2 = [−1.6, 1.6] × [−1.6, 1.6],
for the computation of the particular solution ūN2 . We will use the Cartesian meshes
2n1 × 2n1 ≡ 2n2 × 2n2 with n1 ≡ n2 = 10
for the domains Ω0G1 and Ω0G2 , and we will select two different discrete norms (3.9) with α = 0.5 and α = 1.0
for the construction of the G1 (ṽΓ1 ) and G2 (ṽΓ2 ). The results are reported in Table 5.11 (α = 0.5) and in
Table 5.12 (α = 1.0). As demonstrated, the results are not affected by the choice of α. The errors that are
reported in Tables 5.11 - 5.12 are smaller than the ones in Tables 5.2 - 5.9. This is again expected due to
the less oscillatory behavior of the function u(2) in (5.8), and due to the choice of the auxiliary problems.
Remark: In the numerical experiments presented in Sections 5.1 and 5.2 we observed the overall second
order convergence of the scheme in space. Let us note that according to the general theoretical results for
a second order differential operator approximated by a discrete one with a second order accuracy [44] (see
also [49]), one would need to consider Taylor expansion in (3.20) with the derivative of order 2 + 2 = 4 to
maintain a second order accuracy of the approximation of the continuous potential by a difference potential.
However, it is also established that in reality this condition can be relaxed (see for example [34] and formula
(3.20) in this paper). We believe that the choice of the norm in the variational formulation (3.13) plays an
important role for the obtained convergence of our method in the considered numerical tests.
21
7
8
9
L2 (L∞ ) error of u(x, y)
0.00130
0.00032
0.00008
Ratio
4.06
4.00
L∞ (L∞ ) error of u(x, y)
0.01523
0.00382
0.00096
Ratio
3.99
3.98
Table 5.3. Errors as functions of the mesh size m1 = m2 ; number of the basis functions 2L = 30; kθ = 0, Test problem 1
7
8
9
L2 (L∞ ) error of u(x, y)
0.00131
0.00033
0.00008
Ratio
3.97
4.13
L∞ (L∞ ) error of u(x, y)
0.01534
0.00389
0.00097
Ratio
3.94
4.01
Table 5.4. Errors as functions of the mesh size m1 = m2 ; number of the basis functions 2L = 22; kθ = 2, Test problem 2
7
8
9
L2 (L∞ ) error of u(x, y)
0.00131
0.00033
0.00008
Ratio
3.97
4.13
L∞ (L∞ ) error of u(x, y)
0.01534
0.00389
0.00097
Ratio
3.94
4.01
Table 5.5. Errors as functions of the mesh size m1 = m2 ; number of the basis functions 2L = 30; kθ = 2, Test problem 2
7
8
9
L2 (L∞ ) error of u(x, y)
0.00131
0.00033
0.00008
Ratio
3.97
4.13
L∞ (L∞ ) error of u(x, y)
0.01540
0.00387
0.00097
Ratio
3.98
3.99
Table 5.6. Errors as functions of the mesh size m1 = m2 ; number of the basis functions 2L = 22; kθ = 3, Test problem 3
7
8
9
L2 (L∞ ) error of u(x, y)
0.00131
0.00033
0.00008
Ratio
3.97
4.13
L∞ (L∞ ) error of u(x, y)
0.01540
0.00387
0.00097
Ratio
3.98
3.99
Table 5.7. Errors as functions of the mesh size m1 = m2 ; number of the basis functions 2L = 30; kθ = 3, Test problem 3
7
8
9
L2 (L∞ ) error of u(x, y)
0.00138
0.00035
0.00009
Ratio
3.94
3.89
