A Refined Galerkin Error and Stability Analysis for Highly Indefinite Variational Problems

Abstract

Recently, a refined finite element analysis for highly indefinite Helmholtz problems was introduced by the second author. We generalize the analysis to the Galerkin method applied to an abstract highly indefinite variational problem. In the refined analysis, the condition for stability and a quasi‐optimal error estimate are expressed in terms of approximation properties ${\cal T}(S) \approx S$ and ${\cal T}(u+S) \approx S$. Here, u is the solution of the original variational problem, ${\cal T}$ is a certain continuous solution operator, and S is the finite dimensional test and trial space. The abstract analysis can be applied to both finite and boundary element solutions of high‐frequency Helmholtz problems. We apply the analysis to investigate the properties of the Brakhage–Werner boundary integral formulation of the Helmholtz problem, discretized by a standard Galerkin boundary element method. In the case of scattering by the unit sphere, we derive the explicit dependence of the error and of the stability condition on the wave number k. We show that $hk \lesssim 1$ is a sufficient condition for stability and a quasi‐optimal error estimate. Further, we show that the constant of quasioptimality is independent of k, which is an improvement over previously available results. Thus, the boundary element method does not suffer from the pollution effect.