c Pleiades Publishing, Ltd., 2023.
ISSN 1995-4239, Numerical Analysis and Applications, 2023, Vol. 16, No. 1, pp. 34–44. c The Author(s), 2023, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2023, Vol. 26, No. 1, pp. 43–54.
Russian Text A Posteriori Error Majorants for FEM Solutions
of Plate Bending Problem upon Winkler Subgrade1
V. G. Korneev*
St. Petersburg State University, St. Petersburg, Russia
Received October 10, 2022; in final form, November 9, 2022; accepted November 23, 2022
Abstract—The paper is devoted to the mixed finite element method for the equation ΔΔu + κ2 u =
f , x ∈ Ω, with boundary conditions u = ∂u/∂ν = 0 on ∂Ω, where ν is the normal to the boundary
and κ ≥ 0 is an arbitrary constant on each finite element. At κ ≡ 0 residual type a posteriori error
bounds for the mixed Ciarlet–Raviart method were derived by several authors at the use of different
error norms. The bounds, termed sometimes a posteriori functional error majorants, seem to be less
dependent on the constants in the general approximation bounds and are more flexible and adaptable
for attaining higher accuracy at practical implementation. In this paper, we present a posteriori
functional error majorants for the mixed Ciarlet–Raviart method in the case of κ = 0 and having
large jumps. Robustness and sharpness of the bounds are approved by the lower bounds of local
efficiency.
DOI: 10.1134/S1995423923010044
Keywords: a posteriori error bounds, singularly perturbed elliptic equations of 4th order,
mixed finite element method, lower error bounds.
1. INTRODUCTION
The paper is devoted to the error control at the implementation of the finite element methods to the
equation
ΔΔu + σu = f (x) in Ω ,
u = ∂u/∂ν = 0
(1.1)
on ∂Ω ,
where ν is the internal normal to the boundary ∂Ω, σ = κ2 and κ is an arbitrary element-wise constant
nonnegative function. Sufficiently simple and locally efficient a posteriori error bound for the mixed
Ciarlet–Raviart FEM, applied to the problem in a polygonal domain Ω at σ ≡ 0, was derived by Gudi [8]
with the use of the residual technique. Herewith no assumption on the elliptic regularity result nor quasiuniformity of the triangulation were used. There are a few other papers devoted to the residual optimal
and suboptimal a posteriori bounds of different error norms for the mixed Ciarlet–Raviart FEM for the
same problem, see, e.g., Charbonneau et al. [6], Verfürth [11] and references there. Quite recently Du et
al. [7] considered mixed FEM’s for the problem ε2 Δ2 u − Δu = f (x), u = ∂u/∂ν = 0 on ∂Ω, with small
parameter ε > 0. Taking into account differential properties of the solution, they introduced a different
from [8] norm for the error of the FEM solution (ε|v − vh |H 1 (Ω) + |v − vh |L2 (Ω) + |u − uh |H 1 (Ω) )1/2 , in
which v = Δu and (vh , uh ) is the mixed FEM solution. They derived residual a posteriori bounds of
this error norm, robust with respect to ε, and approved their accuracy by the lower bounds, which are
commonly termed in the literature as efficiency bounds.
Residual a posteriori error bounds are characterized by a relatively strong dependence on constants
in the local and, sometimes, global approximation estimates. There is another type of a posteriori error
1
*
The work was presented at the International Conference “Marchuk Scientific Readings-2022.”
E-mail: vad.korneev2011@yandex.ru
34
A POSTERIORI ERROR MAJORANTS FOR FEM SOLUTIONS
35
bounds, in which the accuracy of the approximate solution is measured by means of testing fields of
flaxes, stresses, moments, etc. of a free choice. It includes the bounds, occasionally called functional
error majorants, recovery based bounds, and some others and allows to avoid intensive use of the
approximation estimates. In concurrence with this, a variety of techniques for obtaining differently
defined testing fields, including fields produced by the superconvergent recovery procedures, equilibrated
fields etc. have been developed and used in practice.
