Hindawi Publishing Corporation
Advances in Numerical Analysis
Volume 2013, Article ID 303952, 13 pages
http://dx.doi.org/10.1155/2013/303952
Research Article
Parallel Nonoverlapping DDM Combined with
the Characteristic Method for Incompressible Miscible
Displacements in Porous Media
Keying Ma and Tongjun Sun
School of Mathematics, Shandong University, Jinan, Shandong 250100, China
Correspondence should be addressed to Keying Ma; makeying@sdu.edu.cn
Received 28 May 2012; Revised 27 October 2012; Accepted 1 November 2012
Academic Editor: J. Rappaz
Copyright © 2013 K. Ma and T. Sun. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Two types of approximation schemes are established for incompressible miscible displacements in porous media. First, standard
mixed finite element method is used to approximate the velocity and pressure. And then parallel non-overlapping domain
decomposition methods combined with the characteristics method are presented for the concentration. These methods use the
characteristic method to handle the material derivative term of the concentration equation in the subdomains and explicit flux
calculations on the interdomain boundaries by integral mean method or extrapolation method to predict the inner-boundary
conditions. Thus, the velocity and pressure can be approximated simultaneously, and the parallelism can be achieved for the
concentration equation. The explicit nature of the flux prediction induces a time step limitation that is necessary to preserve stability.
These schemes hold the advantages of nonoverlapping domain decomposition methods and the characteristic method. Optimal
error estimates in 𝐿2 -norm are derived for these two schemes, respectively.
u ⋅ 𝜈 = (𝐷 (u) ∇𝑐) ⋅ 𝜈 = 0,
1. Introduction
The two-phase fluid displacements in porous media is one
of the most important basic problems in the oil reservoir
numerical simulation. It is governed by a nonlinear coupled system of partial differential equations with initial and
boundary values. In this paper, we will consider the following
incompressible miscible case: the pressure is governed by
an elliptic equation and the concentration is governed by a
convection-diffusion equation [1–5].
−∇ ⋅ (
𝑘 (𝑥)
(∇𝑝−𝛾 (𝑐) ∇𝑑 (𝑥))) ≡ ∇ ⋅ u = 𝑞,
𝜇 (𝑐)
𝑥 ∈ Ω, 𝑡 ∈ 𝐽,
(1a)
𝜙
𝜕𝑐
+u ⋅ ∇𝑐 − ∇ ⋅ (𝐷 (u) ∇𝑐) = (̃𝑐 − 𝑐) 𝑞̃,
𝜕𝑡
𝑥 ∈ Ω, 𝑡 ∈ 𝐽,
(1b)
𝑐 (𝑥, 0) = 𝑐0 (𝑥) ,
𝑥 ∈ 𝜕Ω, 𝑡 ∈ 𝐽,
(1c)
𝑥 ∈ Ω,
(1d)
2
where Ω is a bounded domain in R , 𝐽 = (0, 𝑇], and 𝑞̃ =
max{𝑞, 0} is nonzero at injection wells only. The variables in
(1a)–(1d) are the pressure 𝑝(𝑥, 𝑡) in the fluid mixture, the
Darcy velocity u = (𝑢1 , 𝑢2 )󸀠 , and the relative concentration
𝑐(𝑥, 𝑡) of the injected fluid. The 𝜈 is the unit outward normal
vector on boundary 𝜕Ω.
The coefficients and data in (1a)–(1d) are 𝑘(𝑥): the
permeability of the porous media; 𝜇(𝑐), the viscosity of the
fluid mixture; 𝑞(𝑥, 𝑡): representing flow rates at wells; 𝛾(𝑐)
and 𝑑(𝑥), the gravity coefficient and vertical coordinate; 𝜙(𝑥):
the porosity of the rock; ̃c(𝑥, 𝑡), the injected concentration
at injection wells (𝑞 > 0) and the resident concentration at
production wells (𝑞 < 0). Here, 𝐷(u) is a tensor 2 × 2 matrix
and generally has the form
𝐷 (u) = 𝜙 (𝑥) {𝑑𝑚 𝐼 + |u| (𝑑𝑙 𝐸 (u) + 𝑑𝑡 𝐸⊥ (u))} ,
(2)
2
where 2 × 2 matrix 𝐸 = (𝑒𝑖𝑗 ) satisfies 𝑒𝑖𝑗 = 𝑢𝑖 𝑢𝑗 /|u|2 ,
𝐸⊥ = 𝐼 − 𝐸, 𝑑𝑚 is the molecular diffusivity, and 𝑑𝑙 , 𝑑𝑡
are longitudinal and transverse dispersivities, respectively.
Furthermore, a compatibility condition ∫Ω 𝑞(𝑥, 𝑡)𝑑𝑥 = 0
must be imposed to determine the pressure.
The pressure equation is elliptic and easily handled by
standard mixed finite element method, which has been
proven to be an effective numerical method for solving fluid
problems. It has an advantage to approximate the unknown
variable and its diffusive flux simultaneously. There are many
research articles on this method [6–9]. The concentration
equation is parabolic and normally convection dominated.
It is well known that standard Galerkin scheme applied
to the convection dominated problems does not work well
and produces excessive numerical diffusion or nonphysical
oscillation. A variety of numerical techniques have been
introduced to obtain better approximations for (1a)–(1d),
such as characteristic finite difference method [10], characteristic finite element method [11], the modified of characteristic finite element method (MMOC-Galerkin) [12], and
the Eulerian-Lagrangian localized adjoint method (ELLAM)
[13].
It is well known that parallel algorithms, based upon
overlapping or non-overlapping domain decompositions, are
effective ways to solve the large scale of PDE systems for
most practical problems in engineering (e.g., see [14–17]).
We have presented parallel Galerkin domain decomposition
procedures for parabolic equation [18–20]. These procedures
use implicit Galerkin methods in the subdomains and explicit
flux calculations on the interdomain boundaries by integral
mean method or extrapolation-integral mean method. Some
constraints for time step are still needed for these procedures
to preserve stability, but less severe than that for fully explicit
methods. With respect to the accuracy order of ℎ, 𝐿2 -norm
error estimates are optimal for higher-order finite element
spaces and almost optimal for linear finite element space in
twodimensional domain. Compared with Dawson-Dupont’s
schemes [21], these 𝐿2 -norm error estimates avoid the loss of
𝐻−1/2 factor.
We also have considered using the procedures in [18] for
wave equation [22] and convection-diffusion equation [23].
This paper is one of our sequent research papers. The main
purpose is to use parallel Galerkin domain decomposition
procedures in [18] combined with the characteristic method
for the concentration equation of incompressible miscible
displacements in porous media. This paper is organized as
follows. In Section 2, we first present mixed finite element
method for the velocity and pressure and then formulate
parallel non-overlapping domain decomposition methods
combined with the characteristics method for the concentration. We establish the combined approximation schemes.
Auxiliary lemmas are listed in Section 3, which show some
properties of finite element spaces and projections. In Section 4, we derive the optimal-order 𝐿2 -norm error estimates.
Finally in Section 5, we extend the consideration for another
approximate scheme by using extrapolation method. It is
worthwhile to specially emphasize that the research of this
paper is creative. No former researchers discussed it. These
Advances in Numerical Analysis
schemes not only hold the advantages of non-overlapping
domain decomposition methods, but also hold the advantages of the characteristic method.
2. Formulation of the Methods
In this paper, we adopt notations and norms of usual Sobolev
spaces [3]:
𝐻𝑚 (Ω) = {𝑓 :
𝜕|𝛼| 𝑓
∈ 𝐿2 (Ω) for |𝛼| ≤ 𝑚} ,
𝜕𝑥𝛼
󵄩󵄩 |𝛼| 󵄩󵄩2
󵄩𝜕 𝑓󵄩
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑓󵄩󵄩𝑚 = ∑ 󵄩󵄩󵄩󵄩 𝛼 󵄩󵄩󵄩󵄩 ,
󵄩 𝜕𝑥 󵄩󵄩
|𝛼|≤𝑚 󵄩
𝑚 ≥ 0,
𝜕|𝛼| 𝑓
𝑚
𝑊∞
∈ 𝐿∞ (Ω) for |𝛼| ≤ 𝑚} ,
(Ω) = {𝑓 :
𝜕𝑥𝛼
󵄩󵄩 |𝛼| 󵄩󵄩
󵄩𝜕 𝑓󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑓󵄩󵄩𝑊∞𝑚 = max󵄩󵄩󵄩󵄩 𝛼 󵄩󵄩󵄩󵄩 ,
|𝛼|≤𝑚󵄩
󵄩 𝜕𝑥 󵄩󵄩∞
(3)
𝑚 ≥ 0.
0
(Ω) = 𝐿∞ (Ω). The inner proIn particular, 𝐻0 (Ω) and 𝑊∞
duct on 𝐿2 (Ω) is denoted by (⋅, ⋅).
We also use the following spaces that incorporate time
dependence. Let [𝑎, 𝑏] ⊂ 𝐽 and let 𝑋 be any of the spaces just
defined. For 𝑓(𝑥, 𝑡) suitably smooth on Ω × [𝑎, 𝑏], we let
𝑏󵄩
󵄩󵄩 𝜕𝛼 𝑓 (⋅, 𝑡) 󵄩󵄩󵄩
󵄩󵄩 𝑑𝑡 < ∞, for 𝛼 ≤ 𝑚} ,
𝐻𝑚 (𝑎, 𝑏; 𝑋) = {𝑓 : ∫ 󵄩󵄩󵄩
󵄩
𝑎 󵄩
󵄩 𝜕𝑡𝛼 󵄩󵄩𝑋
𝑚
𝑏
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑓󵄩󵄩𝐻𝑚 (𝑎,𝑏;𝑋) = ( ∑ ∫
𝑎
𝛼=0
1/2
󵄩󵄩 𝜕𝛼 𝑓 (⋅, 𝑡) 󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑑𝑡)
,
󵄩󵄩
󵄩
󵄩󵄩 𝜕𝑡𝛼 󵄩󵄩󵄩𝑋
𝑚 ≥ 0.
(4)
𝑚
(𝑎, 𝑏; 𝑋) and the norm ‖𝑓‖𝑊∞𝑚 (𝑎,𝑏;𝑋) are defined.
Similarly, 𝑊∞
1
If [𝑎, 𝑏] = 𝐽, we simplify our notation and write 𝐿∞ (𝑊∞
) for
∞
1
𝐿 (0, 𝑇; 𝑊∞ (Ω)), and so forth.
If 𝑓 = (𝑓1 , 𝑓2 ) is a vector function, we note that 𝑓 ∈ 𝑋 if
𝑓1 ∈ 𝑋 and 𝑓2 ∈ 𝑋. We also use the vector-function spaces
and norms:
𝐻𝑚 (div; Ω) = {𝑓 : 𝑓1 , 𝑓2 , ∇ ⋅ 𝑓 ∈ 𝐻𝑚 (Ω)} ,
󵄩2
󵄩 󵄩2 󵄩 󵄩2 󵄩
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑓󵄩󵄩𝐻𝑚 (div;Ω) = 󵄩󵄩󵄩𝑓1 󵄩󵄩󵄩𝑚 + 󵄩󵄩󵄩𝑓2 󵄩󵄩󵄩𝑚 + 󵄩󵄩󵄩∇ ⋅ 𝑓󵄩󵄩󵄩𝑚 ,
𝑚 ≥ 0,
(5)
𝐻 (div; Ω) = 𝐻0 (div; Ω) .
