Infinite element methods for Helmholtz equation problems on unbounded domains

In this dissertation two methods for improving the conditioning of infinite element stiffness matrices are proposed and examined for one-dimensional, two-dimensional and three-dimensional infinite elements. The first method is a preconditioning technique which is based on a Gram-Schmidt-like transformation induced by general bilinear and sesquilinear forms. Although this preconditioning method can be applied to many types of infinite elements, it will be applied here to improve the conditioning of the popular multipole infinite element. The second method improves the numerical conditioning of infinite element stiffness matrices by replacing the characteristic (eigenfunction) basis functions which have global support with basis functions which have local support. These new infinite elements employing basis functions with compact support are called piecewise multipole infinite elements. In this dissertation the conditioning and convergence properties of these new infinite elements are presented in solving one-dimensional and two-dimensional elliptic problems on unbounded domains as well as two-dimensional and three-dimensional exterior problems for the Helmholtz equation. The improved conditioning and convergence properties of these new infinite elements are demonstrated in various numerical examples