A Multigrid Method Based on Incomplete Gaussian Elimination

Reusken, Arnold

doi:10.1002/(sici)1099-1506(199609/10)3:5<369::aid-nla89>3.0.co;2-m

Cited by 29 publications

(13 citation statements)

References 16 publications

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“…The convergence of MG3 is slower than that of the other algorithms but is in accordance with the performance in [12]. Their performance is similar for this test problem.…”

Section: Milu Preconditionersupporting

confidence: 76%

“…For this test problem we investigate the eigenvalue spectrum of the Richardson iteration matrix I − K −1 A (5). For MG3 this has also been observed in [12]. The spectra are presented in Figure 5.…”

Section: Milu Preconditionersupporting

confidence: 70%

“…The number of iterations as well as the elapsed time are reduced compared to Table 1 for many cases. With MG3 as a preconditioner the methods also converge on the 514 2 grid with linear interpolation P 12 in (41), i.e., the original method from [12]. It is interesting to see that the performance of MG3 as a preconditioner is much better and more robust than that of MG3 as a solver.…”

Section: Milu Preconditionermentioning

confidence: 97%

“…The robustness of this algorithm has been presented in [12] by means of two-grid Fourier analysis for the convection-diffusion equation. In the numerical results in [12] parameter ω was set to 0.7.…”

Section: Multigrid Based On the Block Lu Decompositionmentioning

confidence: 99%

“…The first problem investigated is an anisotropic diffusion problem from [12]: The first problem investigated is an anisotropic diffusion problem from [12]: …”

Section: Milu Preconditionermentioning

confidence: 99%

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An Evaluation of Parallel Multigrid as a Solver and a Preconditioner for Singularly Perturbed Problems

Oosterlee¹,

Washio²

1998

SIAM J. Sci. Comput.

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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 multigrid methods differ only in the transfer operators. One uses standard matrixdependent prolongation operators from []. Both employ the Galerkin coarse grid approximation and an alternating zebra line Gauss-Seidel smoother. The third method is based on the block LU decomposition of a matrix and on an approximate Schur complement. This multigrid variant is presented in [A. Reusken, A Multigrid Method Based on Incomplete Gaussian Elimination, University of Eindhoven, the Netherlands, 1995]. All three multigrid algorithms are algebraic methods.The eigenvalue spectra of the three multigrid iteration matrices are analyzed for the equations solved in order to understand the convergence of the three algorithms. Furthermore, the construction of the search directions for the multigrid preconditioned GMRES solvers is investigated by the calculation and solution of the minimal residual polynomials.For Poisson and convection-diffusion problems all solution methods are investigated and evaluated for finite volume discretizations on fine grids. The methods have been parallelized with a grid partitioning technique and are compared on an MIMD machine. ). C. W. OOSTERLEE AND T. WASHIOwith jumping coefficients, can be solved efficiently. The first algorithm includes wellknown matrix-dependent operators from the literature ([3], [6]). In [3] the operators have been designed so that problems on grids with arbitrary mesh sizes, not just powers of two (+1), can be solved with similar efficiency. Although in [3] these operators are mainly used for symmetric interface problems, we will also consider them here for unsymmetric problems.The second algorithm uses the prolongation operators introduced by de Zeeuw ([25]), which are specially designed for solving unsymmetric problems. Both algorithms employ Galerkin coarsening ([6], [24]) for building the matrices on coarser grids. A robust smoother is a necessary requirement in standard multigrid for achieving grid-size-independent convergence for many problems. The alternating zebra line Gauss-Seidel relaxation method was found to be a robust smoother for 2D model equations ([15], [24]). This smoother is used since it is possible to parallelize a line solver efficiently on a parallel machine ([23]). Constructing robust parallel solvers is an important purpose of our work.The third multigrid algorithm in our comparison is based on the block Gaussian elimination of a matrix and an approximate Schur complement. This method has been recently introduced by Reusken ([12]). An interesting aspect ...

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“…The convergence of MG3 is slower than that of the other algorithms but is in accordance with the performance in [12]. Their performance is similar for this test problem.…”

Section: Milu Preconditionersupporting

confidence: 76%

Section: Milu Preconditionersupporting

confidence: 70%

Section: Milu Preconditionermentioning

confidence: 97%

Section: Multigrid Based On the Block Lu Decompositionmentioning

confidence: 99%

“…The first problem investigated is an anisotropic diffusion problem from [12]: The first problem investigated is an anisotropic diffusion problem from [12]: …”

Section: Milu Preconditionermentioning

confidence: 99%

See 3 more Smart Citations

An Evaluation of Parallel Multigrid as a Solver and a Preconditioner for Singularly Perturbed Problems

Oosterlee¹,

Washio²

1998

SIAM J. Sci. Comput.

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Optimal v-cycle algebraic multilevel preconditioning

Notay

1998

Numer. Linear Algebra Appl.

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We consider algebraic multilevel preconditioning methods based on the recursive use of a 2 × 2 block incomplete factorization procedure in which the Schur complement is approximated by a coarse grid matrix. As is well known, for discrete second‐order elliptic PDEs, optimal convergence properties are proved for such basic two‐level schemes under mild assumptions on the PDE coefficients, but their recursive use in a simple V‐cycle algorithm does not generally lead to optimal order convergence. In the present paper, we analyse the combination of these techniques with a smoothing procedure much the same as the one used in standard multigrid algorithms, except that smoothing is not required on the finest grid. Theoretical results prove optimal convergence properties for the V‐cycle under an assumption similar to the ‘approximation property’ of the classical multigrid convergence theory. On the other hand, numerical experiments made on both 2D and 3D problems show that the condition number is close to that of the two‐level method. Further, the method appears robust in the presence of discontinuity and anisotropy, even when the material interfaces are not aligned with the coarse grid. Copyright © 1999 John Wiley & Sons, Ltd.

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Model case analysis of an algebraic multilevel method

Shapira

1999

Numer. Linear Algebra Appl.

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A Multigrid Method Based on Incomplete Gaussian Elimination

Cited by 29 publications

References 16 publications

An Evaluation of Parallel Multigrid as a Solver and a Preconditioner for Singularly Perturbed Problems

An Evaluation of Parallel Multigrid as a Solver and a Preconditioner for Singularly Perturbed Problems

Optimal v-cycle algebraic multilevel preconditioning

Model case analysis of an algebraic multilevel method

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