Summary: 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.