On optimal L^{2}- and surface flux convergence in FEM (extended version)

On optimal L2- and surface flux convergence in FEM (extended version)

Abstract

We show that optimal -convergence in the finite element method on quasi-uniform meshes can be achieved if, for some , the boundary value problem has the mapping property for . The lack of full elliptic regularity in the dual problem has to be compensated by additional regularity of the exact solution. Furthermore, we analyze for a Dirichlet problem the approximation of the normal derivative on the boundary without convexity assumption on the domain. We show that (up to logarithmic factors) the optimal rate is obtained.

Keywords:
a priori bounds duality argument reentrant corners
7

1 Introduction

The finite element method (FEM) is a widely used numerical technique for approximating solutions of boundary value problems. It is based on approximating the solution by piecewise polynomials of degree . In the classical case of second order elliptic equations with an -coercive bilinear form, the method is of optimal convergence order in the -norm. An important tool for the convergence analysis in other norms such as the -norm are duality arguments (“Nitsche trick”). The textbook procedure for optimal order convergence in is to exploit full elliptic regularity for the dual problem. Conversely, this procedure suggests a loss of the optimal convergence rate in if -regularity fails to hold. This occurs, for example, in polygonal domains with reentrant corners.

Nevertheless, it is possible to recover the optimal convergence rate in , if the exact solution has additional regularity to compensate for the lack of full regularity of the dual problem. More precisely: In this note, we consider a setting where an elliptic shift theorem holds for both the dual and the bidual problem in the range for some (see Assumption 1.1) and show that if the solution is in the Sobolev space , then the extra regularity can be exploited to recover the optimal convergence rate in (up to a logarithmic factor in the lowest order case ).

In the second part of this note, we consider the convergence in of the normal derivative on the boundary. We show that the optimal rate (up to a logarithmic factor in the lowest order case) can be achieved, if the solution is sufficiently smooth. The proof is based on a local error analysis of the FEM as discussed, e.g., in wahlbin91 (); wahlbin95 (). Here, we extract error bounds for the flux on the boundary from an optimal FEM estimate on a strip of width near the boundary. Although we present the convergence of the flux for an -conforming discretization, the techniques are applicable to mixed methods, melenk-rezaijafari-wohlmuth14 (), FEM-BEM coupling, melenk-praetorius-wohlmuth14 (), and mortar and DG methods, melenk-wohlmuth12 (); waluga-wohlmuth14 (). In fact, the results of the present work lead to a sharpening of melenk-wohlmuth12 (), where convexity of the domain was assumed to avoid the analysis of a suitable additional dual problem. The techniques employed here are in part similar to those developed in melenk-wohlmuth12 (). Nevertheless, they are also significantly different since we have opted to forego the direct use of anisotropic norms and instead rely on weighted Sobolev norms and the embedding result of Lemma 1.

The analysis of the optimal convergence of fluxes has attracted some attention recently. Besides our own contributions melenk-praetorius-wohlmuth14 (); melenk-rezaijafari-wohlmuth14 (); melenk-wohlmuth12 (), we mention the works apel-pfefferer-roesch12 (); apel-pfefferer-roesch14 (); larson-massing14 () where similar results have been obtained by different methods.

1.1 Notation

For bounded Lipschitz domains with boundary , we employ standard notation for Sobolev spaces , adams75a (); tartar07 (). We will formulate certain regularity results in terms of Besov space: for , , and the Besov space is defined by interpolation (the “real” method, also known as -method as described, e.g., in tartar07 (); triebel95 ()) as

 Bs2,q(Ω)=(H⌊s⌋(Ω),H⌈s⌉(Ω))θ,q,θ=s−⌊s⌋.

Integer order Besov spaces with are also defined by interpolation:

 Bn2,q(Ω)=(Hn−1(Ω),Hn+1(Ω))1/2,q,n∈N.

To give some indication of the relevance of the second parameter in the definition of the Besov spaces, we recall the following (continuous) embeddings:

 Hs+ε(Ω)⊂Bs2,1(Ω)⊂Hs(Ω)⊂Bs2,∞(Ω)⊂Hs−ε(Ω)∀ε>0.

