Wavenumber-explicit analysis for the Helmholtz $h$-BEM: error estimates and iteration counts for the Dirichlet problem

Jeffrey Galkowski, Eike H. Müller, Euan A. Spence

Published 2016-08-03Version 1

We consider solving the exterior Dirichlet problem for the Helmholtz equation with the $h$-version of the boundary element method using the standard second-kind combined-field integral equations. We sharpen previously-existing results on how $h$ must decrease with $k$ to maintain $k$-independent quasi-optimality of the Galerkin solutions, and we prove new bounds on how the number of GMRES iterations must grow with $k$ in order to have the error in the iterative solution bounded independently of $k$. Despite the fact that all the integral-operator bounds used in these arguments are sharp in their $k$-dependence, numerical experiments demonstrate that although the bounds on $h$ and the number of iterations are sufficient, they are not necessary for many geometries. We prove these results by proving new, sharp bounds on norms of the Helmholtz single- and double-layer boundary integral operators as mappings from $L^2(\Gamma)\rightarrow H^1(\Gamma)$ (where $\Gamma$ is the boundary of the obstacle), and then using these in conjunction with existing results. The new $L^2(\Gamma)\rightarrow H^1(\Gamma)$ bounds are obtained using estimates on the restriction to the boundary of eigenfunctions of the Laplacian, building on recent work by the first author and collaborators.