An attempt to expand results for the mixed FEM’s to the problem (1.1) with σ ≥ 0 was made by
Korneev [9, 10]. However, in the contrast to the case of the conform finite element methods, see [9], its
outcome were the bounds of the functional error majorant type accurate for σ, mildly changing between
finite elements, but not robust for the growing jumps. Additional restrictive features of the previous
bounds is that they assumed, sometimes not explicitly, the elliptic regularity condition on the boundary
and the quasi-uniformity of the finite element mesh. In the current paper, we present sharp error bounds
for the mixed FEM-s, introduced for the problem (1.1) with σ ≡ 0 by Ciarlet and Raviart [5], in the
form of the recovery-based a posteriori error majorants, which are robust in case of σ, having significant
jumps. They are applicable for a wider range of domains since the elliptic regularity conditions are not
used at their derivation. It is worth to mention also the simplicity of their derivation, and weak restrictions
on the range of meshes. For certifying the efficiency of the presented a posteriori error bounds, we prove
the corresponding lower bounds.
The paper contains four sections. In Section 2 we formulate the problem and the mixed Ciarlet–
Raviart finite element method for its numerical solution and discuss some assumptions. The a posteriori
error bounds are derived in Section 3, and their sharpness under a stronger, than in Section 3,
assumption on the finite element meshes is approved in Section 4.
For the norms and quasi-norms in Sobolev’s spaces H k (Ω) = W2k (Ω), standard notations · k , | · |k
or · k,Ω , | · |k,Ω are used with the agreement that | · |0 = · 0 = · . At description of the primal and
mixed problems, we use additionally the spaces H02 (Ω) = {φ ∈ H 2 (Ω) : u(y) = ∂u/∂ν(y) = 0 for y ∈
∂Ω}, H̊ 1 (Ω) = {φ ∈ H 1 (Ω) : u(y) = 0 for y ∈ ∂Ω} and H02 (Ω, Δ2 ) = {φ ∈ H02 (Ω) : Δ2 φ ∈ L2 (Ω)}.
2. PRELIMINARIES
The weak form of problem (1.1) reads as: find u ∈ H02 (Ω) such that
a(u, v) + (σu, v) = (f, v) ∀ v ∈ H02 (Ω) ,
(2.1)
where
a(w, v) = (Δw, Δv) ,
(w, v) : = (w, v)Ω =
wv dx .
Ω
The reaction coefficient σ is assumed to be a piecewise constant function, i.e., σ = σr = κ2r = const ≥ 0
for x ∈ τr on the assemblage of conform finite elements on subdomains τr , r = 1, 2, . . . , R, which
closures cover the sufficiently smooth domain Ω,
Ω=
R
τr ,
τr ∩ τr = ∅ for
r = r .
r=1
The weak mixed Ciarlet–Raviart type formulation of problem (1.1) is to find the vector-function
w = (v, u) ∈ H 1 (Ω) × H̊ 1 (Ω) satisfying the system of the integral identities
(v, q) + ∇u, ∇q
∀ q ∈ H 1 (Ω) ,
=0
− ∇v, ∇g + (σu, g) = (f, g)
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(2.2)
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where H̊ 1 (Ω) = {φ ∈ H 1 (Ω) : φ = 0 on ∂Ω} and ·, · is the scalar product of vector-functions z, y =
(z1 , y1 ) + (z2 , y2 ) with y = (y1 , y2 ) and z = (z1 , z2 ) .
Suppose Ω is polygonal and Th is its triangulation. Then for solving Eqs. (2.2), two finite element
assemblages, denoted Kh,u and Kh,v and defined on Th , with triangular Lagrange finite elements having
straight edges can be employed. They induce the spaces
Uh (Ω) = {φh ∈ C(Ω) : φh τr ∈ Ppu , pu ≥ 2, r = 1, 2, . . . , R} ,
Vh (Ω) = {ψh ∈ C(Ω) : ψh τr ∈ Ppv , pv ≥ 2, r = 1, 2, . . . , R} ,
respectively. These spaces, as it is known, possess the following approximation properties: for any
= Gh,u w ∈ Uh (Ω) there are hold the approximation estimates
w ∈ H l (Ω) and w
|w
− w|k,τr ≤ cr,k,l hl−k |w|l,τr ,
cr,k,l = const ,
(2.3)
where Gh,u : C(Ω) → Uh (Ω) is the linear Lagrange interpolation operator of the function values on the
set of the finite element nodes and k ≤ l , 2 ≤ l ≤ pu + 1 and r = 1, 2, . . . , R. Naturally, Gh,u w|∂Ω = 0,
in other words Gh,u w ∈ Ůh (Ω) := Uh (Ω) ∩ H̊ 1 (Ω), if w ∈ H̊ l (Ω) : = H̊ 1 (Ω) ∩ H l (Ω). Similar approxi = Gh,v w ∈ Vh (Ω), Gh,v : C(Ω) → Vh (Ω).
