A Simple and Efficient Segregated Smoother for the Discrete Stokes Equations

Gaspar, Francisco José; Notay, Yvan; Oosterlee, Cornelis W.; Rodrigo, Carmen

doi:10.1137/130920630

Cited by 39 publications

(77 citation statements)

References 21 publications

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“…The symmetric Gauss-Seidel method consists of one forward and one backward sweep for all velocities in the computational domain. Numerical experiments in [23] revealed that, for essentially the same cost, the convergence associated with M A in (31) is most efficient. So, this variant is the one that we extend to the Darcy equation.…”

Section: S643mentioning

confidence: 98%

“…The approximation of (σ xy ) w can be calculated in a similar way. The discrete equation for the vertical velocities for the Stokes problem at the interface is thus obtained by substituting (20), (23), and (25) into (19), giving…”

Section: Discretization Of the Interfacementioning

confidence: 99%

“…An Uzawa smoother, which was proposed for the Stokes problem in [23], is considered for the coupled system. The Uzawa smoother.…”

Section: Multigrid Based On Uzawa Smoothermentioning

confidence: 99%

“…An Uzawa smoother using two forward Gauss-Seidel sweeps as M A was suggested in [35] for the Stokes problem. In [23] another variant was considered, where M A was based on the symmetric Gauss-Seidel iterations for A; i.e.,…”

Section: S643mentioning

confidence: 99%

“…So, this variant is the one that we extend to the Darcy equation. The efficiency of the proposed Uzawa smoother for threedimensional Stokes and Biot poroelasticity equations was presented in [23] and [34], respectively. In addition, M A has two important properties needed in our theoretical analysis, in addition to its efficiency as a smoother.…”

Section: S643mentioning

confidence: 99%

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Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

Luo¹,

Rodrigo²,

Gaspar³

et al. 2017

SIAM J. Sci. Comput.

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Abstract. The multigrid solution of coupled porous media and Stokes flow problems is considered. The Darcy equation as the saturated porous medium model is coupled to the Stokes equations by means of appropriate interface conditions. We focus on an efficient multigrid solution technique for the coupled problem, which is discretized by finite volumes on staggered grids, giving rise to a saddle point linear system. Special treatment is required regarding the discretization at the interface. An Uzawa smoother is employed in multigrid, which is a decoupled procedure based on symmetric Gauss-Seidel smoothing for velocity components and a simple Richardson iteration for the pressure field. Since a relaxation parameter is part of a Richardson iteration, local Fourier analysis is applied to determine the optimal parameters. Highly satisfactory multigrid convergence is reported, and, moreover, the algorithm performs very well for small values of the hydraulic conductivity and fluid viscosity, which are relevant for applications.

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Section: S643mentioning

confidence: 98%

Section: Discretization Of the Interfacementioning

confidence: 99%

“…An Uzawa smoother, which was proposed for the Stokes problem in [23], is considered for the coupled system. The Uzawa smoother.…”

Section: Multigrid Based On Uzawa Smoothermentioning

confidence: 99%

Section: S643mentioning

confidence: 99%

Section: S643mentioning

confidence: 99%

See 3 more Smart Citations

Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

Luo¹,

Rodrigo²,

Gaspar³

et al. 2017

SIAM J. Sci. Comput.

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An all‐at‐once algebraic multigrid method for finite element discretizations of Stokes problem

Bacq

Gounand

Napov

et al. 2022

Numerical Methods in Fluids

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We consider numerical solution of finite element discretizations of the Stokes problem. We focus on the transform-then-solve approach, which amounts to first apply a specific algebraic transformation to the linear system of equations arising from the discretization, and then solve the transformed system with an algebraic multigrid method. The approach has recently been applied to finite differences discretizations of the Stokes problem with constant viscosity, and has recommended itself as a robust and competitive solution method. In this work, we examine the extension of the approach to standard finite element discretizations of the Stokes problem, including problems with variable viscosity. The extension relies, on one hand, on the use of the successive over-relaxation method as a multigrid smoother for some finite element schemes. On the other hand, we present strategies that allow us to limit the complexity increase induced by the transformation. Numerical experiments show that our method is competitive compared to a state-of-the-art solver based on a block diagonal preconditioner and MINRES, and suggest that the transform-thensolve approach is also more robust. In particular, for problems with variable viscosity, the transform-then-solve approach demonstrates significant speed-up with respect to the block diagonal preconditioner.

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On an Uzawa smoother in multigrid for poroelasticity equations

Luo

Rodrigo

Gaspar

et al. 2016

Numerical Linear Algebra App

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SummaryA poroelastic saturated medium can be modeled by means of Biot's theory of consolidation. It describes the time-dependent interaction between the deformation of porous material and the fluid flow inside of it. Here, for the efficient solution of the poroelastic equations, a multigrid method is employed with an Uzawa-type iteration as the smoother. The Uzawa smoother is an equation-wise procedure. It shall be interpreted as a combination of the symmetric Gauss-Seidel smoothing for displacements, together with a Richardson iteration for the Schur complement in the pressure field. The Richardson iteration involves a relaxation parameter which affects the convergence speed, and has to be carefully determined. The analysis of the smoother is based on the framework of local Fourier analysis (LFA) and it allows us to provide an analytic bound of the smoothing factor of the Uzawa smoother as well as an optimal value of the relaxation parameter. Numerical experiments show that our upper bound provides a satisfactory estimate of the exact smoothing factor, and the selected relaxation parameter is optimal. In order to improve the convergence performance, the acceleration of multigrid by iterant recombination is taken into account. Numerical results confirm the efficiency and robustness of the acceleration scheme.

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A Simple and Efficient Segregated Smoother for the Discrete Stokes Equations

Cited by 39 publications

References 21 publications

Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

An all‐at‐once algebraic multigrid method for finite element discretizations of Stokes problem

On an Uzawa smoother in multigrid for poroelasticity equations

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