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Fourier analysis of GMRES(m) preconditioned by multigrid. (English) Zbl 0967.65101
The paper presents convergence estimates for the restarted generalized minimal residual (GMRES) methods preconditioned with multigrid when used for the solution of linear systems arising from the discretization of scalar transport equations by finite-difference methods. The estimates are based on a rigorous Fourier analysis and apply for simple uniform Cartesian grids. The theoretical etimates are tested as against numerical findings for simple model problems. It turns out that optimal multigrid methods cannot be further improved by additional outer GMRES iterations, however, if an optimal multigrid procedure is not easily possible to derive, its use as a preconditioner within the restarted GMRES method appears to be an attractive alternative, since convergence rates comparable to those for optimal multigrid can be achieved.

MSC:
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
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