Adaptive boundary element methods with convergence rates

TL;DR

This paper develops adaptive boundary element methods for various operator equations, proving their convergence rates are quasi-optimal without saturation assumptions, based on new a posteriori error estimates and inverse inequalities.

Contribution

It introduces new convergence proofs for adaptive boundary element methods that do not rely on saturation assumptions, using novel a posteriori error estimates and inverse inequalities.

Findings

Convergence rates are quasi-optimal under mild assumptions.

New a posteriori error estimates for boundary element methods.

Inverse inequalities involving boundary integral operators on refined spaces.

Abstract

This paper presents adaptive boundary element methods for positive, negative, as well as zero order operator equations, together with proofs that they converge at certain rates. The convergence rates are quasi-optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods. In particular, no saturation-type assumption is used. The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.