Invited Lecture - Abstract

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 (FEG), orthogonal spline collocation and spectral methods are applied to Poisson problems in the unit square. These algorithms depend on knowledge of the eigensystems of discrete second derivative operators subject to certain boundary conditions. When such an eigensystem is known for a particular method, fast Fourier transforms can be employed to solve the corresponding linear system in O(N² log N) operations on an N x N uniform partition of the unit square.

In this talk, we describe new matrix decomposition algorithms for the FEG method with piecewise Hermite bicubics and for modified spline collocation with C² bicubic splines, for various boundary conditions. For modified spline collocation, we present the results of numerical experiments which confirm the published analysis of Dirichlet and Neumann problems and indicate that similar results hold for mixed and periodic boundary conditions. These results also exhibit superconvergence phenomena not reported in earlier studies.

This is joint work with Bernard Bialecki, Andreas Karageorghis, D. Abram Lipman and Gadalia M. Weinberg.