Convergence of ABEM and adaptive FEM-BEM coupling

# Convergence of adaptive BEM and adaptive FEM-BEM coupling for estimators without h-weighting factor

## Abstract.

We analyze adaptive mesh-refining algorithms in the frame of boundary element methods (BEM) and the coupling of finite elements and boundary elements (FEM-BEM). Adaptivity is driven by the two-level error estimator proposed by Ernst P. Stephan, Norbert Heuer, and coworkers in the frame of BEM and FEM-BEM or by the residual error estimator introduced by Birgit Faermann for BEM for weakly-singular integral equations. We prove that in either case the usual adaptive algorithm drives the associated error estimator to zero. Emphasis is put on the fact that the error estimators considered are not even globally equivalent to weighted-residual error estimators for which recently convergence with quasi-optimal algebraic rates has been derived.

###### Key words and phrases:
boundary element method (BEM), FEM-BEM coupling, a posteriori error estimate, adaptive algorithm, convergence
###### 2000 Mathematics Subject Classification:
65N12, 65N38, 65N30, 65N50

## 1. Introduction

A posteriori error estimation and related adaptive mesh-refining algorithms are one important basement of modern scientific computing. Starting from an initial mesh and based on a computable a posteriori error estimator, such algorithms iterate the loop

 ~{}solve~{}→~{}% estimate~{}→~{}mark~{}→~{}refine~{} (1)

to create a sequence of successive locally refined meshes , corresponding discrete solutions , as well as a posteriori error estimators . We consider the frame of conforming Galerkin discretizations, where is linked to a finite-dimensional subspace of a Hilbert space with corresponding Galerkin solution , where successive refinement guarantees nestedness for all .

Convergence of this type of adaptive algorithm in the sense of

 limℓ→∞∥u−Uℓ∥H=0 (2)

has first been addressed in [BV84] for 1D FEM and [Dör96] for 2D FEM. We note that already the pioneering work [BV84] observed that validity of some Céa-type quasi-optimality and nestedness for all imply a priori convergence

 limℓ→∞∥U∞−Uℓ∥H=0, (3)

where is the unique Galerkin solution in . From a conceptual point of view, it thus only remained to identify the limit . Based on such an a priori convergence result (3), a general theory of convergence of adaptive FEM is devised in [MSV08, Sie11], where the analytical focus is on estimator convergence

 limℓ→∞μℓ=0. (4)

Moreover, the recent work [CFPP14] gives an analytical frame to guarantee convergence with optimal convergence rates; see also the overview article [FFH14] for the current state of the art of adaptive BEM. Throughout, it is however implicitly assumed that the local contributions of the error estimator are weighted with the local mesh-size, i.e., for some appropriate , or that is locally equivalent to a mesh-size weighted error estimator.

In this work, we consider two particular error estimators whose local contributions are not weighted by the local mesh-size. We devise a joint analytical frame which proves estimator convergence (4). First, we let be the Faermann error estimator [Fae00, Fae02, CF01] for BEM for the weakly-singular integral equation with . The local contributions of are overlapping -seminorms of the residual . The striking point of is that it is the only a posteriori BEM error estimator which is known to be both reliable and efficient without any further assumptions on the given data, i.e., it holds

 C−1effμℓ≤∥u−Uℓ∥H≤Crelμℓ (5)

with -independent constants . We note that is not equivalent to an -weighted error estimator which prevents to follow the arguments from the available literature.

Second, our analysis covers the two-level error estimators for BEM [MSW98, MMS97, MS00, HMS01, Heu02, EH06] or the adaptive FEM-BEM coupling [MS99, KMS10, GMS12, AFKP12]. The local contributions are projections of the computable error between two Galerkin solutions onto one-dimensional spaces, spanned by hierarchical basis functions. These estimators are known to be efficient. On the other hand, reliability is only proven under an appropriate saturation assumption which is even equivalent to reliability for the symmetric BEM operators [EFLFP09, EFGP13, AFF14]. However, such a saturation assumption is formally equivalent to asymptotic convergence of the adaptive algorithm [FLP08] which cannot be guaranteed mathematically in general and is expected to fail on coarse meshes.

