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Preconditioning spectral element schemes for definite and indefinite problems. (English) Zbl 0939.65128

Authors’ abstract: Spectral element schemes for the solution of elliptic boundary value problems are considered. Preconditioning methods based on finite difference and finite element schemes are implemented. Numerical experiments show that inverting the preconditioner by a single multigrid iteration is most efficient and that the finite difference preconditioner is superior to the finite element one for both definite and indefinite problems. A multigrid preconditioner is also derived from the finite difference preconditioner and is found suitable for the CGS acceleration method. It is pointed out that, for the finite difference and finite element preconditioners, CGS does not always converge to the accurate algebraic solution.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
65N06 Finite difference methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J25 Boundary value problems for second-order elliptic equations
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