Effective numerical algorithms for the solution of algebraic systems arising in spectral methods

Abstract

This work addresses the algorithmical aspects of spectral methods for elliptic equations. We focus on the vectorization properties of widely used algorithms (e.g., direct factorization methods and the Richardson method) as well as other algorithms which are less popular within the spectral community (e.g., GMRES, CGS and Bi-CGSTAB, the latter two are variants of the conjugate gradient method for nonsymmetric systems). The GMRES, CGS and Bi-CGSTAB generally perform better than direct factorization and the Richardson methods. We show that the spectral collocation approximation in the weak form for boundary value problems with Neumann conditions is more accurate and easier to precondition than the usual strong form. We also introduce two diagonal preconditioners that dramatically reduce the condition number of the spectral matrix. Finally, we address the issue of domain decomposition algorithms, and show several results concerning accuracy and convergence of iteration-by-subdomain procedures. The numerical calculations have been performed on the CRAY X/MP-EA, the CRAY Y-MP 8/432, the IBM 3090/200S VF and NCUBE2 mod. 6401 with 16 processors.