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

TL;DR

This paper proves the convergence of adaptive BEM and FEM-BEM algorithms driven by specific error estimators, even without the common h-weighting factor, ensuring the estimators tend to zero.

Contribution

It demonstrates convergence of adaptive algorithms using certain error estimators without requiring h-weighting factors, a novel aspect in boundary element methods.

Findings

Adaptive algorithms drive error estimators to zero.

Convergence holds without global equivalence to weighted-residual estimators.

Results apply to both BEM and FEM-BEM coupling methods.

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.