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 high-order, tensor-product orthogonal spline collocation applied to separable, second order, linear, elliptic partial differential 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 finite element Galerkin equations.