Convergence of Adaptive BEM and Adaptive FEM-BEM Coupling for Estimators Without h-Weighting Factor

Abstract

Abstract:We analyze adaptive mesh-re ning algorithms in the frame of boundary element methods (BEM) and the coupling of nite 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.

Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations

“…For standard BEM with piecewise polynomials, [18] shows reliability and efficiency for , whereas [16] proves reliability. [25, Section 5.4] proves optimal convergence of adaptive -refinement for , while the estimate as well as plain convergence for -based adaptivity is analyzed in [26] . The transfer of the mentioned results from standard BEM to adaptive IGABEM leaves interesting and challenging questions for future research.…”

“…However, it is questionable if an analogous version of the reduction property on refined element domains [25, (A2)] , can be proved for the Faermann estimator . Indeed, this is yet an open problem even for standard BEM with piecewise polynomials; see [26] , where at least convergence of an -adaptive algorithm is analyzed. For the weighted-residual error estimator proposed in [16] , the axioms of [25] are satisfied for standard BEM with piecewise polynomials, see [25, Section 5.4] .…”

Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations

“…We aim to apply Lemma 6.1 and show in the following that (A1)–(A2) hold for all even with and that (A3) holds for all . Then, Lemma 6.1 shows convergence (3.10) of the Faermann estimator.Of Lemma 6.1 follows immediately from [24, Theorem 4.3], where the constant depends only on , and .Can be proved exactly as in [23, Section 2.4] as is efficient (see [25, Theorem 3.1]) and has a semi-norm structure. The constant depends only on .…”

Adaptive BEM for elliptic PDE systems, part I: abstract framework, for weakly-singular integral equations

“…The aim of this section is to formulate an adaptive algorithm (Algorithm 3.3) for conforming BEM discretizations of our model problem (24), where adaptivity is driven by the residual a posteriori error estimator (see (37) below). We identify conditions for the underlying meshes, the mesh-refinement, as well as the boundary element spaces which ensure that the residual error estimator is reliable and fits into the general framework of [6] and which hence guarantee optimal convergence behavior of the adaptive algorithm.…”

“…However, due to the lack of an h-weighting factor, it is unclear whether the reduction property (E2) of Section 4.2 is satisfied. [24,Theorem 7] proves at least plain convergence of even for f ∈ H 1/2 ( ) D if one uses piecewise constants on affine triangulations of as ansatz space. The proof immediately extends to our current situation, where the assumptions (M1)-(M5), (R2)-(R3), and (S1)-(S2) are employed.…”