Of importance will be the distance function and the regularized distance function given by

 δΓ(x):=dist(x,Γ),˜δΓ(x):=h+dist(x,Γ). (1)

Later on, the parameter will be the mesh size of the quasi-uniform triangulation. Also of importance will be neighborhoods of the boundary given by

 SD:={x∈Ω|δΓ(x)

with particular emphasis on the case .

1.2 Model problem

We let , , be a bounded Lipschitz domain with a polygonal/polyhedral boundary and let (3) be our model problem:

 −∇⋅(A(x)∇u)=f in Ω,u=0 on ∂Ω. (3)

We assume that and are sufficiently smooth. Moreover is pointwise symmetric positive definite, and for some and all . As usual, (3) is understood in a weak sense, i.e., for a right-hand side the boundary value problem (3) reads: Find such that

 a(u,v):=∫ΩA∇u⋅∇v=⟨f,v⟩∀v∈H10(Ω). (4)

We denote by the solution operator. We emphasize that the choice of boundary conditions (here: homogeneous Dirichlet boundary conditions) is not essential for our purposes. Essential, however, is the following assumption:

Assumption 1.1

There exists such that the solution operator for (4) satisfies

 ∥Tf∥H1+s0(Ω)≤C∥f∥(H1−s00(Ω))′≤C∥f∥L2(Ω).

Here and in the following , denote generic constants that do not depend on the mesh-size but possibly depend on . We also use to abbreviate .

Remark 1

The present problem is symmetric. As a consequence certain dual problems that will be needed below coincide with the primal problem. This will simplify the presentation but is not essential. Inspection of the procedure below shows that we need Assumption 1.1 for the dual problem and the bidual problem with weighted right-hand side.

Let be an affine simplicial quasi-uniform triangulation of with mesh size and the continuous space of piecewise polynomials of degree . This space has the following well-known properties:

1. Existence of a quasi-local approximation operator: The Scott-Zhang operator of scott-zhang90 () satisfies:

• If then .

• is quasi-local and stable: , where is the patch of elements sharing a node with .

• has approximation properties:

 ∥∇j(u−Ikhu)∥L2(K)≲hl+1−j∥∇l+1u∥L2(ωK),j∈{0,1},0≤l≤k. (5)
2. For every there holds

 infz∈Vh∥v−z∥H1(Ω)≤h1/2∥v∥B3/22,∞(Ω).

(This follows from property (i) and an interpolation argument using the -method).

3. The space satisfies standard elementwise inverse estimates: for integer

 |v|Hm(K)≤Ch−(m−j)|v|Hj(K)∀v∈Vh. (6)

The Galerkin method for (4) is then: Find such that

 a(uh,v)=⟨f,v⟩∀v∈Vh. (7)
Remark 2

The restriction to simplicial triangulations is not essential. Our primary motivation for this restriction is that in this case the space is known to have the above approximation properties, the inverse estimates, and moreover it has the “superapproximation property” that underlies the local error analysis as presented in (wahlbin95, , Sec. 5.4).

2 Regularity

2.1 Preliminaries

A key mechanism in our arguments that will allow us to exploit additional regularity of a function is the following embedding theorem.

Lemma 1

The following estimates hold, if is a bounded Lipschitz domain and sufficiently regular.

 ∥˜δ−1/2+εΓz∥L2(Ω) ≤ ∥δ−1/2+εΓz∥L2(Ω)≤Cε∥z∥H1/2−ε(Ω),ε∈(0,1/2], (8) ∥˜δ−1/2Γz∥L2(Ω) ≤ C|lnh|1/2∥z∥B1/22,1(Ω), (9) ∥˜δ−1/2−εΓz∥L2(Ω) ≤ Cεh−ε∥z∥B1/22,1(Ω),ε>0, (10) ∥z∥L2(Sh) ≤ Ch1/2∥z∥B1/22,1(Ω),h>0, (11) ∥z∥L2(Γ) ≤ C∥z∥B1/22,1(Ω). (12)
Proof