L∞ (L∞ ) error of u(x, y)
0.01588
0.00398
0.0010
Ratio
3.99
3.98
Table 5.8. Errors as functions of the mesh size m1 = m2 ; number of the basis functions 2L = 22; kθ = 5, Test problem 4
7
8
9
L2 (L∞ ) error of u(x, y)
0.00138
0.00035
0.00009
Ratio
3.94
3.89
L∞ (L∞ ) error of u(x, y)
0.01588
0.00398
0.0010
Ratio
3.99
3.98
Table 5.9. Errors as functions of the mesh size m1 = m2 ; number of the basis functions 2L = 30; kθ = 5, Test problem 4
22
7
8
9
L2 (L∞ ) error of u(x, y)
0.00131
0.00033
0.00008
Ratio
3.97
4.13
L∞ (L∞ ) error of u(x, y)
0.01540
0.00387
0.00097
Ratio
3.98
3.99
Table 5.10. Errors as functions of the mesh size m2 ; m1 = 7 is fixed. number of the basis functions 2L = 22; kθ = 3,
Test problem 3
8
9
L2 (L∞ ) error of u(x, y)
0.00001
2.5e − 6
Ratio
4.0
L∞ (L∞ ) error of u(x, y)
0.00020
0.00005
Ratio
4.0
Table 5.11. Errors as functions of the mesh size m1 = m2 ; number of the basis functions 2L = 22; kθ = 3, α = 0.5, Test
problem 3
5.3. Third Example
As the third test problem we will again consider the heat equation (2.5), but with the exact solution
given below:
−t
e uΩ1 (x, y) = e−t u(1) , (x, y) ∈ Ω1 ,
u(x, y, t) =
(5.9)
e−t uΩ2 (x, y) = e−t (u(1) + u(2) + 1.5u(3) ), (x, y) ∈ Ω2 .
We consider here c1 = c2 = 1 in (5.3). This test problem (5.9) is more challenging since β(s) 6= 1 in the
interface conditions (3.4) and it will be calculated as
β(s) =
∂uΩ1 ∂uΩ2 −1
∂n
∂n
(5.10)
As before, we set time step dt = 1.e − 6 for the test in Section 5.3, and we consider the same time interval
[0, 0.01]. Here, for the construction of the G1 (ṽΓ1 ), as well as for the computation of the particular solution
ūN1 we will choose the same auxiliary domains (as in the Sections 5.1 - 5.2)
Ω0G1 ≡ Ω0ūN1 := [−2, 2] × [−2, 2].
Also, for the construction of the G2 (ṽΓ2 ) we will select the auxiliary domain
Ω0G2 = [−1.6, 1.6] × [−1.6, 1.6],
and for the computation of the particular solution ūN2 we will consider the auxiliary domain
Ω0ūN2 = [−1.7, 1.7] × [−1.7, 1.7].
For the domains Ω0G1 and Ω0G2 , we will use the Cartesian meshes
2n1 × 2n1 ≡ 2n2 × 2n2 with n1 ≡ n2 = 10,
and we will select the discrete norm (3.9) with α = 0.5 for the construction of the G1 (ṽΓ1 ) and G2 (ṽΓ2 ). The
results are reported in Table 5.13. For this test problem (5.9) we observe that the convergence of our scheme
drops. This could be explained by the jump in the normal derivative of the solution (5.9) at the interface,
by the accuracy limitations of the considered second order finite difference space discretization (used away
from the interface), by the choice of the extension operator (formula (3.23)) , as well as by the first order
time discretization scheme used in this paper within the algorithms composition framework. As the part of
the future research, this result will be improved by considering different time and space discretization within
the developed approach, as well as different choices of the extension operators.