We need some assumptions. The regularity assumptions
on the solution of (1a)–(1d) are noted by [4]:
1
𝑐 ∈ 𝐿∞ (𝐻2 ) ∩ 𝐻1 (𝐻1 ) ∩ 𝐿∞ (𝑊∞
) ∩ 𝐻2 (𝐿2 ) ,
{
{
∞
1
(𝑃) : {𝑝 ∈ 𝐿 (𝐻 ) ,
{
∞
1
∞
1
1
∞
2
2
{u ∈ 𝐿 (𝐻 (div))∩𝐿 (𝑊∞ )∩𝑊∞ (𝐿 )∩𝐻 (𝐿 ) .
(6)
Advances in Numerical Analysis
3
We also require the following assumptions on the coefficients in (1a)–(1d) [4]. Let 𝑎0 , 𝑎1 , 𝜙0 , 𝜙1 , 𝑑0 , 𝑑1 , 𝐾1 , and 𝐾2 be
positive constants such that
𝑘 (𝑥)
{
≤ 𝑎1 ,
0 < 𝑎0 ≤
0 < 𝜙0 ≤ 𝜙 (𝑥) ≤ 𝜙1 ,
{
{
𝜇 (𝑐)
{
{
󵄨
󵄨
{
󵄨
󵄨
󵄨󵄨 𝜕 (𝑘/𝜇)
󵄨󵄨 󵄨󵄨 𝜕𝛾
󵄨󵄨 󵄨
{
󵄨
{
󵄨󵄨
{
(𝑥, 𝑐)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨 (𝑥, 𝑐)󵄨󵄨󵄨 + 󵄨󵄨󵄨∇𝜙 (𝑥)󵄨󵄨󵄨
󵄨󵄨
{
{
󵄨
󵄨󵄨
𝜕𝑐
𝜕𝑐
󵄨
󵄨
{󵄨
󵄨
󵄨
{
󵄨
󵄨󵄨
{
{
󵄨 󵄨󵄨 𝜕̃𝑞
󵄨
󵄨
(𝑄) : { + 󵄨󵄨󵄨𝑞̃ (𝑥, 𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨 𝜕𝑡 (𝑥, 𝑡)󵄨󵄨󵄨󵄨 ≤ 𝐾1 ,
󵄨
󵄨
{
{
󵄨󵄨
{
󵄨󵄨 𝜕𝐷
󵄨󵄨 󵄨󵄨󵄨 𝜕𝐷−1
{
󵄨
󵄨
󵄨
{
{
{󵄨󵄨󵄨󵄨 𝜕u (u)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨 𝜕u (u)󵄨󵄨󵄨󵄨 ≤ 𝐾2 ,
{
󵄨󵄨
{
󵄨
󵄨 󵄨󵄨
{
{
2
{
2
󵄩
󵄩
{
{𝑑0 󵄩󵄩󵄩𝜉󵄩󵄩󵄩 ≤ ∑ 𝐷 (u) 𝜉𝑖 𝜉𝑗 ≤ 𝑑1 󵄩󵄩󵄩󵄩𝜉󵄩󵄩󵄩󵄩2 ,
∀𝜉 ∈ R2 .
𝑖,𝑗=1
{
(7)
Further assumptions will be made in individual theorems as
necessary.
For convenience, we assume that (1a)–(1d) is Ω periodic
(see [4]); that is, all functions will be assumed to be spatially Ω-periodic throughout the rest of this paper. This is
physically reasonable, because no-flow conditions (1c) are
generally treated by reflection, and in general, interior flow
patterns are much more important than boundary effects in
reservoir simulation. Thus, the boundary conditions (1c) can
be dropped.
Throughout the analysis, 𝐶 and 𝐾𝑖 (𝑖 = 3, . . . , 6) will
denote generic positive constants, independent of ℎ𝑐 , ℎ𝑝 ,
Δ𝑡𝑐 , and Δ𝑡𝑝 , but possibly depending on constants in (𝑄),
norms in (𝑃). Similarly, 𝜀 will denote a generic small positive
constant.
2.1. A Mixed Finite Element Method for the Pressure and
Velocity. Let
𝐻 = {𝜒 ∈ 𝐻 (div; Ω) | 𝜒 ⋅ 𝜈 = 0 on 𝜕Ω} ,
𝑊=
𝐿2 (Ω)
.
{𝑤 ≡ constant on Ω}
(8)
(9)
As in [4], the pressure equation (1a) is equal to the
following saddle-point problem of finding a map (u, 𝑝) : 𝐽 →
𝐻 × 𝑊 such that
(a) 𝐴 (𝑐; u, 𝑣) + 𝐵 (𝑣, 𝑝)
= (𝛾 (𝑐) ∇𝑑, 𝑣) ,
∀𝑣 ∈ 𝐻 (div; Ω) ,
(b) 𝐵 (u, 𝑤) = − (𝑞, 𝑤) ,
(10)
∀𝑤 ∈ 𝐿2 (Ω) ,
𝜇 (𝑐)
u, 𝑣) ,
𝑘 (𝑥)
𝐵 (𝑣, 𝑝) = − (∇ ⋅ 𝑣, 𝑝) ,
𝐴 (𝐶; 𝑈, 𝑣) + 𝐵 (𝑣, 𝑃) = (𝛾 (𝑐) ∇𝑑, 𝑣) ,
𝐵 (𝑈, 𝑤) = − (𝑞, 𝑤) ,
∀𝑣 ∈ 𝑉ℎ𝑝 ,
∀𝑤 ∈ 𝑊ℎ𝑝 .
(11)
𝐵 (u, 𝑤) = − (∇ ⋅ u, 𝑤) .
For ℎ𝑝 > 0, we discrete (10) in space on a quasiuniform mesh Tℎ𝑝 of Ω with diameter of element ≤ ℎ𝑝 . Let
(12)
Existence and uniqueness of 𝑈 and 𝑃 are proved in [1], based
on ideas of [24].
2.2. A Characteristic DDM for the Concentration. In this
section, we assume 𝑢 in (1a)–(1d) is given. Define
𝑀 = {𝑣 ∈ 𝐿2 (Ω) | 𝑣 is piecewise constant} ,
𝑀 is dense in 𝐿2 (Ω) .
(13)
Let
2 1/2
𝜓 (𝑥) = [|u (𝑥)|2 + (𝜙 (𝑥)) ]
2
2
2 1/2
= [(𝑢1 (𝑥)) + (𝑢2 (𝑥)) + (𝜙 (𝑥)) ]
(14)
,
and let the characteristic direction associated with the operator 𝜙𝜕𝑐/𝜕𝑡 + u ⋅ ∇𝑐 be denoted by 𝜏, where
𝜓 (𝑥)
𝜕𝑐
𝜕𝑐
= 𝜙 (𝑥)
+ u ⋅ ∇𝑐.
𝜕𝜏
𝜕𝑡
(15)
The weak form of (1a)–(1d) is finding a map 𝑐 : 𝐽 →
𝐿2 (Ω) such that
(a) (𝜓
𝜕𝑐
, 𝜑) + ((𝐷 (u) ∇𝑐) , ∇𝜑)
𝜕𝜏
= ((̃𝑐 − 𝑐) 𝑞̃, 𝜑) ,
(b) 𝑐 (𝑥, 0) = 𝑐0 (𝑥) ,
∀𝜑 ∈ 𝐿2 (Ω) ,
(16)
∀𝑥 ∈ Ω.
Part 𝐽 into 0 = 𝑡0 < 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑁 = 𝑇, with Δ𝑡𝑐𝑛 = 𝑡𝑛 −𝑡𝑛−1 .
The analysis is valid for variable time steps, but we drop the
superscript from Δ𝑡𝑐 for convenience. For functions 𝑓 on
Ω×𝐽, we write 𝑓𝑛 (𝑥) for 𝑓(𝑥, 𝑡𝑛 ). Approximate (𝜕𝑐𝑛 /𝜕𝜏)(𝑥) =
(𝜕𝑐/𝜕𝜏)(𝑥, 𝑡𝑛 ) by a backward difference quotient in the 𝜏direction,
𝑐𝑛 (𝑥) − 𝑐𝑛−1 (𝑥 − (u (𝑥) /𝜙 (𝑥)) Δ𝑡𝑐 )
𝜕𝑐𝑛
.
(𝑥) ≃
2
𝜕𝜏
Δ𝑡𝑐 √1 + |u (𝑥)|2 /(𝜙 (𝑥))
where the bilinear forms
𝐴 (𝑐; u, 𝑣) = (
𝑉ℎ𝑝 × 𝑊ℎ𝑝 ⊂ 𝐻 × 𝑊 be Raviart-Thomas spaces [6, 7] of index
𝑙 = 0 for this mesh.
The mixed method for pressure and velocity, given a
concentration approximation 𝐶 at a time 𝑡 ∈ 𝐽, consists of
𝑈 ∈ 𝑉ℎ𝑝 and 𝑃 ∈ 𝑊ℎ𝑝 such that
(17)
If we let 𝑥 = 𝑥 − (u(𝑥)/𝜙(𝑥))Δ𝑡𝑐 and 𝑓(𝑥) = 𝑓(𝑥), then
we get
𝜓
𝜕𝑐𝑛
𝑐𝑛 − 𝑐𝑛−1
.
≃𝜙
𝜕𝜏
Δ𝑡𝑐
(18)
Since the problem is Ω-periodic [4], 𝑐𝑛−1 is always defined
and the tangent to the characteristic (i.e., the 𝜏 segment)
4
Advances in Numerical Analysis
ν1
νΓ
Ω1
Γ1
ν2
Ω2
Γ
Γ2
Figure 1: The domain Ω with the interdomain boundary Γ.
cannot cross a boundary to an undefined location. The
difference quotient relates the concentration at a given 𝑥 at
time 𝑡𝑛 to the concentration that would flow to 𝑥 from time
𝑡𝑛−1 if the problem were purely hyperbolic.
The time difference (18) will be combined with a characteristic DDM in the space variables. We recall the domain
decomposition procedures in [18] here. For simplicity and
without losing generality, we only discuss the case of two
subdomains. But the algorithms and theories can be extended
to the case of many subdomains. Divide Ω into two subdomains Ω𝑗 (𝑗 = 1, 2) by an interdomain boundary Γ, which
is a surface of dimension 𝑑 − 1, see Figure 1. We denote by
Γ𝑗 = 𝜕Ω𝑗 ⋂ 𝜕Ω the part of the boundary of the subdomains
which coincides with 𝜕Ω. Denote the unit vector normal to Γ
as 𝜈Γ , which points from Ω1 toward Ω2 .
