CONVERGENCE AND NUMERICS OF A MULTISECTIONMETHOD FOR SCATTERING BY THREE-DIMENSIONALROUGH SURFACES

摘要

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 replaced by some appropriate half-space Green's function. The basic idea of the multisection method is to solve an integral equation A_φ = f by approximately solving the equation P_QAP_τφ = P_Qffor some positive constants Q, τ. Here P_Q is a projection operator that truncates a function to a ball with radius Q > 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 φ0 of the operator equation A_φ = 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.