The estimate involving in (8) can be found, e.g., in (grisvard85a, , Thm. 1.4.4.3) and (11) is shown in (li-melenk-wohlmuth-zou10, , Lemma 2.1). The estimates (9), (10), (12) follow from 1D Sobolev embedding theorems for and locally flattening the boundary in the same way as it is done in the proof of (li-melenk-wohlmuth-zou10, , Lemma 2.1). For example, for (12) we note that a local flattening of the boundary and the 1D embedding imply . This implies the estimate by (tartar07, , Lemma 25.3). We recall that for half-spaces, the upper bound (12) can be directly found in (triebel95, , Thm. 2.9.3), see also the comment in (tartar07, , Sec. 32, Eq. (32.8)). ∎

One of several applications of Lemma 1 is that it allows us to transform negative norms into weighted -estimates:

Lemma 2

For and sufficiently regular there holds

 ∥δβΓz∥(H1/2−ε(Ω))′ ≤ Cε∥δβ+1/2−εΓz∥L2(Ω),−1<β≤0, (13) ∥˜δ−1Γz∥(B1/22,1(Ω))′ ≤ C|lnh|1/2∥˜δ−1/2Γz∥L2(Ω). (14)
Proof

Firstly, we show (13):

 ∥δβΓz∥(H1/2−ε(Ω))′ = supv∈H1/2−ε(Ω)⟨δβΓz,v⟩∥v∥H1/2−ε(Ω) = supv∈H1/2−ε(Ω)⟨δβ+1/2−εΓz,δ−1/2+εΓv⟩∥v∥H1/2−ε(Ω)≤Cε∥δβ+1/2−εΓz∥L2(Ω),

where, in the last step, we employed (8) of Lemma 1. Secondly, (14) follows by the same type of arguments, where the application of (8) is replaced with that of (9). ∎

2.2 Regularity

We recall the following variant of interior regularity of elliptic problems:

Lemma 3

Let be a bounded Lipschitz domain and , , solve

 −∇⋅(A∇z)=f in Ω.

Then, for a constant depending only on , , , and

 ∥δ1−βΓ∇2z∥L2(Ω)≤C(∥δ1−βΓf∥L2(Ω)+∥z∥H1+β(Ω)).
Proof

The upper bound follows from local interior regularity for elliptic problems (see (morrey66, , Lemma 5.7.2) or (gilbarg-trudinger77a, , Thm. 8.8)) and a Besicovitch covering argument, see, e.g., (evans98, , Section 1.5.2) and (melenk02, , Chapter 5). We refer also to (khoromskij-melenk03, , Lemma A.3) where a closely related result is worked out in detail. ∎

Refined regularity for polygons and polyhedra

It is worth pointing out that neither the structure of the boundary nor the kind of boundary conditions play a role in Lemma 3. One possible interpretation of Lemma 3 is that could lose the -regularity anywhere near . For certain boundary conditions such as homogeneous Dirichlet conditions and piecewise smooth geometries the solution fails to be in only near the points of nonsmoothness of the geometry. With methods similar to those of Lemma 3 one can show the following, stronger result:

Lemma 4

Let be a (curvilinear) polygon in 2D or a (curvilinear) polyhedron in . Denote by the set of all vertices of in 2D and the set of all edges of in 3D. Let be the distance from . Let , , solve (3). Then, for a constant depending only on , , , and ,

 ∥δ1−βE∇2z∥L2(Ω)≤Cβ(∥δ1−βEf∥L2(Ω)+∥z∥H1+β(Ω)).
Proof

Follows from local considerations as in Lemma 3. The novel aspect is the behavior near the boundary away from the vertices (in 2D) and the edges (in 3D). This is achieved with an adapted covering theorem of the type described in Theorems A.1, A.2. The key feature of these coverings is that they allow us to reduce the considerations to balls and stretched balls (with fixed ) with and the following properties: either with or and is a half-ball. Local elliptic regularity assertions can then be employed for each ball . ∎

Lemma 4 assumes that a loss of -regularity occurs at any point of non-smoothness of . However, the set of “singular” vertices or edges can be further reduced. For example, in 2D for , it is well-known that only the vertices of with interior angle greater than lead to a loss of full -regularity. It will therefore be useful to introduce the closed set of boundary points associated with a loss of -regularity. Before introducing this set, we point out that this set is a subset of the vertices and edges:

Definition 1 (H2-regular part and singular part of the boundary)

Let be a polygon (in 2D) or a polyhedron (in 3D) with vertices and edges .