mation estimates hold for w ∈ H l (Ω), w
The solution wh = (vh , uh ) ∈ Vh (Ω) × Ůh (Ω) of the mixed finite element method satisfies the
system of equations
(vh , qh ) + ∇uh , ∇qh
=0
− ∇vh , ∇gh + (σuh , gh ) = (f, gh )
∀ qh ∈ Vh (Ω) ,
(2.4)
∀ g ∈ Ůh (Ω) ,
and, as a consequence of (2.2) and (2.4), the error of the numerical solution
efem = (ev , eu ) ,
where
eu = uh − u ,
ev = vh + Δu ,
satisfies the equations
(ev , qh ) + ∇eu , ∇qh
=0
∀ qh ∈ Vh (Ω) ,
− ∇ev , ∇gh + (σeu , gh ) = 0
∀ gh ∈ Ůh (Ω) .
(2.5)
v, given on
For brevity, let us introduce the operator Δh , which for a sufficiently smooth function
Ω, produces Δh v = Δv, x ∈ τr , on each finite element. Thus, Δh v2 = Δh v2Ω = Δv2τr . For the
r
measure of the error we will use the norm
1/2
1 .
]||efem ||[= √ ev 2 + Δh eu 2 + 2κeu 2
2
(2.6)
The a posteriori error bounds of the paper are based upon the gradient and deflection recovery
operators. For instance, the relatively cheap and accurate operator Gh : Vh (Ω) → Vh (Ω) can be used
which for any φh ∈ Vh (Ω) produces ψh as L2 -projection of φh onto some smoother finite element space
Vh (Ω) ⊂ C 1 (Ω) ∩ H 2 (Ω). There is vast literature on the gradient recovery operators and their reflection
upon the accuracy of the error bounds, see, e.g, [12, 14] and references there, and only a few papers on
smoothing the space Vh (Ω), see [3, 4]. We will not elaborate on this, since effects of superconvergence
are out of the scope of this paper, and choice of the recovery operator, retaining basic approximation
properties, is sufficient for the purposes of this paper. Due to the importance of simplicity and locality,
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A POSTERIORI ERROR MAJORANTS FOR FEM SOLUTIONS
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the preference is given to the operators Dh : Ůh (Ω) → Vh,0 (Ω) and D h : Vh (Ω) → Vh (Ω) defined by the
averaging technique, which is employed for evaluation of the nodal parameters of the conform finite
element spaces Vh,0 (Ω) ⊂ C 1 (Ω) ∩ H02 (Ω) and Vh (Ω) ⊂ C 1 (Ω) ∩ H 2 (Ω). Let A be any global degree of
freedom of Vh (Ω), i.e., A is either the evaluation of a shape function ψh = D h φh at some node x(j) or
its derivative from the corresponding to the node set Ψj of degrees of freedom, alternatively called nodal
parameters. For φh ∈ Vh , we define
A(D h φh ) =
1
|ωA | τ
A(φh |τr ) ,
r ∈ωA
where ωA is the set of triangles in Th that share the degree of freedom A and |ωA | is its cardinality. Nodal
parameters, found in this way, uniquely define the function ψh .
The operator Dh can be defined differently, as well as the spaces Vh (Ω) of functions recovered from uh
and vh . For simplicity and, in particular, for diminishing number of the FEM spaces, involved in the error
estimation, we will assume that Uh (Ω) = Vh (Ω) and respectively Ůh (Ω) = V̊h (Ω) : = Vh (Ω) ∩ H̊ 1 (Ω).
More over, abstracting from possible differences, we formulate as an assumption the property which
remains invariant and is important for the efficiency lower bounds, derived in Section 4.
