Carlenzoli, C.; Gervasio, P. Effective numerical algorithms for the solution of algebraic systems arising in spectral methods. (English) Zbl 0754.65031 Appl. Numer. Math. 10, No. 2, 87-113 (1992). 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. Reviewer: H.R.Schwarz (Zürich) Cited in 4 Documents MSC: 65F10 Iterative numerical methods for linear systems 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65Y10 Numerical algorithms for specific classes of architectures Keywords:domain decomposition; Richarson method; CGS method; BI-CGSTAB method; GMRES method; CRAY X/MP-EA; CRAY Y-MP 8/432; IBM 3090/200S VF; NCUBE2; spectral collocation method; unsymmetric spectral matrices; preconditioning; vectorization; comparisons; iteration Software:CGS PDFBibTeX XMLCite \textit{C. Carlenzoli} and \textit{P. Gervasio}, Appl. Numer. 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