1. A vertex of is said to be -regular, if there is a ball of radius such that the solution of (3) satisfies whenever together with the a priori estimate .

2. In 3D, an edge of with endpoints , is said to be -regular if the following condition is satisfied: There is such that for the neighborhood of the edge we have the regularity assertion for the solution of (3) whenever together with the a priori estimate .

Denote by the set of -regular vertices and by the set of -regular edges. Correspondingly, let and be the set of vertices and edges, respectively, associated with a loss of -regularity. Define the singular set as

 Ms:=As⋃Es⊂Γ. (15)

With the notion of the singular set in hand, we can formulate the following regularity result:

Lemma 5

Let be a polygon or a polyhedron. Let be the singular set as defined in Definition 1. Then the following is true for any solution of (3): If for some , then with , there holds for some depending only on , , , and ,

 ∥δ1−βMs∇2z∥L2(Ω)≤Cβ(∥δ1−βMsf∥L2(Ω)+∥z∥H1+β(Ω)).
Proof

The proof is based on local considerations as in Lemma 4. We recall that not all vertices and edges (in 3D) are included in the singular set . This is accounted for by a further refinement of the covering employed. We restrict ourselves to the 3D situation. Using finite coverings provided by Theorem A.2, one may restrict the attention to balls and stretched balls (with fixed ) with where one of the following additional properties is satisfied: a) with ; b) and is a solid angle; c) and is a dihedral angle; d) lies in the interior of a face and is a half-ball. We emphasize that we do not need to consider balls with or since the covering provided by Theorem A.2 is such that for every such there is a neighborhood of that is covered by (countably many) balls whose radii tend to as their centers approach .

Shift theorems for locally supported right-hand sides

We have the following continuity results for the solution operator for our model problem (3):

Lemma 6

Let Assumption 1.1 be valid. Then satisfies

 ∥Tf∥B3/22,∞(Ω) ≤ C∥f∥(B1/22,1(Ω))′, (16) ∥Tf∥H3/2+ε(Ω) ≤ Cε∥δ1/2−εΓf∥L2(Ω),0<ε≤s0−1/2. (17)

In particular, if with , then

 ∥Tf∥B3/22,∞(Ω) ≤ Ch1/2∥f∥L2(Ω), (18) ∥Tf∥H3/2+ε(Ω) ≤ Cεh1/2−ε∥f∥L2(Ω),0<ε≤s0−1/2. (19)
Proof

We follow the arguments of (melenk-wohlmuth12, , Lemma 5.2). The starting point for the proof of (16) is that interpolation and Assumption 1.1 yield with

 T:((H1−s00(Ω))′,(H10(Ω))′)θ,∞→(H1+s0(Ω),H1(Ω))θ,∞=B1+s0(1−θ)2,∞(Ω).

Next, we recognize as in (melenk-wohlmuth12, , Lemma 5.2) (cf. (triebel95, , Thm. 1.11.2) or (tartar07, , Lemma 41.3))

 ((H1−s00(Ω))′,(H10(Ω))′)θ,∞ =((H1−s00(Ω),H10(Ω))θ,1)′ ⊃((H1−s0(Ω),H1(Ω))θ,1)′=(B1−s0(1−θ)2,1(Ω))′.