Outline. The remainder of the paper is organized as follows: In Section 2, we introduce an abstract frame which covers both BEM as well as the FEM-BEM coupling. We formally state the adaptive loop (Algorithm 2). Under three assumptions on the error estimator which are later verified for the particular model problems, we prove that the adaptive loop drives the underlying error estimator to zero (Proposition 4 and Proposition 5). Section 3 treats the weakly-singular integral equation associated with the Laplacian. We prove that two-level error estimator (Theorem 6) as well as Faermann error estimator (Theorem 7) fit into the abstract framework. In Section 4, we consider the hyper-singular integral equation associated with the Laplacian. We prove that the two-level error estimator fits into the abstract framework (Theorem 11). The final Section 5 considers a nonlinear Laplace transmission problem which is reformulated by some FEM-BEM coupling. We prove that the two-level error estimator fits into the abstract framework as well (Theorem 13).

Notation. Associated quantities are linked through the same index, i.e., is the discrete solution with respect to the discrete space which corresponds to the triangulation . Throughout, the star is understood as general index and may be accordingly replaced by the level of the adaptive algorithm (e.g., ) or by the infinity symbol (e.g., ). All constants as well as their dependencies are explicitly given in statements and results. In proofs, we shall use to abbreviate with some generic multiplicative constant which is clear from the context. Moreover, abbreviates .

## 2. Abstract setting

### 2.1. Model problem

Let be a Hilbert space with dual space and be a bi-Lipschitz continuous operator, i.e.,

 C−1cont∥w−v∥H≤∥Aw−Av∥H∗≤Ccont∥w−v∥H (6)

for all . Here, denotes the operator norm on ,

 ∥F∥H∗=supv∈H∖{0}|⟨F,v⟩|∥v∥Hfor all F∈H∗. (7)

Suppose that there exists some subspace such that for any given closed subspace and any continuous linear functional on , the Galerkin formulation

 ⟨AU⋆,V⋆⟩=⟨F,V⋆⟩for all V⋆∈X⋆ (8)

admits a unique solution , where denotes the duality bracket between and its dual . Particularly, this implies the existence of a unique solution of

 Au=F. (9)

Moreover, we suppose that there holds the Céa-type estimate

 ∥u−U⋆∥H≤C\rm\scriptsize C\'{e% }aminV⋆∈X⋆∥u−V⋆∥H, (10)

where the constant depends only on the operator (and possibly on ). To be precise, we will write and in the following to indicate that resp.  are the unique solutions with respect to some given right-hand side .

Remark 1.  (i) The assumptions (6)–(10) are particularly satisfied with , , and if is Lipschitz continuous and strongly monotone in the sense

 ∥Aw−Av∥H∗≤˜Ccont∥w−v∥HandCmon∥w−v∥2H≤⟨Aw−Av,w−v⟩ (11)

for all ; see e.g. [Zei90, Section 25.4] for the corresponding proofs. In particular, this also covers linear problems in the frame of the Lax-Milgram lemma, e.g., the symmetric BEM formulations of Section 34.

(ii) The assumptions (6)–(10) are motivated by the FEM-BEM coupling formulations in Section 5.

(iii) For being linear, it is also sufficient if additionally to (6), satisfies a uniform inf-sup-condition along the sequence of discrete subspaces generated by Algorithm 2 below. ∎

We shall assume that is a finite-dimensional subspace of related to some triangulation and that is the corresponding Galerkin solution (8) for . Starting from an initial mesh , the triangulations are successively refined by means of the following realization of (1), where

 μℓ(F):=μℓ(F;Tℓ)withμℓ(F;Eℓ):=(∑T∈Eℓμℓ(F;T)2)1/2<∞for all Eℓ⊆Tℓ (12)

is a computable a posteriori error estimator. Its local contributions measure, at least heuristically, the error locally on each element .

