Efficient based for the galerkin solution of multiple-scattering problems

Abstract

In this thesis we consider high-frequency multiple scattering problems in the exterior of two-dimensional smooth compact scatterers consisting of two disjoint strictly convex obstacles. The motivation in considering this problem is the lack of fast yet rigorous numerical algorithms designed for its solution. Indeed, the only algorithm designed for the solution of this multiple scattering problem is the integral equation method developed by Bruno et al. in 2005 that uses a combination of geometrical optics to determine the phases of multiple scattering iterations, Nystrom discretization to spectrally represent the unknowns, extensions of the stationary phase method to evaluate the arising integrals independent of the frequency, and a matrix free linear algebra solver. Unfortunately this algorithm is not supported with a rigorous convergence analysis. In this thesis, we take an alternative approach and develop two classes of highly ef- cient Galerkin boundary element methods extending the recent single scattering algorithms, namely the frequency-adapted Galerkin boundary element methods and change of variables Galerkin boundary element methods recently developed by Ecevit et al., to multiple scattering problems. In connection with each multiple scattering iterate, in both cases, we prove that the number of degrees of freedom necessary to obtain prescribed error tolerances independent of frequency needs increase as O(k ) (for any > 0) with increasing wavenumber k. Consequently, the theoretical developments supported with the numerical tests in this thesis ll an important gap in the literature.