Let E be an edge of the triangulation Th , he = |E|, and νe be the normal to the edge. The normal νe
can be fixed, e.g., as directed to the right, if one looks along E from its end, which is the node with lesser
∂φ
on the
number than the node at the other end of E. For a given function φ ∈ W1∞ (Ω), the function ∂ν
e
edge E is the jump of the derivative of φ with respect to the normal νe at crossing the edge in the direction
of the normal. Namely we introduce the assumption:
α): for Dh = Dh , D h
uh −
Dh uh 2T
≤
cT h4T
z∈T E∈Ez
1
he
E
∂uh
∂νe
2
ds ,
(2.7)
where T = τr is a triangle of the triangulation, hT = diam[T ], z is a vertex of T , Ez is the set of the
edges, having z for one end, and cT is the constant, depending only on the shape regularity conditions for
triangles of Th . The wish of simplicity and the fact that even for convex polygonal domains at f ∈ L2 (Ω)
the solution u can belong only to H 3 (Ω), see [2], inclined the authors to use the spaces Vh,0 (Ω) for the
deflection recovery, generated by finite elements, the nodal parameters of which do not contain derivatives
of higher order than first. A good example is the Hsieh–Clough–Tocher (HCT) finite element space, see
Ciarlet [5, Ch. 6, Sec. 1] and Gudi [8], defined on the triangulation Th the same for the spaces Vh (Ω) and
Uh (Ω). For the operator Dh , when Vh,0 (Ω) is the HCT space, Brenner et al. [4] proved the bounds (2.7).
This paper stems from the earlier paper of Brenner and Sung [3] where both types of recovery operators
were studied. Herewith the recovery spaces were generated by the C 1 TUBA type finite elements, see
Argyris et al. [1], the nodal parameters of which include second derivatives.
For smooth functions, the HCT space provides at most fourth order approximation in L2 (Ω) that does
not allow us to obtain adequate a posteriori error bounds for more accurate finite element methods. There
is another family of finite triangular elements, curvilinear version of which was studied by Korneev and
Khusanov [18]. Each element has 13 nodal parameters, if sides of τr are straight, the nodal parameters
are values of FE functions at vertices and the barycenter of a triangle, both first derivatives at the vertices,
and normal derivatives at middle node of each edge. This finite element is a modification of Hermit
Triangle, which is a third order complete nonconform element, see [5, Sec. 2.2, p. 68]. It is obtained
by adding to Hermit Triangle additional three nodes at the middles of the edges in order to produce an
element conform in C 1 , see [13, Sec. 10.10] and [18]. Coordinate functions of such elements are not
polynomials, but rational functions.
Finite element assemblage with curvilinear elements can be constructed on the basis of the triangulation Th with the domain Ωh , whose all the boundary ∂Ωh nodes lie on ∂Ω. The triangles of this
triangulation are denoted τr . If τr has one or no vertices on ∂Ω, then τr := τr is accepted for the domain
NUMERICAL ANALYSIS AND APPLICATIONS Vol. 16
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of a finite element. If τr ∈ Th has two vertices on ∂Ω, then (only) one side of τr is curvilinear and is the
part of ∂Ω, joining these two vertices. Obviously, with this definition
Rh
τr = Ω .
r=1
On the curvilinear triangles, Korneev and Khusanov [18] constructed C 1 -conform finite elements of the
same order of accuracy as for the respective finite elements with straight edges. The constructions of C 0
FE spaces Vh (Ω) and Ůh (Ω) with the same finite element subdomains, matching the boundary ∂Ω, are
found in [15–17].
It should be noted that the possibility to use curvilinear finite elements, exactly representing curvilinear boundaries ∂Ω, is important for two reasons. If ∂Ω is smooth and σ is bounded, then the problem
is H 2 -regular [2], however for polygonal ∂Ωh , approximating boundary ∂Ω, this property can be lost.
Besides, even with the regularity retained, the error, caused by the approximation of ∂Ω by ∂Ωh and
by shifting the first boundary conditions from ∂Ω to ∂Ωh , can compromise the accuracy of the finite
elements.
3. ERROR BOUNDS FOR CIARLET–RAVIART MIXED FORMULATION
Let ũ ∈ Vh,0 (Ω) be some approximation of u, recovered from uh , and ẽ = ũ − u. If in ]||efem ||[ to add
and subtract ũ and use the triangle inequality, we get
1/2
1 vh − Δũ2 + κ(uh − ũ)2
+ |||uh − ũ||| + 2|||ẽ||| .
]||efem ||[≤ √
2
(3.1)
In accordance with the duality principle
1/2
a(ẽ, φ) + (σẽ, φ)
1 ,
=
sup
|||ẽ||| = √ Δ(ũ − u)2 + κ(ũ − u)2
|||φ|||
2
2
φ∈H0 (Ω)\0
(3.2)
and the numerator of the fraction can be transformed as
a(ẽ, φ) + (σẽ, φ) = (Δũ − vh , Δφ) + (σ(ũ − uh ), φ) + (vh − Δu, Δφ) + (σ(uh − u), φ) .