Setting , we get and . The assertion (17) follows from the Assumption 1.1 and (13) with . For the bound (18), we argue as in the proof of Lemma 2 and use (11), see also (melenk-wohlmuth12, , Lemma 5.2). Finally, the proof of (19) follows from (17) and the assumed support properties of . ∎

We will also require mapping properties of the solution operator in weighted spaces:

Lemma 7

Let Assumption 1.1 be valid. Then for

 ∥T(˜δ−1Γv)∥B3/22,∞(Ω) ≤ C|lnh|1/2∥˜δ−1/2Γv∥L2(Ω), (20) ∥T(˜δ−1Γv)∥H3/2+ε(Ω) ≤ Cεh−ε∥˜δ−1/2Γv∥L2(Ω),ε∈(0,s0−1/2], (21) ∥T(δ−1+2εΓv)∥H3/2+ε(Ω) ≤ Cε∥δ−1/2+εΓv∥L2(Ω),ε∈(0,s0−1/2]. (22)
Proof

The results follow by combining Lemmas 2 and 6. ∎

For the analysis of the FEM error on the neighborhood , we need a refined version of interior regularity for elliptic problems. The following result is very similar to (melenk-wohlmuth12, , Lemma 5.4) and closely related to Lemma 3:

Lemma 8

Let solve the equation

 −∇⋅(A∇z)=v in Ω.

Then there exist (depending only on , and ) and (depending only on ) such that for , we have

 ∥δ1/2Γ∇2z∥L2(Ω∖Sh)≤CA[√|lnh|∥z∥B3/22,∞(Ω)+∥√δΓv∥L2(Ω∖Sc1h)]. (23)

If the right-hand side satisfies additionally and furthermore , then there are constants (depending only on , , and ) and , (depending only on ) such that for all sufficiently small :

1. If then

2. For every there holds

 ∥δαΓ∇3z∥L2(Ω∖S~ch)≤CA[∥δα−1Γ∇2z∥L2(Ω∖Sc′h)+∥A∥C1,1(Ω)∥δαΓ∇z∥L2(Ω∖Sc′h)].
3. If for some , then for some (depending on , , , and ) there holds

 ∥∇2z∥L2(Ω∖S~ch)≤CA,εh−1/2+ε∥z∥H3/2+ε(Ω).
4. If Assumption 1.1 is valid, then for some depending only on the solution operator and of (iii) we have

Proof

of (23), (i), (ii): (melenk-wohlmuth12, , Lemma 5.4) is formulated for . However, the essential property of the differential operator that is required is just interior regularity. Hence, the result also stands for the present, more general elliptic operator (with the appropriate modifications due to the fact that the coefficient is allowed to be non-constant). In the interest of generality, we have also tracked in (23) the dependence on the right-hand side , which was not done in (melenk-wohlmuth12, , Lemma 5.4). A full proof can be found in Appendix C.

Proof of (iii): This follows again by local considerations similar to those employed in the proof of (melenk-wohlmuth12, , Lemma 5.4) and the obvious bound on . A full proof can be found in Appendix C.

Proof of (iv): In view of (iii), we have to estimate . By the support properties of , the bound (19) yields . Inserting this in (iii) gives the result. ∎

3 Fem L2-error analysis

Let be the FEM approximation and denote by the FEM error. The standard workhorse is the Galerkin orthogonality

 a(e,v)=a(u−uh,v)=0∀v∈Vh. (24)

Lemma 9

Let Assumption 1.1 be valid. Assume that a function satisfies the Galerkin orthogonality

 a(z,v)=0∀v∈Vh.

Then

 ∥δ−1/2+εΓz∥L2(Ω) ≤ Cεh1/2+ε∥z∥H1(Ω),ε∈(0,s0−1/2], (25) ∥˜δ−1/2Γz∥L2(Ω) ≤ Ch1/2|lnh|1/2∥z∥H1(Ω). (26)
Proof

The proof follows standard lines. Define , which solves

 ⟨v,δ−1+2εΓz⟩=a(v,ψ)∀v∈H10(Ω).

Then we have by Galerkin orthogonality for arbitrary

 ∥δ−1/2+εΓz∥2L2(Ω) = a(z,ψ)=a(z,ψ−Iψ)≤C∥z∥H1(Ω)∥ψ−Iψ∥H1(Ω).

