Adaptive methods for time domain boundary integral equations for acoustic scattering

Abstract

This thesis is concerned with the study of transient scattering of acoustic waves by an obstacle in an infinite domain, where the scattered wave is represented in terms of time domain boundary layer potentials. The problem of finding the unknown solution of the scattering problem is thus reduced to the problem of finding the unknown density of the time domain boundary layer operators on the obstacle’s boundary, subject to the boundary data of the known incident wave. Using a Galerkin approach, the unknown density is replaced by a piecewise polynomial approximation, the coefficients of which can be found by solving a linear system. The entries of the system matrix of this linear system involve, for the case of a two dimensional scattering problem, integrals over four dimensional space-time manifolds. An accurate computation of these integrals is crucial for the stability of this method. Using piecewise polynomials of low order, the two temporal integrals can be evaluated analytically, leading to kernel functions for the spatial integrals with complicated domains of piecewise support. These spatial kernel functions are generalised into a class of admissible kernel functions. A quadrature scheme for the approximation of the two dimensional spatial integrals with admissible kernel functions is presented and proven to converge exponentially by using the theory of countably normed spaces. A priori error estimates for the Galerkin approximation scheme are recalled, enhanced and discussed. In particular, the scattered wave’s energy is studied as an alternative error measure. The numerical schemes are presented in such a way that allows the use of non-uniform meshes in space and time, in order to be used with adaptive methods that are based on a posteriori error indicators and which modify the computational domain according to the values of these error indicators. The theoretical analysis of these schemes demands the study of generalised mapping properties of time domain boundary layer potentials and integral operators, analogously to the well known results for elliptic problems. These mapping properties are shown for both two and three space dimensions. Using the generalised mapping properties, three types of a posteriori error estimators are adopted from the literature on elliptic problems and studied within the context of the two dimensional transient problem. Some comments on the three dimensional case are also given. Advantages and disadvantages of each of these a posteriori error estimates are discussed and compared to the a priori error estimates. The thesis concludes with the presentation of two adaptive schemes for the two dimensional scattering problem and some corresponding numerical experiments.