High-order numerical methods for scattering problems

Scattering problems are ubiquitous in scientific and engineering applications. These problems can often be straightforwardly modeled using relatively simple linear partial differential equations, such as the Helmholtz equation. The numerical solution of these equations, however, can be challenging due to several features of the problem, such as ill-conditioning, unbounded domains, corner singularities, and oscillatory solutions. In this thesis we present novel numerical methods for the efficient handling of some of these challenges. We first consider the case of homogeneous scattering on domains with corners and introduce a new solver in this regime, based on results in the rational approximation theory literature. The solver uses an exponentially clustered sum of dipoles to resolve the singularities in the solution and a simple expansion to resolve the smooth part. We show that the convergence is root-exponential with respect to the number of degrees of freedom and illustrate the behavior of the solver in several numerical experiments. We next turn our attention to the case of inhomogeneous scattering and present a new solver for the Lippmann–Schwinger equation, an integral equation reformulation of the problem. The new solver is based on the hierarchically block separable matrix format and exploits the geometry of the discretization and problem for accelerated inversion. The solver is shown to be effective both as a direct solver and preconditioner in a number of numerical experiments. For our last contribution we consider broadband applications in homogeneous scattering. We present a new technique for accelerating existing direct solvers in this regime. The technique works by computing basis matrices for all of the low-rank blocks in a rank-structured matrix format and leverages the observation that the resulting ranks are often comparable to those at just the highest frequency. Again, the viability of the method is illustrated in several numerical experiments.