Matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions

Abstract

Bialecki, B. and G. Fairweather, Matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions, Journal of Computational and Applied Mathematics 46 (1993) 369–386. In recent years, several matrix decomposition algorithms have been developed for the efficient solution of the linear algebraic systems arising when finite-difference, finite-element Galerkin and spectral methods are applied to separable elliptic boundary value problems in a rectangle. The success of these methods depends on knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two-point boundary value problems in one space dimension. The primary purpose of this paper is to provide an overview of matrix decomposition algorithms and show how they can be expressed in terms of a unifying framework. Particular emphasis is placed on algorithms formulated recently by the authors for solving the linear systems arising in orthogonal spline collocation, that is, spline collocation at Gauss points. All of the methods discussed in this paper are modular and possess a great deal of natural parallelism.