23
8
9
L2 (L∞ ) error of u(x, y)
0.00001
2.5e − 6
Ratio
4.0
L∞ (L∞ ) error of u(x, y)
0.00020
0.00005
Ratio
4.0
Table 5.12. Errors as functions of the mesh size m1 = m2 ; number of the basis functions 2L = 22; kθ = 3, α = 1.0, Test
problem 3
7
8
9
L2 (L∞ ) error of u(x, y)
3.3e − 5
8.3e − 6
3.1e − 6
Ratio
3.97
2.68
L∞ (L∞ ) error of u(x, y)
0.00051
0.00013
4.2e − 5
Ratio
3.92
3.10
Table 5.13. Errors as functions of the mesh size m1 = m2 ; number of the basis functions 2L = 22; kθ = 0, α = 0.5, Test
problem 1
6. Concluding Remarks
In this work, we developed an efficient and flexible Algorithms Composition Framework based on the idea
of the difference potentials method (DPM) for parabolic problems in composite domains. We illustrated
the accuracy, efficiency, and flexibility of our method with several numerical examples. Here, the parabolic
equation served as the simplified model, and the first step towards future development of the proposed scheme
for more realistic models of materials, fluids, or chemicals with different properties in different domains.
In this work we considered and tested our approach only on the geometries with smooth curvilinear
boundaries. In the future, we plan to extend the proposed method to problems in domains with arbitrary
smooth boundaries and with boundaries with corners. In this respect, the choice of the different basis
functions (3.15) for different parts of the boundaries, as well as the question of the efficient linear solvers for
the scheme (3.13) will be investigated with the goal of designing even more efficient and accurate methods.
Other future investigations will include the extension and further development of the proposed scheme to
problems in physics and biology (see for example [10, 9, 53, 43, 42], [2, 5, 6, 21]), as well as the development
of the space discretization based on the high-order finite difference (see for example [34]), finite-volume (see
for example [22], [7, 10, 9]), finite element methods (see for example [11]), and spectral methods [16] within
the proposed algorithms composition framework. Higher-order time discretization schemes will be studied
as well.
Acknowledgment: The author would like to express her gratitude to Professors Viktor Ryaben’kii and
Viktor Turchaninov for introducing and teaching her the idea of the Difference Potentials Method many
years ago, and for the warm and encouraging atmosphere of their collaboration. The author also would like
to thank very much Professor Viktor Ryaben’kii for the inspiring discussions about future research.
The author would like to thank Professor Fengyan Li for the helpful discussions. Finally, the author is
grateful to the anonymous referees for the careful reading of the paper and for their most valuable comments
about the method, suggestions that helped to improve the manuscript, as well as questions inspiring future
directions. The research is supported in part by the National Science Foundation Grant # DMS-1112984.
REFERENCES
[1] C. A. Brebbia. The boundary element method for engineers. Halsted Press [John Wiley & Sons], New York, 1978.
[2] Didier Bresch and Jonas Koko. Operator-splitting and Lagrange multiplier domain decomposition methods for numerical
simulation of two coupled Navier-Stokes fluids. Int. J. Appl. Math. Comput. Sci., 16(4):419–429, 2006.
[3] S. Britt, S. Tsynkov, and E. Turkel. A high order numerical method for the helmholtz equation with non-standard
boundary conditions. SISC, 2013. submitted for publication.
[4] A.-P. Calderón. Boundary value problems for elliptic equations. In Outlines Joint Sympos. Partial Differential Equations
(Novosibirsk, 1963), pages 303–304. Acad. Sci. USSR Siberian Branch, Moscow, 1963.
[5] Claire Chainais-Hillairet and Francis Filbet. Asymptotic behaviour of a finite-volume scheme for the transient driftdiffusion model. IMA J. Numer. Anal., 27(4):689–716, 2007.
[6] Claire Chainais-Hillairet, Jian-Guo Liu, and Yue-Jun Peng. Finite volume scheme for multi-dimensional drift-diffusion
equations and convergence analysis. M2AN Math. Model. Numer. Anal., 37(2):319–338, 2003.
24
[7] A. Chertock, Y. Epshteyn, and A. Kurganov. High-order finite-difference and finite-volume methods for chemotaxis
models. 2010. in preparation.
[8] Robert Dautray and Jacques-Louis Lions. Mathematical analysis and numerical methods for science and technology. Vol.