Let T𝑗,ℎ𝑐 be quasi-uniform partitions of Ω𝑗 (𝑗 = 1, 2) and
Tℎ𝑐 = T1,ℎ𝑐 ⋃ T2,ℎ𝑐 . Here, ℎ denotes the maximal element
diameter of Tℎ𝑐 . We construct the finite element space Mℎ𝑐
on Tℎ𝑐 which satisfies the following condition (I)
(1) For 𝑗 = 1, 2, let M𝑗,ℎ𝑐 be a finite element subspace of
𝐻1 (Ω𝑗 ), and let Mℎc ⊂ 𝐿2 (Ω) such that if 𝑣 ∈ Mℎc ,
then 𝑣|Ω𝑗 ∈ M𝑗,ℎ𝑐 .
(2) For 𝑗 = 1, 2, 𝑃𝑟 (Ω𝑗 ) ⊂ M𝑗,ℎ𝑐 , where 𝑃𝑟 (Ω𝑗 )
is a polynomial space of degree at most 𝑟.
(3) For 𝑗 = 1, 2, ℎ ∈ (0, 1], the integer 𝑘 ≥ 1,
and 𝑢 ∈ 𝐻𝑘 (Ω𝑗 ), there exists a positive constant 𝐶
independent of ℎ such that
inf ‖𝑢 − 𝑣‖𝐻𝑠 (Ω𝑗 ) ≤ 𝐶ℎ𝜎 ‖𝑢‖𝐻𝑘 (Ω𝑗 ) ,
𝑣∈M𝑗,ℎ𝑐
0 ≤ 𝑠 ≤ 1,
(19)
where 𝜎 = min(𝑟 + 1 − 𝑠, 𝑘 − 𝑠).
From the definition above, we note that functions 𝑣 in
Mℎc have a well-defined jump [𝑣] on Γ:
[𝑣] (x) = 𝑣 (x+ ) − 𝑣 (x− ) ,
∀x on Γ,
(20)
±
where 𝑣(x ) := lim𝜆 → 0± 𝑣(x + 𝜆𝜈Γ ).
To construct parallel algorithm, for a small constant 0 <
𝐻 < min{diam(Ω1 ), diam(Ω2 )}, we introduce an integral
mean value of a given function 𝑉 ∈ 𝐿2 (Ω) on the interdomain
boundary Γ as
𝑉𝐻 (x) =
1 𝐻
∫ 𝑉 (x+𝜆𝜈Γ ) 𝑑𝜆,
2𝐻 −𝐻
∀x on Γ.
(21)
Furthermore, we define the extrapolation of 𝑉𝐻(x) on Γ as
̂𝐻 (x) =
𝑉
4𝑉𝐻/2 (x) − 𝑉𝐻 (x)
,
3
∀x on Γ.
(22)
Generally, near the intersection of boundary 𝜕Ω and
inner-boundary Γ, the value of 𝑉 outside Ω is needed for 𝑉𝐻
̂𝐻. If x ∉ Ω, let x̃ ∈ Ω denote the symmetric point of x
and 𝑉
with respect to 𝜕Ω. For a given function 𝑢 ∈ 𝐿2 (Ω), we define
𝑢 (x) ,
𝐸𝑢 (x) = {
𝑢 (̃x) ,
if x ∈ Ω,
if x ∉ Ω.
(23)
̂𝐻 have the values on a strip
By (23), we know 𝑉𝐻 and 𝑉
domain 𝐺 = {y | y = x + 𝜆𝜈Γ , 𝜆 ∈ [−𝐻, 𝐻], ∀x on Γ}, see
Figure 2.
Now, we present the non-overlapping characteristic
DDM for the concentration equation:
̂𝑛−1
𝐶𝑛 − 𝐶
, 𝑣) + (D∇𝐶𝑛 , ∇𝑣) + (𝑞𝑛 𝐶𝑛 , 𝑣)
(𝜙
Δ𝑡𝑐
𝑛−1
+ (𝐶𝜇,𝐻, [𝑣]) + (𝑣𝜇,𝐻, [𝐶𝑛−1 ])Γ
Γ
+ 𝑑1 𝐾𝐻−1 ([𝐶𝑛−1 ] , [𝑣])Γ = (𝑞𝑛̃𝑐𝑛 , 𝑣) ,
∀𝑣 ∈ Mℎ𝑐 ,
(24)
where
𝑛
̂𝑛−1 (𝑥) = 𝐶𝑛−1 (𝑥 − 𝑈 (𝑥) Δ𝑡𝑐 ) ,
𝐶
𝜙 (𝑥)
𝐶𝜇𝑛−1
𝑛−1
= (D∇𝐸𝐶
(25)
) ⋅ 𝜈Γ .
The coefficient 𝐾 will be given in Lemma 3.
2.3. The Combined Approximate Schemes. We now present
our sequential time-stepping procedure that combines (24)
and (12). As in [4], let us part 𝐽 into pressure time steps 0 =
𝑡0 < 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑀 = 𝑇, with Δ𝑡𝑝𝑚 = 𝑡𝑚 − 𝑡𝑚−1 . Each pressure
step is also a concentration step; that is, for each 𝑚 there exists
𝑛 such that 𝑡𝑚 = 𝑡𝑛 ; in general, Δ𝑡𝑝 > Δ𝑡𝑐 . We may vary Δ𝑡𝑝 ,
but except for Δ𝑡𝑝1 we drop the superscript. For functions 𝑓
on Ω × 𝐽, we write 𝑓𝑚 (𝑥) for 𝑓(𝑥, 𝑡𝑚 ); thus, subscripts refer to
pressure steps and superscripts to concentration steps.
Advances in Numerical Analysis
5
ν1
νΓ
Ω1
Γ1
Γ
H
Ω2
ν2
Γ2
H
Figure 2: The strip domain 𝐺 with width 2𝐻.
If concentration step 𝑡𝑛 relates to pressure steps by 𝑡𝑚−1 <
𝑡 ≤ 𝑡𝑚 , we require a velocity approximation for (25) based on
𝑈𝑚−1 and earlier values. If 𝑚 ≥ 2, take the linear extrapolation
of 𝑈𝑚−1 and 𝑈𝑚−2 defined by
𝑛
𝐹𝑈𝑛 = (1 +
𝑡𝑛 − 𝑡𝑚−1
𝑡𝑛 − 𝑡𝑚−1
) 𝑈𝑚−1 −
𝑈 . (26)
𝑡𝑚−1 − 𝑡𝑚−2
𝑡𝑚−1 − 𝑡𝑚−2 𝑚−2
If 𝑚 = 1, set
𝐹𝑈𝑛 = 𝑈0 .
(27)
Δ𝑡𝑝1
𝑛
We retain the superscript on
because 𝐹𝑈 is first order
correct in time during the first pressure step and second order
during later steps.
The combined time-stepping procedure is finding a
map 𝐶 : {𝑡0 , 𝑡1 , . . . , 𝑡𝑁} → Mℎ𝑐 and a map (𝑈, 𝑃) :
{𝑡0 , 𝑡1 , . . . , 𝑡𝑀} → 𝑉ℎ𝑝 × 𝑊ℎ𝑝 such that
0
̃,
𝐶0 = 𝐶
(28a)
𝐴 (𝐶𝑚 ; 𝑈𝑚 , 𝑧) + 𝐵 (𝑧, 𝑃𝑚 ) = (𝛾 (𝐶𝑚 ) ∇𝑑, 𝑧) ,
𝐵 (𝑈𝑚 , 𝑤) = − (𝑞𝑚 , 𝑤) ,
(𝜙
𝐶𝑛 − 𝐶̌
Δ𝑡𝑐
𝑛−1
∀𝑧 ∈ 𝑉ℎ𝑝 ,
(28b)
∀𝑤 ∈ 𝑊ℎ𝑝 , 𝑚 ≥ 0,
(28c)
𝑛−1
, 𝑣) + (D∇𝐶𝑛 , 𝑣) + (𝑞𝑛 𝐶𝑛 , 𝑣) + (𝐶𝜇,𝐻, [𝑣])
Γ
+ (𝑣𝜇,𝐻, [𝐶𝑛−1 ])Γ + 𝑑1 𝐾𝐻−1 ([𝐶𝑛−1 ] , [𝑣])Γ
= (𝑞𝑛̃𝑐𝑛 , 𝑣) ,
∀𝑣 ∈ Mℎ𝑐 , 𝑛 ≥ 1,
(28d)
where
𝑛
𝑛−1
𝐹𝑈 (𝑥)
𝐶̌ (𝑥) = 𝐶𝑛−1 (𝑥)̌ = 𝐶𝑛−1 (𝑥 −
Δ𝑡𝑐 ) ,
𝜙 (𝑥)
̃0 is given by (45).
𝐶
(29)
The steps of the above calculation are as follows:
Step 1. 𝐶0 known → solve (𝑈0 , 𝑃0 ) by (28b) and (28c);
Step 2. On each domain, use (28d) to parallelly
𝑛1
compute {𝐶𝑗 }𝑗=1
until 𝑡𝑛1 = 𝑡1 ;
Step 3. Then by (28b) and (28c) for (𝑈1 , 𝑃1 );
Step 4. Calculate the approximations in turn analogically to get the pressure, velocity, and concentration
at other time-step, respectively.
The convergence analysis will make use of an analogue of
̂ defined for the exact velocity u𝑛 . If 𝑓 is a function on Ω, set
𝑥
𝐹u𝑛 (𝑥)
𝑓̌ (𝑥) = 𝑓 (𝑥)̌ = 𝑓 (𝑥 −
Δ𝑡𝑐 ) .
𝜙 (𝑥)
(30)
The time step 𝑡𝑛 will be clear from the context.
Remark 1. In the scheme (28a)–(28d), the numerical flux on Γ
is computed explicitly from 𝐶𝑛−1 , so that 𝐶𝑛 can be computed
on Ω1 and Ω2 fully parallel once 𝐶𝑛−1 has been got.
3. Auxiliary Lemmas
We adopt some auxiliary lemmas about the finite element
spaces, which will be used in the next section.
3.1. For the Pressure and Velocity. The Raviart-Thomas spaces
𝑉ℎ𝑝 × 𝑊ℎ𝑝 in Section 2.1 possess the following approximation
and inverse properties [7]:
󵄩
󵄩
󵄩 󵄩
inf 󵄩󵄩𝑓 − 𝑣󵄩󵄩󵄩 ≤ 𝐾3 ℎ𝑝 󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 ,
{
{
𝑣∈𝑉ℎ𝑝 󵄩
{
{
{
󵄩 󵄩
{ inf 󵄩󵄩𝑓 − 𝑣󵄩󵄩
󵄩󵄩𝐻(div) ≤ 𝐾3 ℎ𝑝 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐻1 (div) ,
󵄩󵄩
(𝐴 𝑝 ) : {𝑣∈𝑉
ℎ
𝑝
{
{
{
󵄩
󵄩
󵄩 󵄩
{
{ inf 󵄩󵄩󵄩𝑔 − 𝑤󵄩󵄩󵄩 ≤ 𝐾3 ℎ𝑝 󵄩󵄩󵄩𝑔󵄩󵄩󵄩1 ,
𝑤∈𝑊ℎ𝑝
{
(31)
∀𝑣 ∈ 𝑉ℎ𝑝 ,
‖𝑣‖𝐿∞ ≤ 𝐾3 ℎ𝑝−1 ‖𝑣‖ ,
(𝐼𝑝 ) : {
−1
‖𝑣‖𝑊∞1 (𝑇𝑝 ) ≤ 𝐾3 ℎ𝑝 ‖𝑣‖𝐿∞ (𝑇𝑝 ) , ∀𝑣 ∈ 𝑉ℎ𝑝 ,
where 𝐾3 is a positive constant independent of ℎ𝑝 and 𝑇𝑝 is
an element of the mesh Tℎp .
