Estimator reduction and convergence of adaptive BEM

Authors:
Markus Aurada;Samuel Ferraz-Leite;Dirk Praetorius
Affiliations:
Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstr. 8-10, 1040 Wien, Austria;Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany;Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstr. 8-10, 1040 Wien, Austria
Venue:
Applied Numerical Mathematics
Year:
2012

Citing 13
Cited 3

A convergent adaptive algorithm for Poisson's equation

SIAM Journal on Numerical Analysis
Data Oscillation and Convergence of Adaptive FEM

SIAM Journal on Numerical Analysis
Adaptive Finite Element Methods with convergence rates

Numerische Mathematik
Averaging Techniques for the Effective Numerical Solution of Symm's Integral Equation of the First Kind

SIAM Journal on Scientific Computing
Optimality of a Standard Adaptive Finite Element Method

Foundations of Computational Mathematics
Averaging Techniques for the A Posteriori BEM Error Control for a Hypersingular Integral Equation in Two Dimensions

SIAM Journal on Scientific Computing
Convergence analysis of a conforming adaptive finite element method for an obstacle problem

Numerische Mathematik
Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method

SIAM Journal on Numerical Analysis
Simple a posteriori error estimators for the h-version of the boundary element method

Computing
Energy norm based a posteriori error estimation for boundary element methods in two dimensions

Applied Numerical Mathematics
A Posteriori Error Estimate and Adaptive Mesh Refinement for the Cell-Centered Finite Volume Method for Elliptic Boundary Value Problems

SIAM Journal on Numerical Analysis
Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators

Numerische Mathematik
Convergence of adaptive BEM for some mixed boundary value problem

Applied Numerical Mathematics

Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity

Computational Mechanics
Convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data

Journal of Computational and Applied Mathematics
Axioms of adaptivity

Computers & Mathematics with Applications

Quantified Score

Hi-index	0.00

Visualization

Abstract

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 error to zero, we only aim to prove estimator convergence in the sense that the adaptive algorithm drives the underlying error estimator to zero. We observe that certain error estimators satisfy an estimator reduction property which is sufficient for estimator convergence. The elementary analysis is only based on Dorfler marking and inverse estimates, but not on reliability and efficiency of the error estimator at hand. In particular, our approach gives a first mathematical justification for the proposed steering of anisotropic mesh-refinements, which is mandatory for optimal convergence behavior in 3D boundary element computations.