Fourier Analysis for Multigrid Methods on Triangular Grids

Gaspar, Francisco José; Gracia, J. L.; Lisbona, F. J.

doi:10.1137/080713483

Cited by 34 publications

(37 citation statements)

References 23 publications

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“…Local Fourier analysis used to be applied to discretizations on rectangular grids, however, in [9] LFA was extended to discretizations on non-rectangular grids, in particular, to triangular grids. The key to this generalization was a two-dimensional Fourier transform using coordinates in non-orthogonal bases.…”

Section: Finite Difference Discretization On Collocated Gridmentioning

confidence: 99%

A Simple and Efficient Segregated Smoother for the Discrete Stokes Equations

Gaspar

Notay

Oosterlee

et al. 2014

SIAM J. Sci. Comput.

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Abstract. We consider the multigrid solution of the generalized Stokes equations from incompressible fluid dynamics. We introduce a segregated (i.e., equation-wise) Gauss-Seidel smoother based on a Uzawa-type iteration. We analyze it in the framework of local Fourier analysis. We obtain an analytic bound on the smoothing factor showing uniform performance for a family of Stokes problems, ranging from stationary to time-dependent with small time step. These results are confirmed by the numerical computation of the two-grid convergence factor for different types of grids and discretizations. Numerical results also show that the actual convergence of the W-cycle is roughly the same as that obtained with the Vanka smoother, despite this latter is significantly more costly per iteration step.

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Section: Finite Difference Discretization On Collocated Gridmentioning

confidence: 99%

A Simple and Efficient Segregated Smoother for the Discrete Stokes Equations

Gaspar

Notay

Oosterlee

et al. 2014

SIAM J. Sci. Comput.

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“…Recently, a generalization to triangular grids-which is based on an expression of the Fourier transform in new coordinate systems in space and frequency variables-has been proposed in [13]. The ideas developed in that paper can be extended to systems of equations similarly as in cartesian grids, see [1,3,7].…”

Section: Fourier Analysis Resultsmentioning

confidence: 99%

“…This technique has been widely used for systems of PDEs in the framework of discretizations on rectangular grids [8][9][10][11]; even for the elasticity problem an alternative approach to Fourier analysis was developed in [12]. Recently a generalization of LFA to triangular grids has been proposed in [13] for a scalar problem, and in [14] it has been extended using a three-coarsening strategy. The key to carrying out this generalization is to write the Fourier transform in new coordinate systems, both in space and in frequency variables, associated with reciprocal bases fitting the structure of the grid.…”

Section: Introductionmentioning

confidence: 99%

Multigrid finite element methods on semi‐structured triangular grids for planar elasticity

Gaspar¹,

Lisbona²,

Gracia³

et al. 2009

Numerical Linear Algebra App

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SUMMARYMultigrid methods for a stencil-based implementation of a finite element method for planar elasticity, using semi-structured triangular grids, are presented. Local Fourier analysis (LFA) is applied to identify the correct multigrid components. To this end, LFA for multigrid methods on regular triangular grids is extended to the case of the problem of planar elasticity, although its application to other systems is straightforward. For the discrete elasticity operator obtained with linear finite elements, different collective smoothers, such as three-color smoother and some zebra-type smoothers, are analyzed, and LFA results for these smoothers are shown. The multigrid method is constructed in a block-wise form. In particular, different smoothers and different numbers of pre-and post-smoothing steps are considered in each triangle of the coarsest triangulation of the domain. Some numerical experiments are presented to illustrate the efficiency of this multigrid algorithm.

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“…Since we are working on triangular grids, the ideas about the recently introduced LFA on triangular meshes [9,22] have to be taken into account. The key fact for this extension is to consider an expression of the Fourier transform in new coordinate systems in space and frequency variables.…”

Section: Notation and Basics Of Lfa For Ilu Smoothersmentioning

confidence: 99%

On the robustness of ILU smoothers on triangular grids

Pinto

Rodrigo

Gaspar

et al. 2016

Applied Numerical Mathematics

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In this work, incomplete factorization techniques are used as smoothers within a geometric multigrid algorithm on triangular grids. A local Fourier analysis is proposed to study the smoothing properties of these methods, as well as the asymptotic convergence of the whole multigrid procedure. With this purpose, two-and three-grid local Fourier analysis are performed. Several two-dimensional diffusion problems, including different kinds of anisotropy are considered to demonstrate the robustness of this type of methods.

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Fourier Analysis for Multigrid Methods on Triangular Grids

Cited by 34 publications

References 23 publications

A Simple and Efficient Segregated Smoother for the Discrete Stokes Equations

A Simple and Efficient Segregated Smoother for the Discrete Stokes Equations

Multigrid finite element methods on semi‐structured triangular grids for planar elasticity

On the robustness of ILU smoothers on triangular grids

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