6
Advances in Numerical Analysis
̃
̃ ∈ Mℎ for 𝑐: ∀𝑣 ∈ M𝑗,ℎ ,
Define an elliptic projection 𝐶
c
c
𝑗 = 1, 2,
̃ 𝑃)
̃ : 𝐽 → 𝑉ℎ × 𝑊ℎ by (see [7])
Define the map (𝑈,
𝑝
𝑝
̃ 𝑣) + 𝐵 (𝑣, 𝑃)
̃ = (𝛾 (𝑐) ∇𝑑, 𝑣) ,
𝐴 (𝑐; 𝑈,
̃ 𝑤) = − (𝑞, 𝑤) ,
𝐵 (𝑈,
∀𝑣 ∈ 𝑉ℎ𝑝 ,
∀𝑤 ∈ 𝑊ℎ𝑝 ,
(32)
where 𝑐 is the exact solution of (1a)–(1d).
̃ 𝑃)
̃ exists and (𝐴 𝑝 )
By arguments in [1, 24], the map (𝑈,
implies that
󵄩󵄩󵄩
󵄩󵄩󵄩𝑢 − 𝑈
̃ 󵄩󵄩󵄩
̃ 󵄩󵄩󵄩󵄩 ∞
󵄩󵄩
󵄩𝐿 (𝐻(div)) + 󵄩󵄩𝑝 − 𝑃󵄩󵄩𝐿∞ (𝐿2 )
̃
̃
̃ (𝑡) , 𝑣)
̃ (𝑡) , ∇𝑣) + (𝑞 (𝑡) 𝐶
(D∇𝐶
= ((D∇𝑐 (𝑡) , ∇𝑣) + (𝑞 (𝑡) 𝑐 (𝑡) , 𝑣))
= (𝑞 (𝑡) 𝑐 (𝑡)−𝜙
𝜕𝑐
(𝑡)−u (𝑡) ⋅ ∇𝑐 (𝑡) , 𝑣)+(−1)𝑗+1 (𝑔, 𝑣)Γ ,
𝜕𝑡
(39)
(33)
̃
̃ From [25–28], we see
where 𝑔 = (D∇𝑐) ⋅ 𝜈Γ . Let 𝜂 = 𝑐 − 𝐶.
the following.
The positive constant 𝐾4 depends on constant in (𝑄) but
independent of ℎ𝑝 , 𝑢, 𝑝, and 𝑐.
The estimate (33) and (𝐼𝑝 ) imply that
Lemma 4. For 𝜂 defined by (39), there holds the following.
𝐿2 -norm error estimate
󵄩 󵄩
󵄩 󵄩
max 󵄩󵄩𝜂󵄩󵄩 2 + 󵄩󵄩󵄩𝜂t 󵄩󵄩󵄩𝐿2 (0,𝑇;𝐿2 (Ω))
0≤t≤T󵄩 󵄩𝐿 (Ω)
(40)
󵄩
󵄩
𝑟+1
󵄩
󵄩
≤ Cℎc {‖𝑢‖L∞ (0,𝑇;𝐻𝑟+1 (Ω)) + 󵄩󵄩𝑢t 󵄩󵄩L2 (0,𝑇;𝐻𝑟+1 (Ω)) } ,
󵄩
󵄩
≤ 𝐾4 { inf ‖𝑢 − 𝑣‖𝐻(div) + inf 󵄩󵄩󵄩𝑝 − 𝑤󵄩󵄩󵄩}
𝑣∈𝑉ℎ
𝑤∈𝑊ℎ
𝑝
𝑝
󵄩 󵄩
≤ 𝐾4 (‖𝑢‖𝐿∞ (𝐻1 (div)) , 󵄩󵄩󵄩𝑝󵄩󵄩󵄩𝐿∞ (𝐻1 ) ) ℎ𝑝 .
󵄩󵄩 ̃ 󵄩󵄩
󵄩󵄩𝑈󵄩󵄩 ∞ ∞ ≤ 𝐾5 .
󵄩 󵄩𝐿 (𝐿 )
(34)
As in [1], comparison of (32) and (12) implies that
󵄩󵄩
󵄩󵄩
󵄩󵄩 ̃ 󵄩󵄩
̃ 󵄩󵄩󵄩󵄩
̃ 󵄩󵄩󵄩
󵄩󵄩𝑈 − 𝑈
󵄩
󵄩 󵄩
󵄩𝐻(div) + 󵄩󵄩𝑃 − 𝑃󵄩󵄩 ≤ 𝐾6 (1 + 󵄩󵄩𝑈󵄩󵄩𝐿∞ ) ‖𝑐 − 𝐶‖ . (35)
󵄩
The estimates (33) and (35) will handle the coupling of
concentration and velocity errors in the convergence analysis.
3.2. For the Concentration. In this section, we adopt some
following lemmas [18].
Lemma 2. For smooth enough function 𝑊, there holds estimates:
󵄩
󵄩󵄩
󵄩󵄩𝑊𝐻 − 𝑊󵄩󵄩󵄩 2 ≤ √2𝐻‖∇𝑊‖𝐿2 (Ω) ,
󵄩𝐿 (Γ)
󵄩
2
󵄩󵄩󵄩𝑊 − 𝑊󵄩󵄩󵄩
󵄩󵄩 𝐻
󵄩󵄩𝐿∞ (Γ) ≤ 𝐶𝐻 ‖𝑊‖𝑊2,∞ (Ω) ,
𝑊 (x) − 𝑊𝐻 (x)
1 4
1
= − 𝐻2 𝑊𝜈Γ2 (x) −
𝐻 𝑊𝜈Γ4 (x) + 𝑜 (𝐻6 ) ,
6
120
if 𝑟 = 1, 𝑑 = 2,
󵄩 󵄩
≤ 𝐶ℎ𝑐𝑟+1 {‖𝑢‖𝑊𝑟+1,∞ (Ω) + 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝑊𝑟+1,∞ (Ω) } , if 𝑟 > 1.
For functions 𝜓 with restrictions in 𝐻1 (Ω1 ) ∪ 𝐻1 (Ω2 ), we
define a norm
󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨2 = (𝐷∇𝜓, ∇𝜓) + 𝑑1 𝐾𝐻−1 ([𝜓] , [𝜓]) .
󵄨󵄨󵄨 󵄨󵄨󵄨
Γ
Lemma 3. Let 𝐺 = {y | y = x + 𝑡𝜈Γ , 𝑡 ∈ [−𝐻, 𝐻], ∀x 𝑜𝑛 Γ}.
If 𝜓 ∈ 𝐻1 (Ω) and 𝐻 > 0 is small, one has
(37)
(42)
Lemma 5. There exists a positive constant 𝐶0 = 1 − (√2/2)
such that for small 𝐻 > 0,
∀x on Γ,
(36)
(41)
󵄩󵄩 󵄩󵄩
󵄩 󵄩
󵄩󵄩𝜂󵄩󵄩𝐿∞ (Ω) + 󵄩󵄩󵄩𝜂t 󵄩󵄩󵄩𝐿∞ (Ω)
󵄨󵄨󵄨 󵄨󵄨󵄨2
𝐶0 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ (D∇𝜓, ∇𝜓) + d1 KH−1 ([𝜓] , [𝜓])Γ
where 𝑊𝜈Γ2 and 𝑊𝜈Γ4 are the second and fourth normal derivative of 𝑊 on Γ, respectively.
󵄩󵄩 󵄩󵄩
󵄩 󵄩
󵄩󵄩𝐸𝜓󵄩󵄩𝐿2 (𝐺) ≤ √𝐾󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐿2 (Ω) ,
󵄩󵄩
󵄩
󵄩 󵄩
󵄩󵄩∇ (𝐸𝜓) ⋅ 𝜈Γ 󵄩󵄩󵄩𝐿2 (𝐺) ≤ √𝐾󵄩󵄩󵄩∇𝜓󵄩󵄩󵄩𝐿2 (Ω) ,
𝐿∞ -norm error estimate
󵄩󵄩󵄩𝜂󵄩󵄩󵄩 ∞ + 󵄩󵄩󵄩𝜂t 󵄩󵄩󵄩 ∞
󵄩 󵄩𝐿 (Ω) 󵄩 󵄩L (Ω)
󵄨
󵄨
󵄩 󵄩
≤ Cℎc2 󵄨󵄨󵄨ln ℎc 󵄨󵄨󵄨 {‖𝑢‖W2,∞ (Ω) + 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩W2,∞ (Ω) } ,
+ 2(𝜓𝜇,H , [𝜓]) ,
Γ
∀𝜓 ∈ Mℎ𝑐 .
(43)
As we have shown, the combined approximation scheme
(28d) includes two terms on the interdomain boundary Γ
by integral mean method to present explicit flux calculation. These terms are distinct ones different from DawsonDupont’s schemes such that the standard elliptic projection
(39) is insufficient for optimal error estimates. To get optimal
error estimates, we need a new elliptic projection including
̃ ∈ Mℎ of the
terms on Γ. This new elliptic projection 𝐶
𝑐
solution 𝑐 is defined as
̃ , ∇𝑣) + ((𝑐 − 𝐶)
̃
̃
(D∇ (𝑐 − 𝐶)
, [𝑣]) + (𝑣𝜇,𝐻, [𝑐 − 𝐶])
𝜇,H
Γ
where
Γ
𝐾={
1, if 𝐺 ⊂ Ω,
2, if 𝐺 ⊄ Ω.
̃ , [𝑣]) = 0,
+ 𝑑1 𝐾𝐻 ([𝑐 − 𝐶]
Γ
−1
(38)
∀𝑣 ∈ Mℎ𝑐 .
(44)
Advances in Numerical Analysis
7
𝑛−1
0
̃.
𝐶0 = 𝐶
Γ
̃
+ 𝑑1 𝐾𝐻−1 ([𝐶
𝑛
𝑛
Γ
𝑛
̃ ] , [𝑣])
−𝐶
Γ
+ (𝑞̃𝑐𝑛 , 𝑣) − (𝑞𝑛 𝑐𝑛 , 𝑣) − (u ⋅ ∇𝑐 + 𝜙
̃ .
