Basic smoothing procedures for the multigrid treatment of elliptic 3D operators

Thole, Clemens-August; Trottenberg, Ulrich

doi:10.1016/0096-3003(86)90112-8

Cited by 45 publications

(19 citation statements)

References 2 publications

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“…The first 3D test problem is a well-known test for robustness. The general anisotropic diffusion equation also tested in [24] and [5] looks like:…”

Section: Flexible Mg-s Cyclementioning

confidence: 99%

“…Storage would be less in methods based on the standard multigrid sequence in three dimensions. However, alternating plane smoothers would be required for robustness [24], [5], which are more expensive. In [5] one V-cycle of a 2D multigrid method was used as a plane smoother for solving 3D anisotropic diffusion equations with strong couplings in two directions simultaneously, in order to keep the plane smoother cheap.…”

mentioning

confidence: 99%

“…Early multigrid work in which semicoarsening plays an important role is [18], where the multigrid reduction (MGR) method is introduced, [24], where robust 3D multigrid methods for Poisson-type equations are considered, and [23], where semicoarsening is investigated on a multiple instruction, multiple data (MIMD) machine.…”

mentioning

confidence: 99%

“…Representative 3D anisotropic diffusion-and convection-dominated test problems are implemented and solved on the NEC Cenju-3 MIMD machine [8]. A well-known 3D test problem is the anisotropic diffusion equation [3], [24], [5], where the anisotropy is along one or two coordinate directions. The convection-dominated problems solved are convection-diffusion equations with spherical and cylindrical solutions.…”

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confidence: 99%

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Flexible Multiple Semicoarsening for Three-Dimensional Singularly Perturbed Problems

Washio¹,

Oosterlee²

1998

SIAM J. Sci. Comput.

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Abstract. We present robust parallel multigrid-based solvers for 3D scalar partial differential equations. The robustness is obtained by combining multiple semicoarsening strategies, matrixdependent transfer operators, and a Krylov subspace acceleration. The basis for the 3D preconditioner is a 2D method with multiple semicoarsened grids based on the MG-S method from [C. W. Oosterlee, Appl. Numer. Math., 19(1995) , 3(1991), pp. 1-75] and is evaluated as a solver and as a preconditioner on a MIMD machine. The robustness of the 3D method is shown for finite volume and finite difference discretizations of 3D anisotropic diffusion equations and convection-dominated convection-diffusion problems.

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“…The first 3D test problem is a well-known test for robustness. The general anisotropic diffusion equation also tested in [24] and [5] looks like:…”

Section: Flexible Mg-s Cyclementioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Flexible Multiple Semicoarsening for Three-Dimensional Singularly Perturbed Problems

Washio¹,

Oosterlee²

1998

SIAM J. Sci. Comput.

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“…Brandt et al, 1980;Brandt, 1984;Thole and Trottenberg, 1985;Barros, 1988). Furthermore, these solvers achieve optimal efficiency in the sense that the total operation count is a linear function of the number of unknowns.…”

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confidence: 99%

A multigrid solver for semi‐implicit global shallow‐water models

1990

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The multigrid principle produces fast solvers for systems of algebraic equations, particularly those that arise from discretizing elliptic boundary-value problems. A multigrid solver is developed for the discretized two-dimensional elliptic equation on the sphere that arises from a semi-implicit time discretization of the global shallow-water equations. We experiment with different formulations of the semi-implicit scheme that result in variable- IntroductionAn efficient discretization of the atmospheric flow equations should permit a choice of time step dependent on the physical properties of the solution, rather than on more stringent computational properties of the numerical scheme. Such a discretization inevitably requires time-implicit treatment of at least those terms in the model equations that give rise to fast gravity-wave propagation. Fully or partially implicit discretization of the evolution equations leads to large, multi-dimensional systems of algebraic equations to be solved at each time step. Highly efficient implementations of semi-and even fully implicit schemes have been achieved with the aid of splitting or factorization techniques, by which the implicit operator is reduced to simple one-dimensional operators that can easily be inverted by direct solvers (Bates, 1984;Cohn et al., 1985). However, the additional truncation error that is due to simplifying the implicit operator may be unacceptably large for time steps that are physically reasonable (Yakimiw and Robert, 1986;Tanguay and Robert, 1986). The semi-implicit scheme in its original formulation for a grid-point model (Kwizak and Robert, 1971) involves the solution of scalar Helmholtz equations. The combination of this scheme with Lagrangian timeintegration techniques enhances the stability of the method (Robert, 1981;1982) and hence its efficiency, as larger time steps are permitted. Semi-Lagrangian semiimplicit schemes that use three time levels and that are second-order accurate in time (Robert et al., 1985) and similar schemes that use two time levels but are first-order accurate in time (McDonald and Bates, 1988) also lead to scalar Helmholtz equations. These can be solved very efficiently by direct, so-called fast solvers based on reduction methods that rely on particular features of the equations such as separability (Sweet, 1977).Recently developed second-order accurate semi-Lagrangian methods that use only two time levels (Temperton and Staniforth, 1987;Bates, 1988) require the solution of more complicated, non-separable elliptic equations. Direct solution methods are inappropriate for such problems. Present implementations employ iterative solution methods that are quite expensive, since they require several calls to a fast, direct solver at each time step. (Temperton and Staniforth, 1987). Variable-coefficient elliptic equations also arise in models that feature variable grid spacing. Such models have been advocated (Staniforth and Mitchell, 1978;Temperton and Staniforth, 1987) as a way to expand the domain of a limit...

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An improved two‐grid preconditioner for the solution of three‐dimensional Helmholtz problems in heterogeneous media

Calandra

Gratton²,

Pinel

et al. 2012

Numerical Linear Algebra App

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SUMMARYIn this paper, we address the solution of three‐dimensional heterogeneous Helmholtz problems discretized with second‐order finite difference methods with application to acoustic waveform inversion in geophysics. In this setting, the numerical simulation of wave propagation phenomena requires the approximate solution of possibly very large indefinite linear systems of equations. For that purpose, we propose and analyse an iterative two‐grid method acting on the Helmholtz operator where the coarse grid problem is solved inaccurately. A cycle of a multigrid method applied to a complex shifted Laplacian operator is used as a preconditioner for the approximate solution of this coarse problem. A single cycle of the new method is then used as a variable preconditioner of a flexible Krylov subspace method. We analyse the properties of the resulting preconditioned operator by Fourier analysis. Numerical results demonstrate the effectiveness of the algorithm on three‐dimensional applications. The proposed numerical method allows us to solve three‐dimensional wave propagation problems even at high frequencies on a reasonable number of cores of a distributed memory computer. Copyright © 2012 John Wiley & Sons, Ltd.

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Basic smoothing procedures for the multigrid treatment of elliptic 3D operators

Cited by 45 publications

References 2 publications

Flexible Multiple Semicoarsening for Three-Dimensional Singularly Perturbed Problems

Flexible Multiple Semicoarsening for Three-Dimensional Singularly Perturbed Problems

A multigrid solver for semi‐implicit global shallow‐water models

An improved two‐grid preconditioner for the solution of three‐dimensional Helmholtz problems in heterogeneous media

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