4. Springer-Verlag, Berlin, 1990. Integral equations and numerical methods, With the collaboration of Michel Artola,
Philippe Bénilan, Michel Bernadou, Michel Cessenat, Jean-Claude Nédélec, Jacques Planchard and Bruno Scheurer,
Translated from the French by John C. Amson.
[9] Y. Epshteyn. Upwind-difference potentials method for chemotaxis models. 2012. to appear in the proceedings of ”European
Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012)”.
[10] Y. Epshteyn. Upwind-difference potentials method for Patlak-Keller-Segel chemotaxis model. Journal of Scientific Computing, 53(3):689 – 713, 2012. DOI: 10.1007/s10915-012-9599-2.
[11] Yekaterina Epshteyn and Alexander Kurganov. New interior penalty discontinuous Galerkin methods for the Keller-Segel
chemotaxis model. SIAM J. Numer. Anal., 47(1):386–408, 2008/09.
[12] Lisa J. Fauci and Charles S. Peskin. A computational model of aquatic animal locomotion. J. Comput. Phys., 77(1):85–108,
1988.
[13] Ronald P. Fedkiw, Tariq Aslam, Barry Merriman, and Stanley Osher. A non-oscillatory Eulerian approach to interfaces
in multimaterial flows (the ghost fluid method). J. Comput. Phys., 152(2):457–492, 1999.
[14] Frédéric Gibou and Ronald Fedkiw. A fourth order accurate discretization for the Laplace and heat equations on arbitrary
domains, with applications to the Stefan problem. J. Comput. Phys., 202(2):577–601, 2005.
[15] Frederic Gibou, Ronald P. Fedkiw, Li-Tien Cheng, and Myungjoo Kang. A second-order-accurate symmetric discretization
of the Poisson equation on irregular domains. J. Comput. Phys., 176(1):205–227, 2002.
[16] David Gottlieb and Steven A. Orszag. Numerical analysis of spectral methods: theory and applications. Society for
Industrial and Applied Mathematics, Philadelphia, Pa., 1977. CBMS-NSF Regional Conference Series in Applied
Mathematics, No. 26.
[17] A. Greenbaum and A. Mayo. Rapid parallel evaluation of integrals in potential theory on general three-dimensional
regions. J. Comput. Phys., 145(2):731–742, 1998.
[18] L. Greengard and V. Rokhlin. A fast algorithm for particle simulations. J. Comput. Phys., 73(2):325–348, 1987.
[19] Boyce E. Griffith and Charles S. Peskin. On the order of accuracy of the immersed boundary method: higher order
convergence rates for sufficiently smooth problems. J. Comput. Phys., 208(1):75–105, 2005.
[20] Myungjoo Kang, Ronald P. Fedkiw, and Xu-Dong Liu. A boundary condition capturing method for multiphase incompressible flow. J. Sci. Comput., 15(3):323–360, 2000.
[21] Heinz-Otto Kreiss and N. Anders Petersson. An embedded boundary method for the wave equation with discontinuous
coefficients. SIAM J. Sci. Comput., 28(6):2054–2074 (electronic), 2006.
[22] Alexander Kurganov and Eitan Tadmor. New high-resolution central schemes for nonlinear conservation laws and
convection-diffusion equations. J. Comput. Phys., 160(1):241–282, 2000.
[23] Randall J. LeVeque and Zhi Lin Li. The immersed interface method for elliptic equations with discontinuous coefficients
and singular sources. SIAM J. Numer. Anal., 31(4):1019–1044, 1994.
[24] Randall J. LeVeque and Zhilin Li. Immersed interface methods for Stokes flow with elastic boundaries or surface tension.
SIAM J. Sci. Comput., 18(3):709–735, 1997.
[25] Zhilin Li and Kazufumi Ito. The immersed interface method, volume 33 of Frontiers in Applied Mathematics. Society
for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. Numerical solutions of PDEs involving
interfaces and irregular domains.
