Multigrid methods for saddle point systems using constrained smoothers

Chen, Long

doi:10.1016/j.camwa.2015.09.020

Cited by 14 publications

(20 citation statements)

References 20 publications

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“…After each sweep of relaxation, the correction

δ \overset{x}{true}

is distributed to the original unknowns,

δ x = scriptP δ \overset{x}{true}

. DGS‐type relaxation has been widely used . One well‐known drawback of DGS is a persistent “gap” between the smoothing factor predicted by LFA and the convergence factors observed in practice for problems with Dirichlet boundary conditions .…”

Section: Distributive Relaxationmentioning

confidence: 99%

“…DGS-type relaxation has been widely used. 25,34 One well-known drawback of DGS is a persistent "gap" between the smoothing factor predicted by LFA and the convergence factors observed in practice for problems with Dirichlet boundary conditions. 28,[35][36][37] In the work by Niestegge et al, 28 it is noted that the LFA predictions are exact for periodic boundary conditions, but extra boundary relaxation is required for Dirichlet boundary conditions (consistent with later analysis of LFA in general in other works 32,33 ).…”

Section: Distributive Relaxationmentioning

confidence: 99%

“…Two further families are based on using block preconditioning strategies as relaxation schemes, yielding the Braess–Sarazin and Uzawa approaches. Each of these families has been further developed in the recent years, including Braess–Sarazin‐type relaxation schemes, Vanka‐type relaxation schemes, Uzawa‐type relaxation schemes, distributive relaxation schemes, and other types of methods . The aim of this paper is to analyze block‐structured relaxation schemes, including distributive, Braess–Sarazin, and Uzawa relaxation.…”

Section: Introductionmentioning

confidence: 99%

“…Each of these families has been further developed in the recent years, including Braess-Sarazin-type relaxation schemes, 5,10,[12][13][14] Vanka-type relaxation schemes, 5,9,[13][14][15][16][17][18] Uzawa-type relaxation schemes, [19][20][21][22] distributive relaxation schemes, 23,24 and other types of methods. 25,26 The aim of this paper is to analyze block-structured relaxation schemes, including distributive, Braess-Sarazin, and Uzawa relaxation.…”

Section: Introductionmentioning

confidence: 99%

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Local Fourier analysis of block‐structured multigrid relaxation schemes for the Stokes equations

MacLachlan

2018

Numerical Linear Algebra App

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Summary Multigrid methods that use block‐structured relaxation schemes have been successfully applied to several saddle‐point problems, including those that arise from the discretization of the Stokes equations. In this paper, we present a local Fourier analysis of block‐structured relaxation schemes for the staggered finite‐difference discretization of the Stokes equations to analyze their convergence behavior. Three block‐structured relaxation schemes are considered: distributive relaxation, Braess–Sarazin relaxation, and Uzawa relaxation. In each case, we consider variants based on weighted‐Jacobi relaxation, as is most suitable for parallel implementation on modern architectures. From this analysis, optimal parameters are proposed, and we compare the efficiency of the presented algorithms with these parameters. Finally, some numerical experiments are presented to validate the two‐grid and multigrid convergence factors.

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“…After each sweep of relaxation, the correction

δ \overset{x}{true}

is distributed to the original unknowns,

δ x = scriptP δ \overset{x}{true}

Section: Distributive Relaxationmentioning

confidence: 99%

Section: Distributive Relaxationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Local Fourier analysis of block‐structured multigrid relaxation schemes for the Stokes equations

MacLachlan

2018

Numerical Linear Algebra App

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“…Recall that the success of multigrid method relies on two ingredients: the high frequency can be damped efficiently by the smoother, and the low frequency can be well approximated by the coarse grid correction. Notice that for saddle point systems, both smoothing and coarse grid corrections can easily violate the constraint [10]. The main difficulty of developing robust and effective multigrid methods for the saddle point system is to design an effective smoother with the consideration of the constraint div u = g. We shall use the Peaceman-Rachford iteration developed in [15] as a smoother since the nonlinearity can be handled efficiently and the constraint is always satisfied after solving a linear saddle point system.…”

Section: Introductionmentioning

confidence: 99%

Multigrid Methods for a Mixed Finite Element Method of the Darcy–Forchheimer Model

2017

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An efficient nonlinear multigrid method for a mixed finite element method of the Darcy-Forchheimer model is constructed in this paper. A Peaceman-Rachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. The nonlinear equation can be solved element-wise with a closed formulae. The linear saddle point system for the constraint is reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the splitting is proposed and the resulting multigrid method is robust to the so-called Forchheimer number which controls the strength of the nonlinearity. By comparing the number of iterations and CPU time of different solvers in several numerical experiments, our multigrid method is shown to convergent with a rate independent of the mesh size and the Forchheimer number and with a nearly linear computational cost.

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Fast Multilevel Solvers for a Class of Discrete Fourth Order Parabolic Problems

Zheng

Chen

et al. 2016

J Sci Comput

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In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to construct efficient preconditioners. We provide eigenvalue analysis for the preconditioned system and estimate the convergence rate of the preconditioned GMRes method. Furthermore, we show that these preconditioners only need to be solved inexactly by optimal multigrid algorithms. Our numerical examples indicate that the proposed preconditioners are very efficient and robust with respect to both discretization parameters and diffusion coefficients. We also investigate the performance of multigrid algorithms with either collective smoothers or distributive smoothers when solving the preconditioner systems.

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Multigrid methods for saddle point systems using constrained smoothers

Cited by 14 publications

References 20 publications

Local Fourier analysis of block‐structured multigrid relaxation schemes for the Stokes equations

Local Fourier analysis of block‐structured multigrid relaxation schemes for the Stokes equations

Multigrid Methods for a Mixed Finite Element Method of the Darcy–Forchheimer Model

Fast Multilevel Solvers for a Class of Discrete Fourth Order Parabolic Problems

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