Convergence and Numerics of a Multisection Method for Scattering by Three-Dimensional Rough Surfaces

Abstract

We introduce a novel multisection method for the solution of integral equations on unbounded domains. The method is applied to the rough surface scattering problem in three dimensions, in particular to a Brakhage-Werner-type integral equation for acoustic scattering by an unbounded rough surface with Dirichlet boundary condition, where the fundamental solution is replaced by some appropriate half-space Green's function. The basic idea of the multisection method is to solve an integral equation $A\varphi = f$ by approximately solving the equation $P_{\varrho}A P_{\tau}\varphi = P_{\varrho} f$ for some positive constants $\varrho, \tau$. Here $P_{\varrho}$ is a projection operator that truncates a function to a ball with radius $\varrho>0$. For a very general class of operators $A$, for which the Brakhage-Werner equation from acoustic scattering is a particular example, we will show existence of approximate solutions to the multisection equation and show that approximate solutions to the multisection equation approximate the true solution $\varphi_0$ of the operator equation $A\varphi = f$. Finally, we describe a numerical implementation of the multisection algorithm and provide numerical examples for the case of rough surface scattering in three dimensions.