Krylov improvements of the Uzawa method for Stokes type operator matrices

Axelsson, Owe; Karátson, János

doi:10.1007/s00211-021-01208-5

Numer. Math.

2021

DOI: 10.1007/s00211-021-01208-5

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Krylov improvements of the Uzawa method for Stokes type operator matrices

Owe Axelsson

János Karátson

Abstract: The paper is devoted to Krylov type modifications of the Uzawa method on the operator level for the Stokes problem in order to accelerate convergence. First block preconditioners and their effect on convergence are studied. Then it is shown that a Krylov–Uzawa iteration produces superlinear convergence on smooth domains, and estimation is given on its speed.

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Cited by 3 publications

References 30 publications

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On the efficient preconditioning of the Stokes equations in tight geometries

Pimanov,

Iliev,

Oseledets

et al. 2024

Numerical Linear Algebra App

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It is known (see, e.g., [SIAM J. Matrix Anal. Appl. 2014;35(1):143‐173]) that the performance of iterative methods for solving the Stokes problem essentially depends on the quality of the preconditioner for the Schur complement matrix, . In this paper, we consider two preconditioners for : the identity one and the SIMPLE one, and numerically study their performance for solving the Stokes problem in tight geometries. The latter are characterized by a high surface‐to‐volume ratio. We show that for such geometries, can become severely ill‐conditioned, having a very large condition number and a significant portion of non‐unit eigenvalues. As a consequence, the identity matrix, which is broadly used as a preconditioner for solving the Stokes problem in simple geometries, becomes very inefficient. We show that there is a correlation between the surface‐to‐volume ratio and the condition number of : the latter increases with the increase of the former. We show that the condition number of the diffusive SIMPLE‐preconditioned Schur complement matrix remains bounded when the surface‐to‐volume ratio increases, which explains the robust performance of this preconditioner for tight geometries. Further on, we use a direct method to calculate the full spectrum of and show that there is a correlation between the number of its non‐unit eigenvalues and the number of grid points at which no‐slip boundary conditions are prescribed. To illustrate the above findings, we examine the Pressure Schur Complement formulation for staggered finite‐difference discretization of the Stokes equations and solve it with the preconditioned conjugate gradient method. The practical problem which is of interest to us is computing the permeability of tight rocks.

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On the efficient preconditioning of the Stokes equations in tight geometries

Pimanov,

Iliev,

Oseledets

et al. 2024

Numerical Linear Algebra App

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Krylov Subspace Based FISTA‐Type Methods for Linear Discrete Ill‐Posed Problems

Buccini,

Chen,

Pasha

et al. 2024

Numerical Linear Algebra App

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Several iterative soft‐thresholding algorithms, such as FISTA, have been proposed in the literature for solving regularized linear discrete inverse problems that arise in various applications in science and engineering. These algorithms are easy to implement, but their rates of convergence may be slow. This paper describes novel approaches to reduce the computations required for each iteration by using Krylov subspace techniques. Specifically, we propose to impose sparsity on the coefficients in the representation of the computed solution in terms of a Krylov subspace basis. Several numerical examples from image deblurring and computerized tomography are used to illustrate the efficiency and accuracy of the proposed methods.

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Multilevel Uzawa and Arrow–Hurwicz Algorithms for General Saddle Point Problems

Badea

2023

J Optim Theory Appl

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Krylov improvements of the Uzawa method for Stokes type operator matrices

Cited by 3 publications

References 30 publications

On the efficient preconditioning of the Stokes equations in tight geometries

On the efficient preconditioning of the Stokes equations in tight geometries

Krylov Subspace Based FISTA‐Type Methods for Linear Discrete Ill‐Posed Problems

Multilevel Uzawa and Arrow–Hurwicz Algorithms for General Saddle Point Problems

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