Convergence of algebraic multigrid based on smoothed aggregation

Brezina, Marian; Mandel, Jan; Vaněk, Petr

doi:10.1007/s211-001-8015-y

Cited by 204 publications

(83 citation statements)

References 41 publications

(53 reference statements)

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“…We employ smoothed aggregation algebraic multigrid (AMG) for these computations because AMG does not require mesh or geometric information, and thus is attractive for problems posed on complex domains or unstructured meshes. More details on AMG can be found in [32,34].…”

Section: Problem and Preconditioner Structurementioning

confidence: 99%

A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations

Elman

Howle

Shadid

et al. 2008

Journal of Computational Physics

141

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Ray Tuminaro ttIn recent years, considerable effoR has been placed on developing efficient and robust solution algorithms for the incompressible Navier-Stokes equations based on preconditioned Krylov methods. These include physicsbased methods. such as SIMPLE. and ourelv aleebraic oreconditioners based on the aovroximation of the Schur . , u .A complement. All these techniques can be represented as approximate block factorization (ABF) type preconditioners. The goal is to decompose the application of the preconditioner into simplified sub-systems in which scalable multi-level type solvers can be applied. In this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the Navier-Stokes system first presented in [25]. This taxonomv illuminates the similarities and differences amone these oreconditioners and the central role olaved bv -. , , efficient approximation of certain Schur complement operators. We then present a parallel computational study that examines the performance of these methods and compares them to an additive Schwarz domain decomposition (DD) algorithm. Results are presented for two and three-dimensional steady state problems for enclosed domains and inflowloutflow systems on both structured and unstructured meshes. The numerical experiments are performed using MPSalsa, a stabilized finite element code.

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Section: Problem and Preconditioner Structurementioning

confidence: 99%

A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations

Elman

Howle

Shadid

et al. 2008

Journal of Computational Physics

141

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show abstract

“…This is critical for anisotropic problems where it is best not to coarsen in directions of weak coupling. Furthermore, for PDE systems it is often advantageous to build a graph corresponding to the block matrix (grouping into blocks all unknowns at a grid point) as opposed to treating each degree of freedom as a separate vertex [12].…”

Section: Smoothed Aggregation Multigrid Methodsmentioning

confidence: 99%

Parallel Smoothed Aggregation Multigrid : Aggregation Strategies on Massively Parallel Machines

Tuminaro

Tong

2000

ACM/IEEE SC 2000 Conference (SC'00)

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lgebraic multigrid methods offer the hope that multigrid convergence can be achieved (for at least some important applications) without a great deal of effort from engineers and scientists wishing to solve linear systems. In this paper we consider parallelization of the smoothed aggregation multigrid method. Smoothed aggregation is one of the most promising algebraic multigrid methods. Therefore, developing parallel variants with both good convergence and efficiency properties is of great importance. However, parallelization is nontrivial due to the somewhat sequential aggregation (or grid coarsening) phase. In this paper, we discuss three different parallel aggregation" algorithms and illustrate the advantages and disadvantages of each variant in terms of parallelism and convergence. Numerical results will be shown on the Intel Teraflop computer for some large problems coming from nontrivial codes: quasi-static electric potential simulation and a fluid flow calculation.

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“…Adopting a coarsening-by-three strategy has the potential advantage of coarsening more quickly, ultimately reducing the number of levels, and thus, communication steps in a parallel implementation. It also brings the method closer to SA-AMG [6], suggesting the potential for a comparative analysis in the future. In addition, coarsening by even larger factors introduces connections to approximation methods such as the Multiscale Finite Element (MsFEM) method [16].…”

Section: Introductionmentioning

confidence: 95%

Black Box Multigrid with coarsening by a factor of three

Dendy

Moulton

2010

Numerical Linear Algebra App

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SUMMARYBlack Box Multigrid (BoxMG) is a robust variational multigrid solver for diffusion equations on logically structured grids. BoxMG standardly uses coarsening by a factor of two. It handles cell-centered discretizations on logically rectangular grids by treating the cell-centers as the unknowns to be coarsened. Such a strategy does not preserve the cell structure. That is, coarse-grid cells are not the union of fine-grid cells. In some applications, such as local grid refinement, it is desirable that the cell structure be preserved. In this paper, we develop a method that employs coarsening by a factor of three. It is a natural generalization of standard BoxMG, using operator-induced interpolation (which approximately preserves the continuity of the normal flux), restriction as the transpose of interpolation, and Galerkin coarsening. In addition, we introduce a new colored block Gauss-Seidel scheme that is motivated by the form of the interpolation operator, dubbed 'pattern' relaxation. We present numerical results that demonstrate robustness of this method with respect to discontinuous diffusion coefficients, boundary conditions, and grid dimension.

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Convergence of algebraic multigrid based on smoothed aggregation

Cited by 204 publications

References 41 publications

A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations

A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations

Parallel Smoothed Aggregation Multigrid : Aggregation Strategies on Massively Parallel Machines

Black Box Multigrid with coarsening by a factor of three

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