From (22) in Lemma 7, we have so that with the approximation properties of we get

 infIψ∈Vh∥ψ−Iψ∥H1(Ω)≤Cεh1/2+ε∥δ−1/2+εΓz∥L2(Ω).

This shows (25). For (26), we proceed similarly using the regularity assertion (20) and the approximation property of . ∎

Corollary 1

Let Assumption 1.1 be valid and the solution be in , . Then the FEM error satisfies for

 ∥δ−1/2+εΓe∥L2(Ω)≤Cεhμ−1/2+ε∥u∥Hμ(Ω),μ:=min{s,k+1}.

The following Theorem 3.1 shows that the optimal rate of the -convergence of the FEM can be achieved also for non-convex geometries if the solution has some additional regularity:

Theorem 3.1

Let Assumption 1.1 be valid. Let the exact solution satisfy the extra regularity . Then the FEM error satisfies

 ∥u−uh∥L2(Ω)≲hk+1∥u∥Hk+2−s0(Ω). (27)

More generally, if , , then

 ∥u−uh∥L2(Ω)≲hs−1+s0∥u∥Hs(Ω),1≤s≤k+2−s0. (28)
Proof

of (27): We proceed along a standard duality argument. To begin with, we note that the case is classical so that we may assume for the remainder of the proof. Set by our assumption . Let and let be its Galerkin approximation. Quasi-optimality and the use of (13) give us the following energy error estimate:

 ∥w−wh∥H1(Ω) ≲ infv∈Vh∥w−v∥H1(Ω)≲h1/2+ε∥w∥H3/2+ε(Ω) (29) ≲ h1/2+ε∥e∥(H1/2−ε(Ω))′≲h1/2+ε∥e∥L2(Ω).

The Galerkin orthogonalities satisfied by and and a weighted Cauchy-Schwarz inequality yield for the Scott-Zhang interpolant

 ∥e∥2L2(Ω) = a(e,w)=a(e,w−wh)=a(u−Ikhu,w−wh) (30) ≤ C∥˜δ−1/2+εΓ∇(u−Iu)∥L2(Ω)∥˜δ1/2−εΓ∇(w−wh)∥L2(Ω). (31)

We get by a covering argument and (13) of Lemma 1

 ∥˜δ−1/2+εΓ∇(u−Ikhu)∥L2(Ω) ≲ hk∥˜δ−1/2+εΓ∇k+1u∥L2(Ω) (32) ≲ hk∥∇k+1u∥H1/2−ε(Ω).

It should also be noted at this point that in (32), the weight can be considered as constant in each element . For the contribution in (31), we have to analyze the Galerkin error in more detail, which will be done with the techniques from the local error analysis of the FEM. We split into where will be selected sufficiently large below. For fixed , the -norm on can easily be bounded with (29) by

 ∥˜δ1/2−εΓ∇(w−wh)∥L2(Sch)≲h1/2−ε∥∇(w−wh)∥L2(Ω)≲h∥e∥L2(Ω). (33)

The term requires more care. Obviously, on . We have to employ the tools from the local error analysis in FEM. The Galerkin orthogonality satisfied by allows us to use techniques as described in (wahlbin95, , Sec. 5.3), which yields the following estimate for arbitrary balls with the same center (implicitly, is assumed in (34))

 ∥∇(w−wh)∥L2(Br)≲∥∇(w−Ikhw)∥L2(Br′)+1r′−r∥w−wh∥L2(Br′). (34)

By a covering argument (which requires , where is the center of the ball , and is sufficiently small) these local estimates can be combined into a global estimate of the following form, where for sufficiently small ( depends only on and the shape regularity of the triangulation but is independent of ):

 ∥δ1/2−εΓ∇(w−wh)∥L2(Ω∖Sch)≲ (35) ∥δ1/2−εΓ∇(w−Ikhw)∥L2(Ω∖Scc1h)+∥δ−1/2−εΓ(w−wh)∥L2(