Convergence of adaptive BEM and adaptive FEMBEM coupling for estimators without weighting factor
Abstract.
We analyze adaptive meshrefining algorithms in the frame of boundary element methods (BEM) and the coupling of finite elements and boundary elements (FEMBEM). Adaptivity is driven by the twolevel error estimator proposed by Ernst P. Stephan, Norbert Heuer, and coworkers in the frame of BEM and FEMBEM or by the residual error estimator introduced by Birgit Faermann for BEM for weaklysingular 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 weightedresidual error estimators for which recently convergence with quasioptimal algebraic rates has been derived.
Key words and phrases:
boundary element method (BEM), FEMBEM coupling, a posteriori error estimate, adaptive algorithm, convergence2000 Mathematics Subject Classification:
65N12, 65N38, 65N30, 65N501. Introduction
A posteriori error estimation and related adaptive meshrefining 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
(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 finitedimensional 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
(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 Céatype quasioptimality and nestedness for all imply a priori convergence
(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
(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 meshsize, i.e., for some appropriate , or that is locally equivalent to a meshsize weighted error estimator.
In this work, we consider two particular error estimators whose local contributions are not weighted by the local meshsize. 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 weaklysingular 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
(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 twolevel error estimators for BEM [MSW98, MMS97, MS00, HMS01, Heu02, EH06] or the adaptive FEMBEM coupling [MS99, KMS10, GMS12, AFKP12]. The local contributions are projections of the computable error between two Galerkin solutions onto onedimensional 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 FEMBEM 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 weaklysingular integral equation associated with the Laplacian. We prove that twolevel error estimator (Theorem 6) as well as Faermann error estimator (Theorem 7) fit into the abstract framework. In Section 4, we consider the hypersingular integral equation associated with the Laplacian. We prove that the twolevel error estimator fits into the abstract framework (Theorem 11). The final Section 5 considers a nonlinear Laplace transmission problem which is reformulated by some FEMBEM coupling. We prove that the twolevel 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 biLipschitz continuous operator, i.e.,
(6) 
for all . Here, denotes the operator norm on ,
(7) 
Suppose that there exists some subspace such that for any given closed subspace and any continuous linear functional on , the Galerkin formulation
(8) 
admits a unique solution , where denotes the duality bracket between and its dual . Particularly, this implies the existence of a unique solution of
(9) 
Moreover, we suppose that there holds the Céatype estimate
(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 righthand side .
Remark 1. (i) The assumptions (6)–(10) are particularly satisfied with , , and if is Lipschitz continuous and strongly monotone in the sense
(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 LaxMilgram lemma, e.g., the symmetric BEM formulations of Section 3–4.
2.2. Adaptive algorithm
We shall assume that is a finitedimensional 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
(12) 
is a computable a posteriori error estimator. Its local contributions measure, at least heuristically, the error locally on each element .
Algorithm 2.
Input: Righthand 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
(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
(14) 
with local contributions . For all , we suppose that there exists some superset which satisfies the following three assumptions (A1)–(A3):

is a local lower bound of : There is a constant such that for all holds
(15) 
is contractive on : There is a constant such that for all and all holds
(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 welldefined for all , but only on a dense subset , we require the following additional assumption:

is stable on with respect to : There is a constant such that for all and holds
(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 twolevel error estimator for BEM [MSW98, MMS97, MS00, HMS01, Heu02, EH06, EFLFP09, EFGP13, AFF14] and the FEMBEM coupling [MS99, GMS12, AFKP12]. In either case, denotes some weightedresidual error estimator, see [CS95b, CS96, Car97, CMS01, CMPS04] for BEM and [CS95a, GMS12, AFF13a] for the FEMBEM coupling.
Necessity of (A3). In these cases, the weightedresidual error estimator imposes additional regularity assumptions on the given righthand side . For instance, the weightedresidual error estimator for the weaklysingular 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 twolevel estimators, (A1) has first been observed in [CF01, CMPS04] for BEM and [AFKP12] for the FEMBEM 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 meshsize function and refinement of marked elements as well as appropriate inversetype estimates, where we shall build on the recent developments of [AFF12]; see e.g. the proof of Theorem 6.
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
(20) 
exists in . Moreover, it holds
(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
Next, we multiply (A2) by and observe
(22) 
with . Let . Because of the boundedness of , we can hence choose and such that
for all and . Together with (22), this shows
(23) 
Let be accumulation points of . First, choose and such that . With (23), this implies
Second, choose and such that to derive
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.
Proof.
Proposition 5.
3. Weaklysingular integral equation
3.1. Model problem
We consider the weaklysingular integral equation
(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,
(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 simplelayer integral operator is a continuous linear operator between the fractionalorder 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 simplelayer integral operator is also elliptic
(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
(27a)  
with being the twodimensional surface measure, whereas for , we impose uniform boundedness of the local meshratio  
(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. Weightedresidual error estimator
According to the Galerkin formulation (8), the residual has piecewise integral mean zero, i.e.,
(28) 
Suppose for the moment that the righthand side has additional regularity . Since is an isomorphism with additional stability for all (We note that is not isomorphic for and .), a Poincarétype inequality in shows
(29) 
see [CS95b, CS96, Car97, CMS01]. Here, denotes the surface gradient, and is the local meshwidth function defined pointwise almost everywhere by for all . Overall, this proves the reliability estimate
(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 weightedresidual error estimator can be localized via
(31) 
Recently, convergence of Algorithm 2 has been shown even with quasioptimal 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. Twolevel error estimator
In the frame of weaklysingular integral equations (24), the twolevel 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 finemesh functions which satisfy and . We note that usually for and for . Typical choices are shown in Figure 1. Then, the local contributions of the twolevel error estimator from [MSW98, MMS97, HMS01, EH06, EFLFP09] read
(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 weightedresidual error estimator from (31), the twolevel error estimator is welldefined under minimal regularity of the given righthand side.
The twolevel estimator is known to be efficient [MSW98, MMS97, HMS01, EH06, EFLFP09]
(33) 
while reliability
(34) 
holds under [MSW98, MMS97, HMS01, EH06] and is even equivalent to [EFLFP09] the saturation assumption
(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 meshrefining 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 meshrefinement strategies for ).
Theorem 6.
Suppose that the twolevel error estimator (32) is used for marking (13). Suppose that the meshrefinement 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
(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 weightedresidual error estimator from (31) and for all righthand sides . Afterward, assumption (A3) is shown for all , and the final claim then follows from density of within .
Proof of Theorem 6.
For given righthand side , the weightedresidual error estimator from (31) is welldefined.
Note that shape regularity (27) implies for the pointwise equivalence
(37) 
where . In the spirit of [CKNS08], we hence use the modified meshwidth function defined pointwise almost everywhere by and note that for . Then, we consider an equivalent weightedresidual error estimator given by
(38) 
It has first been noted in [CF01, Theorem 8.1] for 2D that
(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
(40) 
With the inverse estimate from [GHS05, Theorem 3.6] and norm equivalence, we obtain
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
where the hidden constant depends only on . This proves that the quotient on the righthand 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 meshsize function. Note that each marked element is refined and that the meshsize sequence is pointwise decreasing. With , this implies the pointwise estimate
where denotes the characteristic function of the set . Hence, the estimator from (38) satisfies