Effective numerical algorithms for the solution of algebraic systems arising in spectral methods
The authors first present the weak formulation of the spectral collocation method for elliptic boundary value problems on a rectangular domain with Dirichlet and Neumann conditions. If Legendre nodes and weights are used it is shown that the weak formulation naturally leads to symmetric matrices, whereas the more popular approach by the pointwise strong formulation yields unsymmetric spectral matrices. Next the solution of the algebraic systems with symmetric and unsymmetric spectral matrices is considered and especially the aspects of preconditioning and vectorization are discussed. The effects and results are outlined for several examples, and comparisons are made between the methods of Richardson, CGS, BI-CGSTAB and GMRES obtained on four different supercomputers. Finally, the multidomain approach is considered in which the given domain is splitted into subregions, where the spectral method is combined with an iteration with respect to interface conditions.
scientific article; zbMATH DE number 89534
scientific article