Adaptive boundary element methods for regularized combined field integral equations

TL;DR

This paper develops adaptive boundary element methods with a posteriori error estimators for regularized combined field integral equations, effectively handling resonances and irregular boundaries in Helmholtz problems.

Contribution

It introduces a novel adaptive algorithm with proven convergence for regularized combined field integral equations on Lipschitz domains.

Findings

Adaptive mesh refinement achieves optimal convergence rates.

Regularized equations outperform standard ones near resonances.

Numerical results confirm effectiveness on irregular domains.

Abstract

While the exterior Helmholtz problem with Dirichlet boundary conditions is always well-posed, the associated standard boundary integral equations are not if the squared wavenumber agrees with an eigenvalue of the interior Dirichlet problem. Combined field integral equations are not affected by this spurious resonances but are essentially restricted to sufficiently smooth boundaries. For general Lipschitz domains, the latter integral equations are applicable through suitable regularization. Under fairly general assumptions on the regularizing operator, we propose {\sl a posteriori} computable error estimators for corresponding Galerkin boundary element methods of arbitrary polynomial degree. We show that adaptive mesh-refining algorithms steered by these local estimators converge at optimal algebraic rate with respect to the number of underlying boundary mesh elements. In particular, we consider mixed formulations involving the inverse Laplace--Beltrami as regularizing operator. Numerical examples highlight that in the vicinity of spurious resonances the proposed adaptive algorithm is significantly more performant when applied to the regularized combined field equation rather than the standard one.