Adaptive BEM with optimal convergence rates for the Helmholtz equation

TL;DR

This paper proves that an adaptive boundary element method for Helmholtz equations converges optimally without requiring initial mesh refinement, supported by new inverse estimates for boundary integral operators.

Contribution

It introduces an adaptive algorithm with residual error estimation for Helmholtz BEM and proves its optimal convergence rates independently of initial mesh fineness.

Findings

Convergence of the adaptive BEM is proven with optimal algebraic rates.

The method does not require a priori mesh refinement.

New local inverse estimates for Helmholtz boundary integral operators are established.

Abstract

We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine. We prove convergence of the error estimator with optimal algebraic rates, independently of the (coarse) initial mesh. As a technical contribution, we prove certain local inverse-type estimates for the boundary integral operators associated with the Helmholtz equation.