1998
DOI: 10.1137/s1064827596302825
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An Evaluation of Parallel Multigrid as a Solver and a Preconditioner for Singularly Perturbed Problems

Abstract: In this paper we try to achieve h-independent convergence with preconditioned GMRES ([Y. Saad and M. H. Schultz, SIAM J. Sci. Comput., 7 (1986), pp. 856-869]) and BiCGSTAB ([H. A. Van der Vorst, SIAM J. Sci. Comput., 13 (1992), pp. 63-644]) for two-dimensional (2D) singularly perturbed equations. Three recently developed multigrid methods are adopted as a preconditioner. They are also used as solution methods in order to compare the performance of the methods as solvers and as preconditioners.Two of the multig… Show more

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Cited by 50 publications
(53 citation statements)
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“…Most eigenvalues are clustered around zero for all cases, which is advantageous for the Krylov methods [18]. As b !…”
Section: Spectral Analysismentioning
confidence: 98%
“…Most eigenvalues are clustered around zero for all cases, which is advantageous for the Krylov methods [18]. As b !…”
Section: Spectral Analysismentioning
confidence: 98%
“…As we will subsequently show through numerical experiments in section 7.2, the convergence performance of the GMRES solver with simple pre-conditioner (such as Jacobi or ILU) deteriorates dramatically with increasing mesh size for the problems we are interested in. Such grid-size-dependent convergence performance can be effectively addressed by using a multigrid method as preconditioner for the GMRES solver [33]. As illustrated in [33], a multigrid-preconditioned GMRES solver is very robust and its overall performance is far superior than that of the multigrid method alone.…”
Section: Solution Of the Pressure-correction Equationmentioning
confidence: 99%
“…Such grid-size-dependent convergence performance can be effectively addressed by using a multigrid method as preconditioner for the GMRES solver [33]. As illustrated in [33], a multigrid-preconditioned GMRES solver is very robust and its overall performance is far superior than that of the multigrid method alone. In our work, we employ a cell-centered multigrid method as the preconditioner for the FGMRES solver.…”
Section: Solution Of the Pressure-correction Equationmentioning
confidence: 99%
“…Note that the shape of the stretch in all these grids is different, which implies that in each of these subproblems we have a different grid induced anisotropy. If the subgrids are simply combined without any interpolation, which means that all the evaluated points in every subgrid are added with the binomial coefficients (11).…”
Section: The Sparse Grid Methodsmentioning
confidence: 99%
“…By choosing multigrid as a preconditioner we do not need to search for the ideal under-relaxation, which is grid-anisotropy dependent, for example, but we can stay with a fixed parameter. Important theoretical and experimental insights into multigrid preconditioning of Krylov subspace solvers can be gained from earlier work in this context [11][12][13].…”
Section: Introductionmentioning
confidence: 99%