Estimator reduction and convergence of adaptive BEM

Estimator reduction and convergence of adaptive BEM

A posteriori error estimation and related adaptive mesh-refining algorithms have themselves proven to be powerful tools in nowadays scientific computing. Contrary to adaptive finite element methods, convergence of adaptive boundary element schemes is, however, widely open. We propose a relaxed notion of convergence of adaptive boundary element schemes. Instead of asking for convergence of the e...

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

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-singula...

Quasi-Optimal Adaptive BEM

Convergence of adaptive 3D BEM for weakly singular integral equations based on isotropic mesh-refinement

We consider the adaptive lowest-order boundary element method (ABEM) based on isotropic mesh-refinement for the weakly-singular integral equation for the 3D Laplacian. The proposed scheme resolves both, possible singularities of the solution as well as of the given data. The implementation thus only deals with discrete integral operators, i.e. matrices. We prove that the usual adaptive mesh-ref...

Residual A-Posteriori Error Estimates in BEM: Convergence of h-Adaptive Algorithms

Galerkin methods for FEM and BEM based on uniform mesh refinement have a guaranteed rate of convergence. Unfortunately, this rate may be suboptimal due to singularities present in the exact solution. In numerical experiments, the optimal rate of convergence is regained when algorithms based on a-posteriori error estimation and adaptive mesh-refinement are used. This observation was proved mathe...