Coercive Combined Field Integral Equations

Abstract

Many boundary integral equations for exterior Dirichlet- and Neumann boundary value problems for the Helmholtz equation suffer from a notorious instability for wave numbers related to interior resonances. The so-called combined field integral equations are not affected. However, if the boundary $\Gamma$ is not smooth, the traditional combined field integral equations for the exterior Dirichlet problem do not give rise to an $L^2 ({\Gamma})$-coercive variational formulation. This foils attempts to establish asymptotic quasi-optimality of discrete solutions obtained through conforming Galerkin boundary element schemes. This article presents new combined field integral equations on two-dimensional closed surfaces that possess coercivity in canonical trace spaces. The main idea is to use suitable regularizing operators in the framework of both direct and indirect methods. This permits us to apply the classical convergence theory of conforming Galerkin methods.