###### Algorithm 2.

Input: Right-hand side , initial mesh with , and bulk parameter .
For iterate the following:

• Compute Galerkin solution .

• Compute refinement indicators for all .

• Determine some set of marked elements which satisfies

 θμℓ(F)2≤μℓ(F;Mℓ)2. (13)
• Generate a new mesh and hence an enriched space by refinement of at least all marked elements .

Output: Sequence of successively refined triangulations as well as corresponding Galerkin solutions and error estimators , for .

### 2.3. Auxiliary estimator and assumptions

The following convergence results of Proposition 4 and Proposition 5 require an auxiliary error estimator

 ρℓ(F):=ρℓ(F;Tℓ)% withρℓ(F;Eℓ):=(∑T∈Eℓρℓ(F;T)2)1/2<∞for all Eℓ⊆Tℓ (14)

with local contributions . For all , we suppose that there exists some superset which satisfies the following three assumptions (A1)–(A3):

1. is a local lower bound of : There is a constant such that for all holds

 μℓ(F;Mℓ)≤C???ρℓ(F;Rℓ). (15)
2. is contractive on : There is a constant such that for all and all holds

 C−1???ρℓ(F;Rℓ)2≤ρℓ(F)2−11+δρℓ+k(F)2+(1+δ−1)C???∥Uℓ+k(F)−Uℓ(F)∥2H. (16)

The constants may depend on , but are independent of the step , i.e., in particular independent of the discrete spaces and the corresponding Galerkin solutions . If is not well-defined for all , but only on a dense subset , we require the following additional assumption:

1. is stable on with respect to : There is a constant such that for all and holds

 |μℓ(F;Mℓ)−μℓ(F′;Mℓ)|≤C???∥F−F′∥H∗. (17)

### 2.4. Remarks

Some remarks are in order to relate the abstract assumptions (A1)–(A3) to the applications, we have in mind.

Choice of . Below, we shall verify that assumptions (A1)–(A3) hold with being the Faermann error estimator [Fae00, Fae02, CF01] for BEM resp.  being the two-level error estimator for BEM [MSW98, MMS97, MS00, HMS01, Heu02, EH06, EFLFP09, EFGP13, AFF14] and the FEM-BEM coupling [MS99, GMS12, AFKP12]. In either case, denotes some weighted-residual error estimator, see [CS95b, CS96, Car97, CMS01, CMPS04] for BEM and [CS95a, GMS12, AFF13a] for the FEM-BEM coupling.

Necessity of (A3). In these cases, the weighted-residual error estimator imposes additional regularity assumptions on the given right-hand side . For instance, the weighted-residual error estimator for the weakly-singular integral equation [CS95b, CS96, Car97, CMS01] requires , while the natural space for the residual is , see Section 3 for further details and discussions. Convergence (4) of Algorithm 2 for arbitrary then follows by means of stability (A3).

Verification of (A1)–(A2). For two-level estimators, (A1) has first been observed in [CF01, CMPS04] for BEM and [AFKP12] for the FEM-BEM coupling and follows essentially from scaling arguments for the hierarchical basis functions. For the Faermann error estimator and a simplified 2D BEM setting, (A1) is also proved in [CF01]. Finally, the novel observation (A2) follows from an appropriately constructed mesh-size function and refinement of marked elements as well as appropriate inverse-type estimates, where we shall build on the recent developments of [AFF12]; see e.g. the proof of Theorem 6.

Verification of (A3). Suppose that the operator is linear and is efficient

 μℓ(F)≤Ceff∥u(F)−Uℓ(F)∥Hfor all F∈H∗. (18)

Provided has a semi-norm structure, the corresponding triangle inequality yields

 μℓ(F)≤μℓ(F′)+μℓ(F−F′)≤μℓ(F′)+Ceff∥u(F−F′)−Uℓ(F−F′)∥H≤μℓ(F′)+CeffC\rm\scriptsize C% \'{e}a∥u(F−F′)∥H≤μℓ(F′)+CeffC\rm\scriptsize C% \'{e}a∥A−1∥∥F−F′∥H∗, (19)

where denotes the operator norm of , and the (bounded) inverse exists due to (6). This proves stability (A3) with .

