A new algebraic multigrid approach for Stokes problems

Notay, Yvan

doi:10.1007/s00211-015-0710-0

Cited by 17 publications

(31 citation statements)

References 32 publications

(52 reference statements)

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“…These results complement our previous study in [32], where we propose and theoretically analyze a slightly different approach, that uses a transformation similar in spirit but more complex (two-sided instead of one-sided; see below for details). The stability issue mentioned in [44] is not examined, but a reported numerical experiment suggests that the method can be cost effective with many levels, at least when using (plain) aggregation-based AMG (along the lines of [22,27,29]).…”

Section: S89supporting

confidence: 82%

“…However, with distributive smoothing, the transformation is only used to obtain an efficient smoother, and the coarse grid correction is still based on the original matrix. Opposite to this, with the methods in [32,44], the whole multigrid scheme is applied to the transformed system; differences and similarities with distributive smoothing are further commented on in section 3.1 below.…”

Section: S89mentioning

confidence: 99%

“…Finally, we address the practical effectiveness of both variants from [44] and [32]. To this aim, we present the results of numerical experiments that include twodimensional (2D) and three-dimensional (3D) problems, either stationary or time dependent, on both structured and unstructured grids.…”

Section: S89mentioning

confidence: 99%

“…Accordingly, below we only report numerical results for right-hand transformations. Right-hand and left-hand transformations can also be combined, leading to the two-sided transformation proposed in [32], with matrix…”

Section: Other Transformationsmentioning

confidence: 99%

“…We omit the details of this approach here, since a complete analysis is already available in [32], whereas the numerical results reported below give the advantage to one-sided transformations. As will be seen, the convergence is in fact about the same with both transformations.…”

Section: Other Transformationsmentioning

confidence: 99%

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Algebraic Multigrid for Stokes Equations

Notay¹

2017

SIAM J. Sci. Comput.

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Abstract. A method is investigated for solving stationary or time-dependent discrete Stokes equations. It uses one of the standard flavors of algebraic multigrid for coupled partial differential equations, which, however, is not applied directly to the linear system stemming from discretization, but to an equivalent system obtained with a simple algebraic transformation (which may be seen as a form of preconditioning in the literal sense). A two-grid analysis is provided, showing that the eigenvalues of the preconditioned matrix are within a region of the complex plane that is both bounded and away from the origin, independently of the mesh or grid size, as well as of other main problem parameters. On the other hand, whereas the approach can in principle be combined with any type of algebraic multigrid scheme, an investigation of the properties of the coarse grid matrices reveals that plain aggregation has to be preferred to maintain nice two-grid convergence at coarser levels. Eventually, numerical experiments are reported showing that the resulting method is both robust and cost effective, being significantly faster than a state-of-the-art competitor which combines MINRES with optimal block diagonal preconditioning.

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Section: S89supporting

confidence: 82%

Section: S89mentioning

confidence: 99%

Section: S89mentioning

confidence: 99%

Section: Other Transformationsmentioning

confidence: 99%

Section: Other Transformationsmentioning

confidence: 99%

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Algebraic Multigrid for Stokes Equations

Notay¹

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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CLC in AMG solvers for saddle‐point problems

Webster¹

2018

Numerical Linear Algebra App

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An algebraic multigrid (AMG) solution method for saddle-point problems is described. The indefinite nature of the saddle-point matrix makes it unsuitable for the simple smoothing methods normally used in AMG. Moreover, even if presented in a stabilised form, as is done here, poorly conditioned matrices will be generated when constructing the coarse-grid approximation. This is because, with each successive coarsening step, the off-diagonal matrix blocks (of interfield coupling) reduce in size more slowly than the diagonal blocks (of intrafield coupling). Stabilised smoothing operators are therefore considered. The first is based on an incomplete decomposition of the complete system matrix into the product of lower-triangular (L) and upper-triangular (U) matrices, an ILU factorisation. The second is based on an exact block decomposition of an incomplete (simplified) system matrix into lower and upper block-triangular matrices, a BILU factorisation. However, the degree of stabilisation thus established in the smoothing operators does not guarantee an efficient smoothing at all grid levels. There can still be inefficiency on the least-stable coarser grids. The breakdown in efficiency begins at a grid level where the ratio of the inter-to intrafield coupling strengths exceeds a critical ratio. Provision is thus made for a further conditioning of coarse-grid operators, a coarse-level conditioning (CLC). This is another block-LU factorisation that is only applied at and beyond the critical grid level. It is not applied directly to the operator for any chosen level but to its fine-grid progenitor. Thus, while ILU and BILU use postcoarsening-step factorisations, CLC uses precoarsening-step factorisations. With CLC so deployed, ILU and BILU become efficient at all grid levels, resulting in an AMG convergence that is independent of the total number of grids.

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A new algebraic multigrid approach for Stokes problems

Cited by 17 publications

References 32 publications

Algebraic Multigrid for Stokes Equations

Algebraic Multigrid for Stokes Equations

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

CLC in AMG solvers for saddle‐point problems

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