(3.3)
If to integrate by parts and take into account (2.1), and (2.4), we can write
(vh − Δu, Δφ) + (σ(uh − u), φ) = (vh , Δφ) − a(u, φ) + (σ(uh − u), φ)
= −(∇vh , ∇φ) − (f, φ) + (σuh , φ)
= −(∇vh , ∇(φ − φh )) − (f, (φ − φh )) + (σuh , (φ − φh ))
(3.4)
∀ φh ∈ Ůh .
There are at least two options in estimation of the first term in the right part. In one, we introduce the
vector μ = (μ1 , μ2 ) , whose components μk ∈ Vh approximate respective derivatives ∂vh /∂xk in some
sense. In another, we use the vector ∇ṽ, where function ṽ ∈ Vh (Ω) approximates vh . In the first option,
μk can be defined as the L2 projections of ∂vh /∂xk on the space Vh . The function φh can be defined as
L2 projection of φ on the space Ůh or as Lagrange interpolation of φ from the space Ůh . Now we take
into account (3.4), a particular form of (2.3)-type approximation bounds
|φ − φh |k ≤ ck,2 h2−k |φ|2 ,
k = 0, 1 ,
(3.5)
and get
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A POSTERIORI ERROR MAJORANTS FOR FEM SOLUTIONS
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|(vh − Δu, Δφ) + (σ(uh − u), φ)| = | − (∇vh − μ, ∇(φ − φh ))(μ, ∇(φ − φh ))
−(f, (φ − φh )) + (σuh , (φ − φh ))|
(3.6)
≤ c1,2 h∇vh − μ + c0,2 h2 f − σuh − ∇ · μ |φ|2 .
Since |φ|2 ≤ |||φ|||, from (3.1), (3.2), and (3.6) it follows
1/2
|φ|2
+ c1,2 h∇vh − μ + c0,2 h2 f − σuh − ∇ · μ
|||ẽ||| ≤ Δũ − vh 2 + κ(ũ − uh )2
|||φ|||
1/2
+ c1,2 h∇vh − μ + c0,2 h2 f − σuh − ∇ · μ
Δũ − vh 2 + κ(ũ − uh )2
and
1/2
+ |||uh − ũ|||
]||efem ||[ ≺ vh − Δũ2 + κ(uh − ũ)2
+c1,2 h∇vh − μ + c0,2
h2 f
(3.7)
− σuh − ∇ · μ
with ≺ meaning the inequality hold up to an absolute constant omitted.
If to introduce the notation ẽh = (ẽv , ẽu ) = (vh − Δũ, uh − ũ) , then this bound can be presented in
a shorter form
]||efem ||[ ≺ ]||ẽh ||[ + c1,2 h∇vh − μ + c0,2 h2 f − σuh − ∇ · μ .
(3.8)
Constants ck,l = max(cr,k,l ), being global, and h = max hr , where hr is size of a triangle τr , can cause
r
r
an overestimation of the error. Slightly changing the proof, we come to the bound with the local
constants.
Theorem 1. Let the solution u of (1.1) belong to H02 (Ω, Δ2 ), and ũ and components μk of the
vector-function μ be any functions from the spaces Vh,0 (Ω) and Vh (Ω), respectively. Then
]||efem ||[ ≺ ]||ẽh ||[ +
c21,2,r h2r ∇vh − μ2τr
1/2
c20,2,r h4r f − σuh − ∇ · μ2τr
+
r
1/2
.
(3.9)
r
As it was noted, instead of μ one can use the vector ∇ṽ with some approximation ṽ ∈ Vh (Ω) of vh . In
particular, ṽ can be defined as ṽ = Dh vh . In this case, by application of the standard inverse inequality
to the norms ∇(vh − ṽ)τr , the bound takes the form
]||efem ||[ ≺ ]||ẽh ||[ +
c21,0,r c21,2,r vh − ṽ2τr
1/2 +
r
c20,2,r h4r f − σuh − Δṽ2τr
1/2
.