Marking strategy. In view of optimal convergence rates, one usually asks for in (A1) and minimal cardinality of in (13). We stress, however, that this is not necessary for the present analysis, where our focus is on a first plain convergence result.

### 2.5. Abstract convergence analysis

We start with the observation that (A2) already implies convergence of the auxiliary estimator . We note that the following lemma is, in particular, independent of the marking strategy (13), i.e., we do not use any information about how the sequence is generated.

###### Lemma 3.

Suppose (A2) for some fixed . Under nestedness of the discrete spaces for all , the auxiliary estimator converges, i.e, the limit

 ρ∞(F):=limℓ→∞ρℓ(F) (20)

exists in . Moreover, it holds

 limℓ→∞ρℓ(F;Rℓ)=0. (21)
###### Proof.

First, we prove that (A2) implies boundedness of . We recall that nestedness for all in combination with the Céa lemma (10) implies that the limit exists in , see e.g. [MSV08, CP12, AFLP12] or even the pioneering work [BV84]. For and , assumption (A2) implies

 12ρk(F)2≤ρ0(F)2+2C???supk∈N0∥U0−Uk∥2H≤M<∞.

Next, we multiply  (A2) by and observe

 0≤ρℓ(F;Rℓ)2≲ρℓ(F)2−ρℓ+k(F)2+δρℓ(F)2+C???(δ)∥Uℓ+k(F)−Uℓ(F)∥2H (22)

with . Let . Because of the boundedness of , we can hence choose and such that

 δρℓ(F)2+C???(δ)∥Uℓ+k(F)−Uℓ(F)∥2H≤ε

for all and . Together with (22), this shows

 ρℓ(F)2−ρℓ+k(F)2≥−ε. (23)

Let be accumulation points of . First, choose and such that . With (23), this implies

 a−b≥−3ε.

Second, choose and such that to derive

 b−a≥−3ε.

Since was arbitrary, the last two estimates imply . Altogether, is a bounded sequence in with unique accumulation point. By elementary calculus, is convergent with limit . Continuity of the square root concludes (20). In particular, this and (22) prove as . ∎

###### Proposition 4.

Suppose assumptions (A1)–(A2) for some fixed . Under nestedness of the discrete spaces for all and due to the marking strategy (13), Algorithm 2 guarantees estimator convergence .

###### Proof.

The marking criterion (13) and assumption (A1) show

 θμℓ(F)2≤μℓ(F;Mℓ)2≲ρℓ(F;Rℓ)2.

Hence, the assertion follows from Lemma 3. ∎

###### Proposition 5.

Suppose that is a dense subset of such that assumptions (A1)–(A2) are satisfied for all . In addition, suppose validity of (A3). Under nestedness of the discrete spaces for all and due to the marking strategy (13), Algorithm 2 guarantees convergence for all .

###### Proof.

Let and choose such that . The marking criterion (13) as well as (A3) and (A1) show

 θμℓ(F)≤μℓ(F;Mℓ)≲μℓ(F′;Mℓ)+∥F−F′∥H∗≲ρℓ(F′;Rℓ)+ε.

Lemma 3 yields , whence

 θlimsupℓ→∞μℓ(F)≲ε.

With , elementary calculus concludes the proof. ∎

## 3. Weakly-singular integral equation

### 3.1. Model problem

We consider the weakly-singular integral equation

 Au(x)=∫ΓG(x−y)u(y)dΓ(y)=F(x)for % all x∈Γ (24)

on a relatively open, polygonal part of the boundary of a bounded, polyhedral Lipschitz domain , . For , we assume that the boundary of (a polygonal curve) is Lipschitz itself. Here,

 G(z)=−12πlog|z|resp.G(z)=14π|z|−1 (25)

denotes the fundamental solution of the Laplacian in . The reader is referred to, e.g., the monographs [HW08, McL00, SS11, Ste08] for proofs of and details on the following facts: The simple-layer integral operator is a continuous linear operator between the fractional-order Sobolev space and its dual . Duality is understood with respect to the extended -scalar product . In 2D, we additionally assume which can always be achieved by scaling. Then, the simple-layer integral operator is also elliptic

 ⟨v,Av⟩≥Cell∥v∥2˜H−1/2(Γ)for all v∈H=˜H−1/2(Γ) (26)

with some constant which depends only on . Thus, meets all assumptions of Section 2, and even defines an equivalent Hilbert norm on .

