Convergence and numerics of a multisection method for scattering by three-dimensional rough surfaces. (English) Zbl 1173.65070
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.
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.
MSC:
65N38 | Boundary element methods for boundary value problems involving PDEs |
35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
76Q05 | Hydro- and aero-acoustics |
65R20 | Numerical methods for integral equations |
45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
76M15 | Boundary element methods applied to problems in fluid mechanics |