[26] Xu-Dong Liu, Ronald P. Fedkiw, and Myungjoo Kang. A boundary condition capturing method for Poisson’s equation
on irregular domains. J. Comput. Phys., 160(1):151–178, 2000.
[27] Xu-Dong Liu and Thomas C. Sideris. Convergence of the ghost fluid method for elliptic equations with interfaces. Math.
Comp., 72(244):1731–1746 (electronic), 2003.
[28] J. Loncaric, V. S. Ryaben’kii, and S. V. Tsynkov. Active shielding and control of noise. SIAM J. Appl. Math., 62(2):563–
596 (electronic), 2001.
[29] A. Mayo and A. Greenbaum. Fast parallel iterative solution of Poisson’s and the biharmonic equations on irregular regions.
SIAM J. Sci. Statist. Comput., 13(1):101–118, 1992.
[30] Anita Mayo. The fast solution of Poisson’s and the biharmonic equations on irregular regions. SIAM J. Numer. Anal.,
21(2):285–299, 1984.
[31] Anita Mayo. The rapid evaluation of volume integrals of potential theory on general regions. J. Comput. Phys., 100(2):236–
245, 1992.
[32] A. McKenney, L. Greengard, and A. Mayo. A fast Poisson solver for complex geometries. J. Comput. Phys., 118(2):348–
355, 1995.
[33] David McQueen and Charles Peskin. Shared-memory parallel vector implementation of the immersed boundary method
for the computation of blood flow in the beating mammalian heart. The Journal of Supercomputing, 11(3):213–236,
1997.
[34] M. Medvinsky, S. Tsynkov, and E. Turkel. The method of difference potentials for the helmholtz equation using compact
high order schemes. Journal of Scientific Computing, pages 1–44, May 2012.
[35] Duc Q. Nguyen, Ronald P. Fedkiw, and Myungjoo Kang. A boundary condition capturing method for incompressible
flame discontinuities. J. Comput. Phys., 172(1):71–98, 2001.
[36] Benoı̂t Perthame. Transport equations in biology. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.
[37] Charles S. Peskin. Mathematical aspects of heart physiology. Courant Institute of Mathematical Sciences New York
University, New York, 1975. Notes based on a course given at New York University during the year 1973/74.
[38] Charles S. Peskin. Numerical analysis of blood flow in the heart. J. Computational Phys., 25(3):220–252, 1977.
[39] Charles S. Peskin. The immersed boundary method. Acta Numer., 11:479–517, 2002.
[40] Charles S. Peskin and David M. McQueen. A three-dimensional computational method for blood flow in the heart. I.
25
Immersed elastic fibers in a viscous incompressible fluid. J. Comput. Phys., 81(2):372–405, 1989.
[41] Alfio Quarteroni and Alberto Valli. Domain decomposition methods for partial differential equations. Numerical Mathematics and Scientific Computation. The Clarendon Press Oxford University Press, New York, 1999. Oxford Science
Publications.
[42] Alfio Quarteroni, Alessandro Veneziani, and Paolo Zunino. Mathematical and numerical modeling of solute dynamics in
blood flow and arterial walls. SIAM J. Numer. Anal., 39(5):1488–1511 (electronic), 2001/02.
[43] Alfio Quarteroni, Alessandro Veneziani, and Paolo Zunino. A domain decomposition method for advection-diffusion
processes with application to blood solutes. SIAM J. Sci. Comput., 23(6):1959–1980 (electronic), 2002.
[44] A. A. Reznik. Approximation of surface potentials of elliptic operators by difference potentials. Dokl. Akad. Nauk SSSR,
263(6):1318–1321, 1982.
[45] V. S. Ryaben0 kiı̆. Boundary equations with projectors. Uspekhi Mat. Nauk, 40(2(242)):121–149, 238, 1985.
[46] V. S. Ryaben0 kiı̆. Difference potentials analogous to Cauchy integrals. Uspekhi Mat. Nauk, 67(3(405)):147–172, 2012.
