Optimal convergence of adaptive BEM driven by functional-type error estimators

Abstract

In the present work, we derive functional upper bounds for the potential error arising from boundary element discretizations of the Laplace-Dirichlet problem. These bounds are based on local auxiliary problems on patches of boundary vertices and the resulting a posteriori error estimator is shown to be locally equivalent to the well-studied residual error estimator. This equivalence result allows us to prove R-linear convergence of the functional a posteriori error estimator and, together with a suitable mesh-refining strategy, to establish that the potential error as well as the functional error estimator converge with optimal rates with respect to the number of boundary elements. Numerical experiments affirm the theoretical findings and illustrate the practical performance of the related adaptive algorithm driven by the proposed functional error estimator.