Solution of Exterior Helmholtz Problems Using Finite and Infinite Elements

This dissertation discusses methods for the computation of solutions of the Helmholtz equation in unbounded domains. Two classes of methods are considered: one in which the infinite exterior domain is truncated and finite elements are used to discretize the resultant computational domain, and another in which the exterior domain is discretized by infinite elements. For the first class of methods a generalized Robin boundary condition is imposed on the truncating surface to replace the Sommerfeld radiation condition at infinity and to ensure uniqueness of the solution. Several of these approximate radiation conditions are discussed and a comparison is presented to illustrate their efficacy. For the second class of methods finite elements are used to discretize the exterior domain out to an enclosing circle or prolate spheroid, then infinite elements are used to discretize the remaining unbounded domain. Strikingly different approximation and convergence behavior is observed depending on whether a bilinear or sesquilinear form is chosen for the variational formulation of the problem. Convergence analysis for the infinite element methods is presented for both two and three spatial dimensions. A solution method based on the idea of domain decomposition is also discussed, as are various techniques for obtaining the solution in the far field. Numerical experiments for problems of acoustic scattering by bodies of revolution convincingly demonstrate the superiority in terms of computational expense of the infinite element methods over boundary element methods.