𝜃𝑛 = 𝐶𝑛 − 𝐶
𝜕𝑐𝑛
, 𝑣)
𝜕𝑡
̃𝑛−1
̃𝑛 − 𝐶
𝐶
, 𝑣) + (𝑐𝑛𝜇,𝐻 − 𝑐𝜇𝑛 , [𝑣]) .
+ (𝜙
Γ
Δ𝑡𝑐
(46)
The following lemma gives the bounds of 𝜉𝑛 .
(52)
Lemma 6. There holds a priori estimates:
󵄩󵄩 󵄩󵄩
𝑟+1
1/2 󵄩 󵄩
󵄩󵄩𝜉󵄩󵄩𝐿2 (Ω) ≤ 𝐶 {ℎ𝑐 + 𝐻 󵄩󵄩󵄩𝜂󵄩󵄩󵄩𝐿∞ (Ω) } ,
󵄩󵄩 󵄩󵄩
r+1
1/2 󵄩 󵄩
󵄩󵄩𝜉t 󵄩󵄩𝐿2 (Ω) ≤ C {ℎc + 𝐻 󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩𝐿∞ (Ω) } .
(47)
Subtracting (52) from (28d) and taking 𝑣 = 𝜃𝑛 , we adopted
the skill of [4] to obtain
(48)
(𝜙
4. A Priori Error Estimates in 𝐿2 -Norm
𝜃𝑛 − 𝜃𝑛−1 𝑛
, 𝜃 ) + (D∇𝜃𝑛 , ∇𝜃𝑛 )
Δ𝑡𝑐
𝑛
+ 𝑑1 𝐾𝐻−1 ([𝜃𝑛 ] , [𝜃𝑛 ])Γ + 2(𝜃𝜇,𝐻, [𝜃𝑛 ])
Now, we turn to derive an optimal priori error estimate in 𝐿2 norm for the concentration of approximation (28d). Optimal
error estimates for velocity in 𝐻(div; Ω) and pressure in
𝐿2 (Ω) follow at once from (33) and (35).
It follows from trace theorem in [29] that
󵄩 󵄩󵄩 󵄩
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝜓󵄩󵄩𝐿2 (Γ) ≤ 𝐶2 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 󵄩󵄩󵄩𝜓󵄩󵄩󵄩1 ,
𝑛−1
(45)
Let
̃ ,
𝜉𝑛 = 𝑐𝑛 − 𝐶
𝑛
̃ −𝐶
̃ , [𝑣]) + (𝑣𝜇,𝐻, [𝐶
̃𝑛−1 − 𝐶
̃𝑛 ])
= (𝐶
𝜇,𝐻
𝜇,𝐻
It follows from Lemma 5 that the projection problem (44) has
̃0 to be the
a unique solution for small 𝐻. Choose the initial 𝐶
projections of 𝑐0 (𝑥) by (44) and take the initial conditions
∀𝜓 ∈ 𝐻1 (Ω)
= (𝜙
̌
𝜕𝑐𝑛
𝑐𝑛 − 𝑐𝑛−1
, 𝜃𝑛 )
+ 𝐹u𝑛 ⋅ ∇𝑐𝑛 − 𝜙
𝜕𝑡
Δ𝑡𝑐
+ ((u𝑛 − 𝐹u𝑛 ) ⋅ ∇𝑐𝑛 , 𝜃𝑛 ) + (𝜙
(49)
Γ
𝑛
𝑛−1
𝜉𝑛 − 𝜉𝑛−1 𝑛
,𝜃 )
Δ𝑡𝑐
𝑛
+ {(𝜃𝜇,𝐻 − 𝜃𝜇,𝐻, [𝜃𝑛 ]) + (𝜃𝜇,𝐻, [𝜃𝑛 − 𝜃𝑛−1 ])
which will be used in the following proof. Then, we can derive
an 𝐿2 (Ω)-norm error estimate for 𝜃𝑛 .
Γ
Γ
+ 𝑑1 𝐾𝐻−1 ([𝜃𝑛 − 𝜃𝑛−1 ] , [𝜃𝑛 ])Γ
𝑛
Lemma 7. For 𝜃 defined by (46), there holds the following
error estimate:
𝑛
+ (D∇ (𝜉𝑛 − 𝜉𝑛−1 ) , ∇𝜃𝑛 ) + (𝑐𝑛𝜇,𝐻 − 𝑐𝑛−1
𝜇,𝐻 , [𝜃 ])
Γ
󵄩 󵄩2
max 󵄩󵄩󵄩𝜃n 󵄩󵄩󵄩
+(𝑐𝜇𝑛 − 𝑐𝑛𝜇,𝐻, [𝜃𝑛 ]) }
0≤𝑛≤𝑁
Γ
󵄩 󵄩2
󵄩 󵄩2
󵄩 󵄩2
≤ 𝐶 {𝐻 [ max (󵄩󵄩󵄩𝜂𝑛 󵄩󵄩󵄩𝐿∞ (Ω) + 󵄩󵄩󵄩𝜂𝑛 󵄩󵄩󵄩 ) + 󵄩󵄩󵄩𝜂t 󵄩󵄩󵄩𝐿2 (0,𝑇;L∞ (Ω))
1≤𝑛≤𝑁
2
󵄩 󵄩2
+󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩𝐿2 (0,𝑇;𝐿2 (Ω)) ] + (Δtc ) + 𝐻5 + ℎp2l+2
3
(50)
provided
+ (𝜙
C1 =
𝑑0 (1 − 𝛿)2 𝐶03
,
16𝑑12 K2 𝐶22
𝑛−1
̃
]) + 𝑑1 𝐾𝐻−1 ([𝐶
Γ
(53)
Since
𝑛
𝑛
𝑛−1
(𝜃𝜇,𝐻, [𝜃𝑛 − 𝜃𝑛−1 ]) = (𝜃𝜇,𝐻, [𝜃𝑛 ]) − (𝜃𝜇,𝐻, [𝜃𝑛−1 ])
𝑛−1
̃𝑛 − 𝐶
̃𝑛−1
𝐶
̃ , [𝑣])
̃𝑛 , ∇𝑣) + (𝐶
(𝜙
, 𝑣) + (D∇𝐶
𝜇,𝐻
Δ𝑡𝑐
Γ
̃
+ (𝑣𝜇,𝐻, [𝐶
𝑛−1
𝜃̌ − 𝜃𝑛−1 𝑛
, 𝜃 ) + (𝑞𝑛 (𝜉𝑛 − 𝜃𝑛 ) , 𝜃𝑛 ) .
Δ𝑡𝑐
∀0 < 𝛿 ≪ 1. (51)
Proof. Combining (39) and (44), we have ∀𝑣 ∈ M,
𝑛−1
̂𝜉
̌
̂𝑐𝑛−1 − 𝑐𝑛−1
, 𝜃𝑛 ) − (𝜙
Δ𝑡𝑐
𝑛−1
𝑛−1
𝑛−1
𝜉̌ − 𝜉𝑛−1 𝑛
𝜃̂ − 𝜃̌
𝑛
, 𝜃 ) − (𝜙
,𝜃 )
+ (𝜙
Δ𝑡𝑐
Δ𝑡𝑐
4
+ℎc2r+2 + (Δ𝑡𝑝1 ) + (Δ𝑡𝑝 ) } ,
Δ𝑡 ≤ 𝐶1 𝐻2 ,
𝑛−1
− 𝜉̌
, 𝜃𝑛 )
Δ𝑡𝑐
𝑛−1
+ (𝜙
Γ
Γ
𝑛
] , [𝑣])
Γ
Γ
𝑛−1
− (𝜃𝜇,𝐻 − 𝜃𝜇,𝐻, [𝜃𝑛−1 ]) ,
Γ
(54)
8
Advances in Numerical Analysis
by summing (53) over 𝑛 we have
Furthermore, noting that
𝑛
𝑑𝐾
󵄩
󵄩 󵄩2
󵄩2
2(𝜃𝜇,𝐻, [𝜃𝑛 ]) ≤ 𝜀󵄩󵄩󵄩𝜃𝑛 󵄩󵄩󵄩1,𝐷 + 1 𝐻−1 󵄩󵄩󵄩[𝜃𝑛 ]󵄩󵄩󵄩𝐿2 (Γ) ,
Γ
2𝜀
1
(𝜙𝜃𝑛 , 𝜃𝑛 )
2
where ‖𝑣‖21,𝐷 = (D∇𝑣, ∇𝑣), and taking 0 < 𝜀 = √2/2 < 1, we
have
1 󵄩󵄩 𝑛 󵄩󵄩2 Δ𝑡𝑐
󵄩
󵄩2
(1 − 𝐶0 ) ((D∇𝜃𝑛 , ∇𝜃𝑛 ) + 𝑑1 𝐾𝐻−1 󵄩󵄩󵄩[𝜃𝑛 ]󵄩󵄩󵄩𝐿2 (Γ) )
󵄩𝜃 󵄩 +
2󵄩 󵄩
2
+ Δ𝑡𝑐 [ (D∇𝜃𝑛 , ∇𝜃𝑛 ) + 𝑑1 𝐾𝐻−1 ([𝜃𝑛 ] , [𝜃𝑛 ])Γ
𝑛
+ (𝜃𝜇,𝐻, [𝜃𝑛 ]) ]
Γ