[47] V. S. Ryaben’kii and S. V. Tsynkov. Artificial boundary conditions for the numerical solution of external viscous flow
problems. SIAM J. Numer. Anal., 32(5):1355–1389, 1995.
[48] V. S. Ryaben0 kiı̆, V. I. Turchaninov, and E. Yu. Èpshteı̆n. An algorithm composition scheme for problems in composite
domains based on the method of difference potentials. Zh. Vychisl. Mat. Mat. Fiz., 46(10):1853–1870, 2006.
[49] V.S. Ryaben’kii. Method of difference potentials and its applications. Springer-Verlag, 2001.
[50] V.S. Ryaben’kii, V. I. Turchaninov, and Ye. Yu. Epshteyn. The numerical example of algorithms composition for solution of
the boundary-value problems on compound domain based on difference potential method. Moscow, Keldysh Institute
for Applied Mathematics, Russia Academy of Sciences, (3), 2003.
[51] Andrea Toselli and Olof Widlund. Domain decomposition methods—algorithms and theory, volume 34 of Springer Series
in Computational Mathematics. Springer-Verlag, Berlin, 2005.
[52] S. Tsynkov. Numerical solution of problems on unbounded domains. A review. Appl. Numer. Math., 27(4):465–532, 1998.
Absorbing boundary conditions.
[53] Idan Tuval, Luis Cisneros, Christopher Dombrowski, Charles W. Wolgemuth, John O. Kessler, and Raymond E. Goldstein.
Bacterial swimming and oxygen transport near contact lines. PNAS, 102 (7):2277–2282, 2005.
[54] Luoding Zhu and Charles S. Peskin. Simulation of a flapping flexible filament in a flowing soap film by the immersed
boundary method. J. Comput. Phys., 179(2):452–468, 2002.
7. Appendix: Matrix Computation for System of Boundary Equations As we discussed in
Section 3.3 using the approximation (3.24) for vγp in the system of Boundary Equations (3.8), we obtain
the system of linear equations (3.25) with |γp | equations for 2L unknowns u0p := (u0p1 , ..., u0pL ), u1p :=
(u1p1 , ..., u1pL ):
Bp0 u0p + Bp1 u1p = 0,
(p = 1, 2).
Once again, here, |γp | is the total number of points in the set γp (p = 1, 2) and (u01` , u11` ), (u02` , u12` ) are the
unknown expansion coefficients of (ṽΓ1 , ṽΓ2 ) (3.19). Matrices Bp0 and Bp1 are defined as Bp0 := (b0p1 , ..., b0pL )
and Bp1 := (b1p1 , ..., b1pL ). Columns b0p` and b1p` of these matrices are |γp | dimensional vectors. The components of the vectors are computed as the values at the points of the set |γp | of the grid functions Qγp [Πγp Γp φ` ]
and Qγp [Πγp Γp φ?` ], respectively. These values of the grid functions
Qγp [Πγp Γp φ` ] ≡ Πγp Γp φ` − PNγp [Πγp Γp φ` ]
and
Qγp [Πγp Γp φ?` ] ≡ Πγp Γp φ?` − PNγp [Πγp Γp φ?` ]
are known and obtained by constructing 2L difference potentials: let us recall that the difference potential
uNp = PNγp vγp (Def. 2.4, Section 2.1) can be easily constructed in general. The operator PNγp is the linear
operator of the density vγp . Here, the difference potential
PNγp [Πγp Γp φ` ], or PNγp [Πγp Γp φ?` ]
is constructed by solving the simple auxiliary problem ((DAP), Def. 2.3), with the right-hand side given in
(2.19), where density vγp is set to:
vγp := Πγp Γp φ`? , for (xj , yk ) ∈ γp ,
(7.1)
vγp := Πγp Γp φ?`? , for (xj , yk ) ∈ γp ,
(7.2)
or to
with fixed `? that takes the values from 1 to L. Let us note that for the detailed discussion on the general
construction of the difference potential, one may refer to [49, 10].
26