Matrix Decomposition Algorithms in Orthogonal Spline Collocation for Separable Elliptic Boundary Value Problems

Fast direct methods are presented for the solution of linear systems arising in highorder, tensor-product orthogonal spline collocation applied to separable, second order, linear, elliptic partial di erential equations on rectangles. The methods, which are based on a matrix decomposition approach, involve the solution of a generalized eigenvalue problem corresponding to the orthogonal spline collocation discretization of a two-point boundary value problem. The solution of the original linear system is reduced to solving a collection of independent almost block diagonal linear systems which arise in orthogonal spline collocation applied to one-dimensional boundary value problems. The results of numerical experiments are presented which compare an implementation of the orthogonal spline collocation approach with a recently developed matrix decomposition code for solving nite element Galerkin equations.