𝑛−1
+ Δ𝑡𝑐 ∑ [ (D∇𝜃𝑘 , ∇𝜃𝑘 ) + 𝑑1 𝐾𝐻−1 ([𝜃𝑘 ] , [𝜃𝑘 ])Γ
𝑘=1
𝑛
+ Δ𝑡𝑐 ∑ [
𝑘=1
𝑘
+2(𝜃𝜇,𝐻, [𝜃𝑘 ])
1
] = (𝜙𝜃0 , 𝜃0 )
2
Γ
󵄩
󵄩2
+𝑑1 𝐾𝐻−1 󵄩󵄩󵄩󵄩[𝜃𝑘 ]󵄩󵄩󵄩󵄩𝐿2 (Γ) ) ]
0
Γ
𝑘
󵄨󵄨 0
󵄨󵄨
1 󵄩 󵄩2
≤ 󵄩󵄩󵄩󵄩𝜃0 󵄩󵄩󵄩󵄩 + Δ𝑡𝑐 󵄨󵄨󵄨(𝜃𝜇,𝐻, [𝜃0 ]) 󵄨󵄨󵄨
󵄨
Γ󵄨
2
𝑘−1
+ Δ𝑡𝑐 ∑ {(𝜃𝜇,𝐻 − 𝜃𝜇,𝐻, [𝜃𝑘 − 𝜃𝑘−1 ])
Γ
𝑘=1
𝑛
𝑘
𝑘=1
𝑘
+ (𝑐𝑘𝜇,𝐻 − 𝑐𝑘−1
𝜇,𝐻 , [𝜃 ])
+ (D∇ (𝜉𝑘 − 𝜉𝑘−1 ) , ∇𝜃𝑘 )
𝑘
+ (u𝑘𝜇,𝐻 − u𝑘−1
𝜇,𝐻 , [𝜃 ])
Γ
𝑘
𝑘
+(u𝑘𝜇
𝑘
− ∑ (𝑞 (𝜉 − 𝜃 ) , 𝜃 ) Δ𝑡𝑐
𝑘=1
̌
𝜕𝑐𝑛
𝑐𝑛 − 𝑐𝑛−1
+ ∑ ([𝜙
, 𝜃𝑘 ) Δ𝑡𝑐
+ 𝐹u𝑛 ⋅ ∇𝑐𝑛 ] − 𝜙
𝜕𝑡
Δ𝑡
𝑐
𝑘=1
𝑛
𝑘
𝑘
𝑘
𝑘
𝑛
𝑛
󵄨
󵄨
󵄨
󵄨
+ [ ∑ 󵄨󵄨󵄨󵄨(𝑞𝑘 𝜃𝑘 , 𝜃𝑘 )󵄨󵄨󵄨󵄨 Δ𝑡𝑐 + ∑ 󵄨󵄨󵄨󵄨(𝑞𝑘 𝜉𝑘 , 𝜃𝑘 )󵄨󵄨󵄨󵄨 Δ𝑡𝑐 ]
𝑘=1
+ ∑ ([𝜙
𝑘=1
𝜉𝑘 − 𝜉𝑘−1 𝑘
+ ∑ (𝜙
, 𝜃 ) Δ𝑡𝑐
Δ𝑡𝑐
𝑘=1
+ ∑ (𝜙
̌
̂𝑐𝑛−1 − 𝑐𝑛−1
+ ∑ (𝜙
, 𝜃𝑛 ) Δ𝑡𝑐
Δ𝑡
𝑐
𝑘=1
+ ∑ (𝜙
𝑛
𝜉𝑘 − 𝜉𝑘−1 𝑘
, 𝜃 ) Δ𝑡𝑐
Δ𝑡𝑐
𝑘=1
𝑛
𝑛
𝑘=1
̂𝜉𝑘−1 − 𝜉̌𝑘−1 𝑘
− ∑ (𝜙
, 𝜃 ) Δ𝑡𝑐
Δ𝑡𝑐
𝑘=1
̌
̂𝑐𝑘−1 − 𝑐𝑘−1
, 𝜃𝑘 ) Δ𝑡𝑐
Δ𝑡𝑐
󵄨󵄨
󵄨
󵄨󵄨 ̂𝜉𝑘−1 − 𝜉̌𝑘−1 𝑘 󵄨󵄨󵄨
󵄨
+ ∑ 󵄨󵄨󵄨(𝜙
, 𝜃 )󵄨󵄨󵄨󵄨 Δ𝑡𝑐
Δ𝑡
󵄨
󵄨󵄨
𝑐
󵄨
𝑘=1 󵄨
󵄨
𝑛
𝑛
𝑘−1
𝑘−1
𝜃̂ − 𝜃̌
+ ∑ (𝜙
, 𝜃𝑘 ) Δ𝑡𝑐
Δ𝑡𝑐
𝑘=1
𝑛
𝑘=1
̌
𝜕𝑐
𝑐𝑘 − 𝑐𝑘−1
, 𝜃𝑘 ) Δ𝑡𝑐
+ 𝐹u𝑘 ⋅ ∇𝑐𝑘 ] − 𝜙
𝜕𝑡
Δ𝑡𝑐
𝑘
𝑘=1
𝑛
− ∑ (𝜙
𝑘=1
𝑛
+ ∑ ((u𝑘 − 𝐹u𝑘 ) ⋅ ∇𝑐𝑘 , 𝜃𝑘 ) Δ𝑡𝑐
𝑘=1
𝜉̌
−
Γ
u𝑘𝜇,𝐻, [𝜃𝑘 ]) ]
Γ
𝑛
+ ∑ ((u − 𝐹u ) ⋅ ∇𝑐 , 𝜃 ) Δ𝑡𝑐
𝑛
𝑑1 𝐾 𝑘
([𝜃 − 𝜃𝑘−1 ] , [𝜃𝑘 ])Γ
𝐻
+
+(𝑐𝜇𝑘 − 𝑐𝑘𝜇,𝐻, [𝜃𝑘 ]) }
𝑛
Γ
+ (D∇ (𝜉𝑘 − 𝜉𝑘−1 ) , ∇𝜃𝑘 )
Γ
𝑘
𝑘−1
+ Δ𝑡𝑐 ∑ [(𝜃𝜇,𝐻 − 𝜃𝜇,𝐻, [𝜃𝑘 − 𝜃𝑘−1 ])
𝑑𝐾
+ 1 ([𝜃𝑘 − 𝜃𝑘−1 ] , [𝜃𝑘 ])Γ
𝐻
𝑛
Δ𝑡𝑐 󵄩󵄩 𝑘 󵄩󵄩2
󵄩󵄩𝜕 𝜃 󵄩󵄩
2 󵄩 𝑡 󵄩
+ 𝐶0 ( (D∇𝜃𝑘 , ∇𝜃𝑘 )
− Δ𝑡𝑐 (𝜃𝜇,𝐻, [𝜃0 ])
𝑛
(56)
𝑛
+ ∑ (𝜙
𝑘=1
𝑘−1
𝑘−1
𝑘−1
𝜃̂ − 𝜃̌
, 𝜃𝑘 ) Δ𝑡𝑐
Δ𝑡𝑐
𝑘−1
𝜉̌ − 𝜉𝑘−1 𝑘
+ ∑ (𝜙
, 𝜃 ) Δ𝑡𝑐
Δ𝑡𝑐
𝑘=1
𝑛
− 𝜉𝑘−1 𝑘
, 𝜃 ) Δ𝑡𝑐
Δ𝑡𝑐
𝑘−1
𝜃̌ − 𝜃𝑘−1 𝑘
+ ∑ (𝜙
, 𝜃 ) Δ𝑡𝑐 .
Δ𝑡𝑐
𝑘=1
𝑛
𝑛
+ ∑ (𝜙
𝑘=1
(55)
𝑘−1
𝜃̌ − 𝜃𝑘−1 𝑘
, 𝜃 ) Δ𝑡𝑐 .
Δ𝑡𝑐
(57)
Advances in Numerical Analysis
9
We estimate the terms on the right-hand side of (57) one by
one. We denote the terms as 𝑇1 , 𝑇2 , 𝑇3 , . . . , 𝑇10 from the third
term on the right-hand side of (57). We turn to analyze them
one by one.
For the term 𝑇1 , similarly as that of [18], by taking
Δ𝑡𝑐 ≤
𝑑0 (1 − 𝛿)2 𝐶03 𝐻2
,
16𝑑12 𝐾2 𝐶22
∀0 < 𝛿 ≪ 1,
Combining the above analyses, we have
1 󵄩󵄩 𝑛 󵄩󵄩2 Δ𝑡𝑐
󵄩
󵄩 󵄩2
󵄩2
(1 − 𝐶0 ) (󵄩󵄩󵄩𝜃𝑛 󵄩󵄩󵄩1,𝐷 + 𝑑1 𝐾𝐻−1 󵄩󵄩󵄩[𝜃𝑛 ]󵄩󵄩󵄩𝐿2 (Γ) )
󵄩𝜃 󵄩 +
2󵄩 󵄩
2
󵄨󵄨 0
󵄨󵄨
1 󵄩 󵄩2
≤ 󵄩󵄩󵄩󵄩𝜃0 󵄩󵄩󵄩󵄩 + Δ𝑡𝑐 󵄨󵄨󵄨(𝜃𝜇,𝐻, [𝜃0 ]) 󵄨󵄨󵄨
󵄨
Γ󵄨
2
(58)
𝑛
󵄩2
󵄩 󵄩2
󵄩 󵄩2 󵄩
+𝐶Δ𝑡𝑐 ∑ [Δ𝑡𝑐 (󵄩󵄩󵄩󵄩𝜃𝑘 󵄩󵄩󵄩󵄩1,𝐷 + 󵄩󵄩󵄩󵄩𝜕𝑡 𝜉𝑘 󵄩󵄩󵄩󵄩1,𝐷)+𝐻5 󵄩󵄩󵄩󵄩u𝑘 󵄩󵄩󵄩󵄩𝐻2,∞ (Ω)]
𝑘=1
we can obtain
3
2
4
+ 𝐶 (ℎ𝑝2𝑙+2 + ℎ𝑐2𝑟+2 + (Δ𝑡𝑐 ) + (Δ𝑡𝑝1 ) + (Δ𝑡𝑝 ) ) .
2
𝑛
󵄩
󵄩2
󵄨󵄨 󵄨󵄨 (Δ𝑡𝑐 )
∑ 󵄩󵄩󵄩󵄩𝜕𝑡 𝜃𝑘 󵄩󵄩󵄩󵄩
󵄨󵄨𝑇1 󵄨󵄨 ≤
2 𝑘=1
(61)
Taking 𝜃0 = 0, applying the Gronwall Lemma and Lemmas 5
and 6, we can derive (50).
𝑛
󵄩 󵄩2
󵄩 󵄩2
+ 𝐶0 Δ𝑡𝑐 ∑ [󵄩󵄩󵄩󵄩𝜃𝑘 󵄩󵄩󵄩󵄩1,𝐷 + 𝑑0 󵄩󵄩󵄩󵄩𝜃𝑘 󵄩󵄩󵄩󵄩 ]
𝑘=1
Applying the triangle inequality, Lemmas 4 and 7, we
have the following error theore.
𝑛
󵄩 󵄩2
+ (1 − 𝛿) 𝐶0 Δ𝑡𝑐 ∑ 󵄩󵄩󵄩󵄩𝜃𝑘 󵄩󵄩󵄩󵄩1,𝐷
𝑘=1
𝑑𝐾 𝑛 󵄩
󵄩2
+ (1 − 𝛿) 𝐶0 Δ𝑡𝑐 1 ∑ 󵄩󵄩󵄩󵄩[𝜃𝑘 ]󵄩󵄩󵄩󵄩𝐿2 (Γ)
𝐻 𝑘=1
(59)
Theorem 8. Suppose that the assumptions (𝑃), (𝑄), (𝐼), (𝐴 𝑝 ),
(𝐼𝑝 ) and 𝑟 ≥ 1, 𝑙 ≥ 0 hold. Assume that the discretization
parameters obey the relations
Δ𝑡 󵄩 󵄩2
󵄩2
󵄩
+ Δ𝑡𝑐 ∑ [ 𝑐 (󵄩󵄩󵄩󵄩𝜃𝑘 󵄩󵄩󵄩󵄩1,𝐷 + 󵄩󵄩󵄩󵄩𝜕𝑡 𝜉𝑘 󵄩󵄩󵄩󵄩1,𝐷)
2
𝑘=1
3/2
(Δ𝑡𝑝1 )
󵄩
󵄩2
+ 𝐶󵄩󵄩󵄩󵄩𝑐𝑘 − 𝑐𝑘−1 󵄩󵄩󵄩󵄩1,𝐷
= O (ℎ𝑝 ) ,
(62)
2
(Δ𝑡𝑝 ) = O (ℎ𝑝 ) .