### 3.2. Discretization

Let be a -shape regular triangulation of into affine line segments for resp. plane surface triangles for . For , -shape regularity means

 supT∈T⋆diam(T)2|T|≤γ<∞ (27a) with |⋅| being the two-dimensional surface measure, whereas for d=2, we impose uniform boundedness of the local mesh-ratio diam(T)diam(T′)≤γ<∞for all T,T′∈T⋆ with T∩T′≠∅. (27b)

To abbreviate notation, we shall write for . In addition, we assume that is regular in the sense of Ciarlet for , i.e., there are no hanging nodes.

With being the space of -piecewise constant functions, we now consider the Galerkin formulation (8).

### 3.3. Weighted-residual error estimator

According to the Galerkin formulation (8), the residual has -piecewise integral mean zero, i.e.,

 ∫T(F−AU⋆(F))dΓ=0for all T∈T⋆. (28)

Suppose for the moment that the right-hand side has additional regularity . Since is an isomorphism with additional stability for all (We note that is not isomorphic for and .), a Poincaré-type inequality in shows

 ∥u(F)−U⋆(F)∥˜H−1/2(Γ)≃∥F−AU⋆(F)∥H1/2(Γ)≲∥h1/2⋆∇Γ(F−AU⋆(F))∥L2(Γ)=:η⋆(F), (29)

see [CS95b, CS96, Car97, CMS01]. Here, denotes the surface gradient, and is the local mesh-width function defined pointwise almost everywhere by for all . Overall, this proves the reliability estimate

 ∥u(F)−U⋆(F)∥˜H−1/2(Γ)≤˜Crelη⋆(F), (30)

and the constant depends only on and the -shape regularity (27) of ; see [CMS01]. In 2D, it holds that , where depends only on ; see [Car97]. In particular, the weighted-residual error estimator can be localized via

 η⋆(F)=(∑T∈T⋆η⋆(F;T)2)1/2 with η⋆(F;T)=diam(T)1/2∥∇Γ(F−AU⋆(F))∥L2(T). (31)

Recently, convergence of Algorithm 2 has been shown even with quasi-optimal rates, if is used for marking (13); see [FKMP13, FFK13a]. We stress that our approach with would also give convergence as . Since this is, however, a much weaker result than that of [FKMP13], we omit the details.

### 3.4. Two-level error estimator

In the frame of weakly-singular integral equations (24), the two-level error estimator was introduced in [MSW98]. Let denote the uniform refinement of . For each element , let denote the set of sons of . Let be a basis of with fine-mesh functions which satisfy and . We note that usually for and for . Typical choices are shown in Figure 1. Then, the local contributions of the two-level error estimator from [MSW98, MMS97, HMS01, EH06, EFLFP09] read

 μ⋆(F;T)2=D∑j=1μ⋆,j(F;T)2withμ⋆,j(F;T)=⟨F−AU⋆(F),φT,j⟩⟨AφT,j,φT,j⟩1/2. (32)

Put differently, we test the residual with the additional basis functions from . This quantity is appropriately scaled by the corresponding energy norm . Note that unlike the weighted-residual error estimator from (31), the two-level error estimator is well-defined under minimal regularity of the given right-hand side.

