Smoothed Combined Field Integral Equations for Exterior Helmholtz Problems

Abstract

This paper presents smoothed combined field integral equations for the solution of Dirichlet and Neumann exterior Helmholtz problems. The integral equations introduced in this paper are smooth in the sense that they only involve continuously differentiable integrands in both Dirichlet and Neumann cases. These integral equations coincide with the well-known combined field equations and are therefore uniquely solvable for all frequencies. In particular, a novel regularization of the hypersingular operator is obtained, which, unlike regularizations based on Maue's integration-by-parts formula, does not give rise to involved Cauchy principal value integrals. The smoothed integral operators and layer potentials, on the other hand, can be numerically evaluated at target points that are arbitrarily close to the boundary without severely compromising their accuracy. A variety of numerical examples in two spatial dimensions that consider three different Nyström discretizations for smooth domains and domains with corners---one of which is based on direct application of the trapezoidal rule---demonstrates the effectiveness of the proposed integral approach. In certain aspects, this work extends to the uniquely solvable Dirichlet and Neumann combined field integral equations, the ideas presented in the recent contribution R. Soc. Open Sci. 2(140520), 2015.