A class of iterative methods for solving saddle point problems

Bank, Randolph E.; Welfert, Bruno D.; Yserentant, Harry

doi:10.1007/bf01405194

Cited by 200 publications

(148 citation statements)

References 16 publications

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“…There are quite many works developing eigenvalue analyzes for these types of preconditioner; see [5,10] for block diagonal preconditioners, [7,32] for block triangular preconditioners, [8,15,36] for inexact Uzawa preconditioners, and [2,3,36] for block approximate factorization preconditioners -to mention just a few; we refer to [4] for many more references and historical remarks.…”

Section: 2)mentioning

confidence: 99%

A New Analysis of Block Preconditioners for Saddle Point Problems

Notay

2014

SIAM J. Matrix Anal. & Appl.

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We consider symmetric saddle point matrices. We analyze block preconditioners based on the knowledge of a good approximation for both the top left block and the Schur complement resulting from its elimination. We obtain bounds on the eigenvalues of the preconditioned matrix that depend only of the quality of these approximations, as measured by the related condition numbers. Our analysis applies to indefinite block diagonal preconditioners, block triangular preconditioners, inexact Uzawa preconditioners, block approximate factorization preconditioners, and a further enhancement of these latter based on symmetric block Gauss-Seidel type iterations. The analysis is unified and allows the comparison of these different approaches. In particular, it reveals that block triangular and inexact Uzawa preconditioners lead to identical eigenvalue distributions. These theoretical results are illustrated on the discrete Stokes problem. It turns out that the provided bounds allow to localize accurately both real and non real eigenvalues. The relative quality of the different types of preconditioners is also as expected from the theory.

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Section: 2)mentioning

confidence: 99%

A New Analysis of Block Preconditioners for Saddle Point Problems

Notay

2014

SIAM J. Matrix Anal. & Appl.

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“…We restrict ourselves to iterative methods of inexact Uzawa type. For this class of methods applied to the stationary Stokes problem, convergence analyses are known (e.g., [1], [3]). Also for other iterative methods based on conjugate or minimal residual techniques there are convergence analyses available (cf.…”

Section: Preconditioning the Discrete Problemmentioning

confidence: 99%

Grad-div stablilization for Stokes equations

2003

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Abstract. In this paper a stabilizing augmented Lagrangian technique for the Stokes equations is studied. The method is consistent and hence does not change the continuous solution. We show that this stabilization improves the well-posedness of the continuous problem for small values of the viscosity coefficient. We analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact Uzawa method.

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“…Using these norms involves some equivalence results and representations from [18,22,24] for the discrete and continuous pressure Schur complement operators. In appropriate norms we consider smoothing properties of distributive iterations and coupled iterations similar to the methods of Braess and Sarazin [6] and Bank et al [2]. From these results the convergence of the two-grid method follows immediately.…”

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confidence: 99%

Multigrid analysis for the time dependent Stokes problem

Olshanskii

2012

Math. Comp.

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Abstract. Certain implicit time stepping procedures for the incompressible Stokes or Navier-Stokes equations lead to a singular-perturbed Stokes type problem at each time step. The paper presents a convergence analysis of a geometric multigrid solver for the system of linear algebraic equations resulting from the disretization of the problem using a finite element method. Several smoothing iterative methods are considered: a smoother based on distributive iterations, the Braess-Sarazin and inexact Uzawa smoother. Convergence analysis is based on smoothing and approximation properties in special norms. A robust (independent of time step and mesh parameter) estimate is proved for the two-grid and multigrid W-cycle convergence factors.

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A class of iterative methods for solving saddle point problems

Cited by 200 publications

References 16 publications

A New Analysis of Block Preconditioners for Saddle Point Problems

A New Analysis of Block Preconditioners for Saddle Point Problems

Grad-div stablilization for Stokes equations

Multigrid analysis for the time dependent Stokes problem

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