The two-level estimator is known to be efficient [MSW98, MMS97, HMS01, EH06, EFLFP09]

 μ⋆(F)≤Ceff∥u(F)−U⋆(F)∥˜H−1/2(Γ), (33)

while reliability

 ∥u(F)−U⋆(F)∥˜H−1/2(Γ)≤Crelμ⋆(F) (34)

holds under [MSW98, MMS97, HMS01, EH06] and is even equivalent to [EFLFP09] the saturation assumption

 ∥u(F)−ˆU⋆(F)∥A≤qsat∥u(F)−U⋆(F)∥A (35)

in the energy norm . Here, is a uniform constant, and is the Galerkin solution with respect to the uniform refinement of . The constant depends only on and -shape regularity of , while additionally depends on the saturation constant .

With the help of Proposition 4 and Proposition 5, we aim to prove the following convergence result for the related adaptive mesh-refining algorithm. Recall that for , refinement of an element does not necessarily imply that for the sons of . However, it is reasonable to assume that each marked element is refined into at least two sons which satisfy with some uniform (and for usual mesh-refinement strategies for ).

###### Theorem 6.

Suppose that the two-level error estimator (32) is used for marking (13). Suppose that the mesh-refinement guarantees uniform -shape regularity (27) of themeshes generated, as well as that all marked elements are refined into sons with with some uniform constant . Then, Algorithm 2 guarantees

 μℓ(F)→0as ℓ→∞ (36)

for all .

The claim of Theorem 6 follows from Proposition 5 as soon as we have verified the abstract assumptions (A1)–(A3). We will show (A1)–(A2) for a slight variant of the weighted-residual error estimator from (31) and for all right-hand sides . Afterward, assumption (A3) is shown for all , and the final claim then follows from density of within .

###### Proof of Theorem 6.

For given right-hand side , the weighted-residual error estimator from (31) is well-defined.

Note that -shape regularity (27) implies for the pointwise equivalence

 C−1meshdiam(T)≤|T|1/2≤diam(T)for all T∈Tℓ, (37)

where . In the spirit of [CKNS08], we hence use the modified mesh-width function defined pointwise almost everywhere by and note that for . Then, we consider an equivalent weighted-residual error estimator given by

 C−1/2meshηℓ(F;T)≤ρℓ(F;T):=∥˜h1/2ℓ∇Γ(F−AUℓ(F))∥L2(T)≤ηℓ(F;T) (38)

It has first been noted in [CF01, Theorem 8.1] for 2D that

 μℓ,j(F;T)≤Clocηℓ(F;T)for % all T∈Tℓ, (39)

where the constant depends only on -shape regularity of , and the proof transfers to 3D as well. For completeness, we include the short argument: With , we infer

 μℓ,j(F;T)=⟨F−AUℓ(F),φT,j⟩∥φT,j∥A≤∥h−1/2ℓ(F−AUℓ(F))∥L2(T)∥h1/2ℓφT,j∥L2(T)∥φT,j∥A. (40)

With the inverse estimate from [GHS05, Theorem 3.6] and norm equivalence, we obtain

 ∥h1/2ℓφT,j∥L2(T)=∥h1/2ℓφT,j∥L2(Γ)≲∥φT,j∥˜H−1/2(Γ)≃∥φT,j∥A,

where the hidden constants depend only on and -shape regularity (27) of . We note that the assumption together with the approximation result of [CP06, Theorem 4.1] also proves the converse estimate

 ∥φT,j∥A≃∥φT,j∥˜H−1/2(Γ)≲∥h1/2ℓφT,j∥L2(T),

where the hidden constant depends only on . This proves that the quotient on the right-hand side of (40) remains bounded. Due to (28), the Poincaré estimate yields . This concludes (39). Together with (38), this proves (A1) with and .

The verification of (A2) hinges on the use of the equivalent mesh-size function. Note that each marked element is refined and that the mesh-size sequence is pointwise decreasing. With , this implies the pointwise estimate

 ˜hℓ−˜hℓ+k≥˜hℓ−˜hℓ+1≥(1−q)˜hℓχ⋃Rℓfor all ℓ,k∈N,

where denotes the characteristic function of the set . Hence, the estimator from (38) satisfies

 (1−q)ρℓ(F;Rℓ)2 =(1−q)∫⋃Rℓ˜hℓ|∇Γ(F−AUℓ(F))|2dΓ ≤