An Alternating-Direction Sinc-Galerkin method for elliptic problems on finite and infinite domains

Abstract

Alternating-Direction Implicit (ADI) schemes are a class of very efficient algorithms for the numerical solution of differential equations. Sinc-Galerkin schemes employ a sinc basis to produce exponentially accurate approximate solutions to differential equations even in the presence of singularities. In this dissertation we begin with a broad overview of sinc methods for problems posed on both finite and infinite, one- and two-dimensional domains. We then present a variety of finite difference methods that lead to the introduction of a new Alternating-Direction Sinc-Galerkin scheme based on the classic ADI scheme for a linear matrix system. We note that when a Sinc-Galerkin method is used to solve a Poisson equation, the resulting matrix system is a Sylvester equation. We discuss ADI model problems in general and then prove that when a symmetric Sinc-Galerkin method is employed, the resulting Sylvester equation can be classified as an ADI model problem. Finally, we derive our Alternating-Direction Sinc-Galerkin (ADSG) method to solve this resulting Sylvester equation, specifying the use of a constant iteration parameter to avoid costly eigen-value computations. We end by applying ADSG to a variety of problems, comparing its performance to the standard technique that uses the Kronecker product, the Kronecker sum, and the concatenation operator.