(3.10)
r
Let us underline that bounds (3.9) and (3.10) do not imply quasiuniformity of the mesh and, in
particular, they hold for an arbitrary shape regular mesh. This is important for the use of a posteriori
error bounds at creating adaptive algorithms.
We introduce also the norm []efem []:
1
[]efem = ]||efem
he
E∈Γe
E
1
ev 2 + Δh eu 2 + 2κeu 2 +
=
2
[]2
∂eu
∂νe
||[2 +
E∈Γe
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1
he
2023
2
ds
E
∂eu
∂νe
(3.11)
2
ds ,
40
KORNEEV
which is an adaptation of the norm, used in [18], to the case of σ not identical to zero. Obviously,
alongside with (3.10), we have the a posteriori bound
[]efem [] ≺ η(uh , vh , ũ, ṽ)
(3.12)
where
η(uh , vh , ũ, ṽ) =
]||ẽh ||[ +
2
E∈Γe
1
he
∂uh
∂νe
E
1/2
2
ds
1/2
c21,0,r c21,2,r vh
+
−
ṽ2τr
r
1/2
c20,2,r h4r f
+
− σuh −
Δṽ2τr
.
r
4. EFFICIENCY OF A POSTERIORI ERROR BOUNDS
Now let us prove a lower bound, commonly termed the efficiency bound in the literature. For simplicity
we restrict consideration to the quasiuniform finite element meshes under the assumption that the
inequality (2.7) is fulfilled for the corresponding space Vh (Ω). Inequality (2.7) directly leads to the bound
for the “reaction” term:
κ(uh − ũ) ≤ c (max σ)h
2
4
x∈Ω
E ∈ Γe E
1 ∂uh
he ∂νe
2
ds,
(4.1)
where Γe is the interelement boundary, including edges of triangles and their vertices lying in Ω. If we
take into account the inequality
Δh (uh − ũ) ≤ c
2
E ∈Γe E
1 ∂uh 2
ds,
he ∂νe (4.2)
arising from (4.1) and the inverse inequality for polynomial functions, as well as the triangle inequality
vh − Δũ|| ≤ vh − Δu + Δh (u − uh ) + Δh (uh − ũ) ,
we obtain
2 ]||ẽh ||[2 = ẽv 2 + Δh ẽu 2 + 2κẽu 2
vh − Δu2 + Δh (u − uh )2 + Δh (uh − ũ)2 + 2 κ(uh − ũ)2
2
1 ∂uh 2
2
4
.
vh − Δu + Δh (u − uh ) + 1 + c max σh
x∈Ω
he ∂νe E
(4.3)
E∈Γe
Here a b means that a ≤ c b and c is the constant depending on the mesh quasiuniformity conditions.
Turning to the residual type term in (3.9) and (3.12), we can write
f − σuh − ΔṽT ≤ κ(u − uh )T + Δh (vh − ṽ)T + f¯ − Δh vh T + f − f¯T ,
(4.4)
where T = τr is any triangle of the triangulation, f = f − σu = Δ2 u and f¯ is the L2 -projection of f
onto piecewise polynomials of degree less than or equal to p, defined by
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f¯φ dx =
T
f φ dx
∀ φ ∈ Pp
and ∀ T ∈ Th .
T
The term f¯ − Δh vh T is estimated by the standard technique of bubble functions, with minor
distinctions from, e.g., [8, Lemma 3.2] caused by the presence of the reaction term in the boundary value
problem. Let βT ∈ H02 (T ) be the polynomial bubble function on triangle T and φ = βT (f¯ − Δh vh ).
Due to the inequalities
1/2
f¯ − Δh vh T βT (f¯ − Δh vh )T ,
βT (f¯ − Δh vh )T f¯ − Δh vh T ,
(4.5)
we have
βT (f¯ − Δh vh )2 dx =
f¯ − Δh vh 2T T
(f¯ − f)φdx +
=
T
(f¯ − Δh vh )φdx
T
(f − Δh vh )φdx .
(4.6)
T
Integration by parts, the inverse inequality, and (4.5) result in the bound
(f − Δh vh )φdx =
T
(Δu − vh )Δφdx c2,0 h−2
T Δu − vh T φT
(4.7)
T
¯
h−2
T Δu − vh T f − Δh vh T .
At the same time according to (4.5)
(f¯ − f)φdx f¯ − fT f¯ − Δh vh T ,
(4.8)
T
and the first multiplier in the right part of (4.8) is easily estimated as
f¯ − fT = f¯ − f T + κT (ū − u)T ≤ f¯ − f T + κT (ũ − u)T .