Then for ℎ𝑐 , ℎ𝑝 , and Δ𝑡𝑐 sufficiently small, the errors of the
approximation (28a)–(28d) for (1a)–(1d) satisfy
󵄩
󵄩2
+ 𝛿𝐶0 𝑑1 𝐾𝐻−1 󵄩󵄩󵄩󵄩[𝜃𝑘 ]󵄩󵄩󵄩󵄩𝐿2 (Γ)
2
3/2
󵄩
󵄩
max 󵄩󵄩󵄩𝑐𝑛 − 𝐶𝑛 󵄩󵄩󵄩 ≤ 𝐶 {Δ𝑡𝑐 + (Δ𝑡𝑝 ) + (Δ𝑡𝑝1 ) + ℎ𝑐𝑟+1
0≤𝑛≤𝑁
󵄩 󵄩2
+𝐶𝐻5 󵄩󵄩󵄩󵄩𝑐𝑘 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) ] .
+ℎ𝑝𝑙+1
The analyses of the terms 𝑇2 , . . . , 𝑇10 are similar to that of [4]
+𝐻
5/2
(63)
},
provided
𝑛
󵄩 𝑘 󵄩2 󵄩 𝑘 󵄩2
󵄨󵄨 󵄨󵄨
󵄨󵄨𝑇2 󵄨󵄨 ≤ 𝐶Δ𝑡𝑐 ∑ (󵄩󵄩󵄩󵄩𝜃 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝜉 󵄩󵄩󵄩󵄩 ) ,
Δ𝑡𝑐 ≤ 𝐶1 𝐻2 ,
𝑘=1
󵄩󵄩 2 󵄩󵄩
󵄩𝜕 𝑐󵄩
󵄩 󵄩2
󵄨󵄨 󵄨󵄨
Δ𝑡𝑐 + 󵄩󵄩󵄩󵄩𝜃𝑘 󵄩󵄩󵄩󵄩 ) ,
󵄨󵄨𝑇3 󵄨󵄨 ≤ 𝐶Δ𝑡𝑐 ∑ (󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
󵄩 𝜕𝜏 󵄩󵄩𝐿2 (𝑡𝑘−1 ,𝑡𝑘 ;𝐿2 )
𝑘=1 󵄩
󵄩 2 󵄩󵄩
𝑛
3󵄩
󵄩𝜕 u󵄩
󵄩 󵄩2
󵄨󵄨 󵄨󵄨
+ 𝐶Δ𝑡𝑐 ∑ 󵄩󵄩󵄩󵄩𝜃𝑘 󵄩󵄩󵄩󵄩 ,
󵄨󵄨𝑇4 󵄨󵄨 ≤ 𝐶Δ𝑡𝑐 (Δ𝑡𝑝 ) 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
󵄩󵄩 𝜕𝑡 󵄩󵄩𝐿2 (𝑡0 ,𝑡𝑛 ;𝐿2 )
𝑘=1
𝑛
(64)
where 𝐶1 is given by (51). Here 𝐶 is a positive constant
dependent on (𝑃), (𝑄), (𝐼), (𝐴 𝑝 ), (𝐼𝑝 ), ‖𝜕2 𝑐/𝜕𝜏2 ‖, ‖𝜕u/𝜕𝑡‖, and
‖𝜕2 u/𝜕𝑡2 ‖, but independent of ℎ𝑐 , ℎ𝑝 , Δ𝑡𝑐 , and Δ𝑡𝑝 .
By combining Theorem 8 with (33) and (35), we obtain at
once the following result.
𝑛
󵄩 𝑘 󵄩2
󵄨󵄨 󵄨󵄨
2𝑟+2
2
󵄨󵄨𝑇5 󵄨󵄨 ≤ 𝐶Δ𝑡𝑐 ∑ (ℎ𝑐 ‖𝑐‖𝐻1 (𝑡𝑘−1 ,𝑡𝑘 ;𝐻𝑟+1 ) + 󵄩󵄩󵄩󵄩𝜃 󵄩󵄩󵄩󵄩 ) ,
Corollary 9. Under the assumption of Theorem 8, the errors
for velocity u, and pressure 𝑝 are obtained by
𝑘=1
2
󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨
2𝑙+2
2𝑟+2
1 3
󵄨󵄨𝑇6 󵄨󵄨 + 󵄨󵄨𝑇7 󵄨󵄨 + 󵄨󵄨𝑇8 󵄨󵄨 ≤ 𝐶Δ𝑡𝑐 (ℎ𝑝 + ℎ𝑐 + (Δ𝑡𝑐 ) (Δ𝑡𝑝 )
󵄩
󵄩
󵄩
󵄩
max (󵄩󵄩󵄩u𝑚 − 𝑈𝑚 󵄩󵄩󵄩𝐻(div;Ω) + 󵄩󵄩󵄩𝑝𝑚 − 𝑃𝑚 󵄩󵄩󵄩)
0≤𝑚≤𝑀
𝑛
4
󵄩 󵄩2
+(Δ𝑡𝑝 ) ) + 𝜖Δ𝑡𝑐 ∑ 󵄩󵄩󵄩󵄩𝜃𝑘 󵄩󵄩󵄩󵄩1 ,
≤ 𝐶 {Δ𝑡𝑐 +(Δ𝑡𝑝 )
𝑘=1
𝑛
𝑛
󵄩 𝑘−1 󵄩2
󵄩 𝑘 󵄩2
󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨
2𝑟+2
󵄨󵄨𝑇9 󵄨󵄨 + 󵄨󵄨𝑇10 󵄨󵄨 ≤ 𝐶Δ𝑡𝑐 (ℎ𝑐 + ∑ 󵄩󵄩󵄩󵄩𝜃 󵄩󵄩󵄩󵄩 ) + 𝜖Δ𝑡𝑐 ∑ 󵄩󵄩󵄩󵄩𝜃 󵄩󵄩󵄩󵄩1 .
𝑘=1
ℎ𝑐𝑟+1 = O (ℎp ) ,
Δ𝑡c = o (ℎp ) ,
𝑛
𝑘=1
(60)
2
3/2
+(Δ𝑡𝑝1 ) +ℎ𝑐𝑟+1 +ℎ𝑝𝑙+1
5/2
(65)
+ 𝐻 }.
Remark 10. As for the accuracy order of ℎc and ℎp , we know
that (63) and (65) are optimal for the concentration 𝑐, the
velocity u and the pressure 𝑝.
10
Advances in Numerical Analysis
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.6
0.4
0.8
1
Figure 3: The exact solution 𝑐(𝑥, 𝑦, 𝑡) = 100𝑡𝑥3 (1 − 𝑥)2 cos(2𝜋𝑦) at time 𝑡 = 0.5.
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.6
0.4
0.8
1
Figure 4: Approximate solution of the characteristic-DDM scheme at 𝑡 = 0.5, ℎ = 1/40.
−6
−6.2
−6.4
−6.6
log e
−6.8
−7
−7.2
−7.4
−7.6
−7.8
−8
−5
−4.5
−4
log h
Figure 5: Convergence order of the characteristic-DDM scheme.
Advances in Numerical Analysis
11
5. Extensions
To get higher-order accuracy with respect to 𝐻, we apply
̂𝐻(x) on the
the definition (22) to an extrapolation value 𝐶
interdomain boundary Γ. Similar to the analysis of [18], we
can present another approximate scheme for (1a)–(1d):
Similarly as the elliptic projection (44), in order to get
optimal error estimates, we introduce an elliptic projection
̃ ∈ Mℎ of the solution 𝑢 as follows:
𝐶
c
̂
̃ , ∇𝑣) + (̂𝑐𝜇,𝐻 − 𝐶
̃𝜇,𝐻, [𝑣]) + (̂𝑣𝜇,𝐻, [𝑐 − 𝐶])
̃
(𝐷∇ (𝑐 − 𝐶)
Γ
Γ
̃ , [𝑣]) = 0,
+ 𝑑1 𝐾𝐻−1 ([𝑐 − 𝐶]
Γ
0
̃,
(a) 𝐶0 = 𝐶
(b) 𝐴 (𝐶𝑚 ; 𝑈𝑚 , 𝑧)+𝐵 (𝑧, 𝑃𝑚 ) = (𝛾 (𝐶𝑚 ) ∇𝑑, 𝑧) ,
(c) 𝐵 (𝑈𝑚 , 𝑤) = − (𝑞𝑚 , 𝑤) ,
𝑛−1
𝐶𝑛 − 𝐶̌
(d) (𝜙
Δ𝑡𝑐
∀𝑧 ∈ 𝑉ℎ𝑝 ,
∀𝑤 ∈ 𝑊ℎ𝑝 , 𝑚 ≥ 0,
𝑛
𝑛
̃𝑛 ,
𝜉𝑛 = 𝑐𝑛 − 𝐶
󵄩󵄩 󵄩󵄩
𝑟+1
1/2 󵄩 󵄩
󵄩󵄩𝜉󵄩󵄩𝐿2 (Ω) ≤ 𝐶 {ℎ𝑐 + 𝐻 󵄩󵄩󵄩𝜂󵄩󵄩󵄩𝐿∞ (Ω) } ,
󵄩󵄩 󵄩󵄩
𝑟+1
1/2 󵄩 󵄩
󵄩󵄩𝜉𝑡 󵄩󵄩𝐿2 (Ω) ≤ 𝐶 {ℎ𝑐 + 𝐻 󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩𝐿∞ (Ω) } .
∀𝑣 ∈ Mℎ𝑐 , 𝑛 ≥ 1.
(66)
Since the differences between two schemes (28a)–(28d)
and (66) are the second and third term to calculate the flux
on the inner-domain boundary Γ, the convergence analysis of
the scheme (66) is similar to that of the scheme (28a)–(28d).
For the sake of brevity, we describe the processes of analysis
for (66) simply.
The proof of Theorem 15 is based on the following four
lemmas. The former three lemmas are adopted from [18].
We omit the proof of Lemma 14, which is similar to that of
Lemma 7 in Section 4.
𝑛
𝜃 .
Lemma 14. For 𝜃𝑛 defined by (71), there exists the following
error estimate
󵄩 󵄩2
󵄩 󵄩2
󵄩 󵄩2
max 󵄩󵄩𝜃n 󵄩󵄩󵄩 ≤ 𝐶 {𝐻 [ max (󵄩󵄩󵄩𝜂𝑛 󵄩󵄩󵄩L∞ (Ω) + 󵄩󵄩󵄩𝜂𝑛 󵄩󵄩󵄩 )
1≤𝑛≤𝑁
0≤𝑛≤𝑁󵄩
󵄩 󵄩2
󵄩 󵄩2
+󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩𝐿2 (0,𝑇;𝐿∞ (Ω)) + 󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩𝐿2 (0,𝑇;𝐿2 (Ω)) ]
2
+ (Δ𝑡𝑐 ) + 𝐻9 + ℎ𝑝2𝑙+2 + ℎc2r+2
provided
(67)
where 𝑊𝜈Γ4 is the fourth normal derivative of 𝑊 on Γ.