(4.9)
Combining (4.6)–(4.9), we get the bound
f¯ − Δh vh T f¯ − f T + κT (ũ − u)T + h−2
T Δu − vh T
f¯ − f T + κ(u − uh )T
1
2
+(h κT )
he
z∈δT E∈Ez E
+ h−2
T Δu − vh T
1/2
∂uh 2
.
∂νe ds
(4.10)
It remains to estimate the second norm in (4.4). For this purpose, we will estimate first vh − ṽT
and then use the inverse inequality
Δ(vh − ṽ)T h−2
T vh − ṽ)T .
(4.11)
In doing this it may be assumed that ṽ = D̄h vh , or that ṽ is defined as L2 -projection of vh on Vh (Ω).
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KORNEEV
h
Let δ e be the union of closures of two triangles with common edge E, φ = ψ1 ψ2 , where ψ1 = ∂v
∂νe
2
1
on E and is constant along the lines orthogonal to E, whereas ψ2 ∈ H0 (δe ) ∩ C (δe ) is an element-wise
polynomial function ψ2 ≥ 0 on E. The obvious inequality
he e (v ) ,
ψ12 dx
∂vh
∂νe
where e (v ) =
h
h
E
δe
2
ds ,
(4.12)
the approximation estimate (2.3), and the inverse inequality allow us to proceed along the lines
∂vh φ ds = Δh vh φ dx + ∇vh · ∇φ dx = Δh vh φ dx − vh Δφ dx
∂νe e
δe
δe
δe
(Δh vh − f)φ dx +
=
δe
δe
(Δu − vh )Δφ dx
δe
h2e Δh vh − fδe + Δu − vh δe h−2
e φ
(4.13)
1/2
he e (v h )
h2e Δh vh − fδe + Δu − vh δe h−2
e
1/2
3
h2e Δh vh − fδe + Δu − vh δe h−3
,
he e (v h )
e
and conclude that
E
1/2
∂vh 2
ds
he h2e Δh vh − fδe + Δu − vh δe .
∂νe 3
Therefore, taking into account (2.7), results in
vh − D̄h vh T 1/2
e (v )
h
h3T
(4.14)
z∈T E∈Ez
h2e Δh vh − f δT + h2e f − f δT + Δu − vh δT ,
where δ T is the union of the closures of triangles having common vertices with the vertices of the triangle
T . The norm Δh vh − f δT was bounded in (4.10), and in the next step we additionally take into
account (4.1):
vh − D̄h vh T h2T f¯ − f δT + h2T κ(ũ − u)δT + Δu − vh δT
h2T f¯ − f δT + h2T κ(uh − u)δT + Δu − vh δT
1/2
1
+ max (h2T κT )
e (uh )
.
T ⊂δT
he
(4.15)
z∈δ T E∈Ez
Since according to (4.11) we have
h2T Δh (vh − ṽ)T vh − ṽT ,
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A POSTERIORI ERROR MAJORANTS FOR FEM SOLUTIONS
43
this term is estimated by similar to (4.15) expression. Combining the resulting bound with (4.4), (4.9),
(4.10), and (4.15), we come to the bound for the residual term
h2 f − σuh − ΔṽT h2T f¯ − f δT + h2T κ(uh − u)δT + Δu − vh δT
+
max (h2T κT )
T ⊂δT
z∈δ T E∈Ez
1/2
1
h
e (u )
.
he
The second term in the error estimator η, see (3.12), is present also in the finite element error
norm (3.11), the rest three terms in η are estimated in (4.3), hold also in a local version, and in (4.15)
and (4.11). Therefore, we have come to the local efficiency bound, given in the theorem below.
Theorem 2. Let u ∈ H02 (Ω, Δ2 ) be the solution of (2.1), wh = (vh , uh ) be the finite element solution satisfying (2.4), and ũ = Dh uh , ṽ = D̄h vh . Let also the triangulation be quasiuniform,
max (h2T κT ) ≤ c = const, and the assumption α) be fulfilled. Then
T ⊂Th
ηT (uh , vh , ũ, ṽ) []efem []δT + h2 f − f¯δT .
ACKNOWLEDGMENTS
The author is grateful to the reviewer for the remarks that contributed to the improvement of the text.
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