For functions 𝜓 with restrictions in 𝐻1 (Ω1 ) ∪ 𝐻1 (Ω2 ), we
use the definition of the norm
(68)
̃0 = 1 − (2√2/3)
Lemma 12. There exists a positive constant 𝐶
such that for small 𝐻 > 0,
̃0 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨2 ≤ (𝐷∇𝜓, ∇𝜓) + 𝑑1 𝐾𝐻−1 ([𝜓] , [𝜓])
𝐶
󵄨󵄨󵄨 󵄨󵄨󵄨
Γ
∀𝑣 ∈ Mℎc .
4
(73)
∀x 𝑜𝑛 Γ,
󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨2
−1
󵄨󵄨󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨󵄨󵄨 = (𝐷∇𝜓, ∇𝜓) + 𝑑1 𝐾𝐻 ([𝜓] , [𝜓])Γ .
3
+(Δ𝑡𝑝1 ) + (Δ𝑡𝑝 ) } ,
2√2 + 1 √
󵄩
󵄩󵄩 ̂
󵄩󵄩𝑊𝐻 − 𝑊󵄩󵄩󵄩 2 ≤
2𝐻‖∇𝑊‖𝐿2 (Ω) ,
󵄩𝐿 (Γ)
󵄩
3
󵄩󵄩
4
󵄩󵄩󵄩𝑊
󵄩󵄩 ̂ 𝐻 − 𝑊󵄩󵄩󵄩𝐿∞ (Γ) ≤ 𝐶𝐻 ‖𝑊‖𝑊4,∞ (Ω) ,
̂ 𝐻 (x) = 1 𝐻4 𝑊𝜈4 (x) + 𝑜 (𝐻6 ) ,
𝑊 (x) − 𝑊
Γ
480
(72)
Now, we turn to derive an 𝐿2 (Ω)-norm error estimate for
Lemma 11. For sufficiently smooth function 𝑊, there holds
estimates:
Γ
(71)
Lemma 13. There holds a priori estimates:
Γ
̂ 𝜇,𝐻, [𝜓]) ,
+ 2(𝜓
̃𝑛 .
𝜃𝑛 = 𝐶𝑛 − 𝐶
The following lemma gives the bounds of 𝜉𝑛 .
̂𝑛−1 , [𝑣]) + (̂𝑣𝜇,𝐻, [𝐶𝑛−1 ])
+ (𝐶
𝜇,𝐻
Γ
+𝑑1 𝐾𝐻−1 ([𝐶𝑛−1 ] , [𝑣])Γ = (𝑞𝑛̃𝑐𝑛 , 𝑣) ,
(70)
It follows from Lemma 12 that the projection problem (70)
has a unique solution for small 𝐻.
Let
𝑛
, 𝑣) + (D∇𝐶 , 𝑣) + (𝑞 𝐶 , 𝑣)
∀𝑣 ∈ Mℎc .
̃ 1 𝐻2 ,
Δ𝑡 ≤ 𝐶
̃1 =
𝐶
∀0 < 𝛿 ≪ 1. (74)
Theorem 15. Suppose that the assumptions (𝑃), (𝑄), (𝐼), (𝐴 𝑝 ),
(𝐼𝑝 ) and 𝑟 ≥ 1, 𝑙 ≥ 0 hold. Assume that the discretization
parameters obey the relations
Δtc = o (ℎ𝑝 ) ,
3/2
(Δ𝑡𝑝1 )
= 𝑂 (ℎp ) ,
ℎ𝑐𝑟+1 = O (ℎ𝑝 ),
2
(75)
(Δ𝑡𝑝 ) = 𝑂 (ℎ𝑝 ) .
Then for ℎc , ℎp and Δ𝑡𝑐 sufficiently small, the errors of the
approximation (66) for (1a)–(1d) satisfy
󵄩
󵄩
max 󵄩󵄩𝑐𝑛 − 𝐶𝑛 󵄩󵄩󵄩
0≤𝑛≤𝑁 󵄩
2
(69)
̃3
𝑎0 (1 − 𝛿)2 𝐶
0
,
16𝑎12 𝐾2 𝐶22
≤ 𝐶 {Δ𝑡𝑐 + (Δ𝑡𝑝 ) + (Δ𝑡𝑝1 )
3/2
+ ℎ𝑐𝑟+1 + ℎ𝑝l+1 + 𝐻9/2 } ,
(76)
12
Advances in Numerical Analysis
Table 1: 𝐿2 -norm error estimate of 𝑒ℎ = 𝑐 − 𝐶 at time 𝑡 = 0.5.
Grids
50 × 50
40 × 40
80 × 80
160 × 160
ℎ
𝐻/ℎ
.5000𝑒 − 01
.2500𝑒 − 01
.1250𝑒 − 01
.6250𝑒 − 02
.1826𝑒 + 01
.2091𝑒 + 01
.2402𝑒 + 01
.2760𝑒 + 01
Characteristic-FEM scheme
‖𝑒ℎ ‖𝐿2
𝛼
.0921𝑒 − 01
.2448𝑒 − 02
1.9100
.6245𝑒 − 03
1.9708
.1583𝑒 − 03
1.9800
provided
Characteristic-DDM scheme
‖𝑒ℎ ‖𝐿2
𝛼
.0878𝑒 − 01
.2282𝑒 − 02
1.9439
.5781𝑒 − 03
1.9809
.1455𝑒 − 03
1.9903
DDM in [18–20, 22, 23]. For simplicity, we consider the
following convection dominated diffusion equation
̃ 1 𝐻2 ,
Δ𝑡𝑐 ≤ 𝐶
(77)
̃1 is given by (74). Here 𝐶 is a positive constant
where 𝐶
dependent on (𝑃), (𝑄), (𝐼), (𝐴 𝑝 ), (𝐼𝑝 ), ‖𝜕2 𝑐/𝜕𝜏2 ‖, ‖𝜕u/𝜕𝑡‖,
‖𝜕2 u/𝜕𝑡2 ‖, but independent of ℎc , ℎp , Δ𝑡𝑐 and Δ𝑡𝑝 .
By combining Theorem 15 with (33) and (35), we obtain
at once the following result.
Corollary 16. Under the assumption of Theorem 15, the errors
for velocity u and pressure 𝑝 are obtained by
󵄩
󵄩
󵄩
󵄩
max (󵄩󵄩󵄩u𝑚 − 𝑈𝑚 󵄩󵄩󵄩𝐻(div;Ω) + 󵄩󵄩󵄩𝑝𝑚 − 𝑃𝑚 󵄩󵄩󵄩)
0≤𝑚≤𝑀
2
3/2
≤ 𝐶 {Δ𝑡𝑐 + (Δ𝑡𝑝 ) + (Δ𝑡𝑝1 )
+ ℎ𝑐𝑟+1 + ℎ𝑝𝑙+1 + 𝐻9/2 } .
(78)
Remark 17. As for the accuracy order of ℎc and ℎ𝑝 , we know
that (76) and (78) are optimal for the concentration 𝑐, the
velocity u and the pressure 𝑝.
Remark 18. From Theorem 15, we can know that the scheme
(66) has the same accuracy order as that of the scheme (28a)–
(28d) with respect to Δ𝑡𝑐 , ℎ𝑐 and ℎ𝑝 , except that has an
accuracy of higher order for 𝐻. This shows that the scheme
(66) can use larger width 𝐻 of middle strip domain than that
of the scheme (28a)–(28d) so that the time step constraint is
more weaker.
6. Numerical Experiments
In this section, we present some numerical experiments
for the procedures described above. All computer programs
below are written by Fortran 90 code and run on a Lenovo
PC with Intel(R) Core i5 3.2 GHz CPU and 4 GB memory.
The resulting linear systems of algebraic equations are solved
by banded Gaussian elimination. Single precision is used for
all calculations.
The main purpose of this paper is to analyze the integral
mean non-overlapping DDM combined with the characteristic method for the concentration equation. There were many
experimental results for the integral mean non-overlapping
𝜙
𝜕𝑐
+ u ⋅ ∇𝑐 − ∇ ⋅ (D∇𝑐) + 𝑞𝑐 = 𝑓,
𝜕𝑡
𝜕𝑐
= 0,
𝜕𝜈
𝑐0 = 0,
(𝑥, 𝑡) ∈ Ω × (0, T] ,
(𝑥, 𝑡) ∈ 𝜕Ω × (0, T] ,
𝑥 ∈ Ω, 𝑡 = 0,
(79)
where, Ω = [0, 1] × [0, 1], 𝑢1 = 2 + 𝑥12 , 𝑢2 = 1 + 𝑥22 , D = 0.05I,
𝜙 = 0.2, 𝑞 = 2. We choose 𝑓(𝑥, 𝑦, 𝑡) suitably so that the exact
solution of (79) is 𝑐(𝑥, 𝑦, 𝑡) = 100𝑡𝑥3 (1 − 𝑥)2 cos(2𝜋𝑦), see
Figure 3.
We consider two scenarios: (1) the characteristic implicit
Galerkin scheme (Charac-teristic-FEM scheme) on uniform
mesh; that is, no domain decomposition; (2) the characteristic integral mean DDM scheme (Characteristic-DDM scheme)
on global uniform mesh with two equal sub-domains Ω1 =
(0, 1/2) × (0, 1), Ω2 = (1/2, 1) × (0, 1), with the interdomain
boundary Γ = {1/2} × (0, 1). In these runs, we approximate
(79) by using linear finite element with 4-node quadrilateral
mesh on 20 × 20, 40 × 40, 80 × 80 and 160 × 160 grids,
respectively. For each dynamic domain decomposition case,
we still take 𝐻5/2 = ℎ2 to balance error accuracy with respect
to ℎ, 𝐻 and mesh ration Δ𝑡 = 𝐻5/2 in order to satisfy the
conditional stability due to the explicit calculation of flux on
the interface.
The following Figure 4 presents the approximate solution
of the characteristic integral mean DDM scheme at time 𝑡 =
0.5 with space step size ℎ = 1/40.
Table 1 shows 𝐿2 -norm error estimate of 𝑒ℎ = 𝑐 − 𝐶
at time 𝑡 = 0.5, where the error order is choosen as 𝛼 =
log(𝑒ℎ𝑙 /𝑒ℎ𝑙−1 )/ log(ℎ𝑙 /ℎ𝑙−1 ).
Figure 5 shows the convergence order of the characteristic
integral mean DDM scheme.
Remark 19. From Table 1, we see that the error estimate of the
characteristic integral mean DDM scheme is better than that
of the characteristic implicit Galerkin scheme. From Table 1
and Figure 5, we see that the convergence order is near 2
which is consisting with the analytical results.
Acknowledgments
The authors thank the referees for their valuable suggestions
and constructive comments which led to great improvement
Advances in Numerical Analysis
of this paper. This paper was supported by the NSF of
China (no. 11271231), Tian Yuan Special Foundation (no.
11126206), the Scientific Research Award Fund for Excellent
Middle-Aged and Young Scientists of Shandong Province
(no. BS2009NJ003) and the NSF of Shandong Province (no.
ZR2010AQ019), China.
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