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A Bramble-Pasciak conjugate gradient method for discrete Stokes problems with lognormal random viscosity

Müller, Christopher and Ullmann, Sebastian and Lang, Jens Schäfer, Michael and Behr, Marek and Mehl, Miriam and Wohlmuth, Barbara (eds.) (2018):
A Bramble-Pasciak conjugate gradient method for discrete Stokes problems with lognormal random viscosity.
In: Lecture Notes in Computational Science and Engineering, In: Recent Advances in Computational Engineering, Cham, Springer International Publishin, pp. 63-87, DOI: 10.1007/978-3-319-93891-2_5,
[Online-Edition: https://link.springer.com/chapter/10.1007/978-3-319-93891-2_...],
[Book Section]

Abstract

We study linear systems of equations arising from a stochastic Galerkin finite element discretization of saddle point problems with random data and its iterative solution. We consider the Stokes flow model with random viscosity described by the exponential of a correlated random process and shortly discuss the discretization framework and the representation of the emerging matrix equation. Due to the high dimensionality and the coupling of the associated symmetric, indefinite, linear system, we resort to iterative solvers and problem-specific preconditioners. As a standard iterative solver for this problem class, we consider the block diagonal preconditioned MINRES method and further introduce the Bramble-Pasciak conjugate gradient method as a promising alternative. This special conjugate gradient method is formulated in a non-standard inner product with a block triangular preconditioner. From a structural point of view, such a block triangular preconditioner enables a better approximation of the original problem than the block diagonal one. We derive eigenvalue estimates to assess the convergence behavior of the two solvers with respect to relevant physical and numerical parameters and verify our findings by the help of a numerical test case. We model Stokes flow in a cavity driven by a moving lid and describe the viscosity by the exponential of a truncated Karhunen-Lo{\`e}ve expansion. Regarding iteration numbers, the Bramble-Pasciak conjugate gradient method with block triangular preconditioner is superior to the MINRES method with block diagonal preconditioner in the considered example.

Item Type: Book Section
Erschienen: 2018
Editors: Schäfer, Michael and Behr, Marek and Mehl, Miriam and Wohlmuth, Barbara
Creators: Müller, Christopher and Ullmann, Sebastian and Lang, Jens
Title: A Bramble-Pasciak conjugate gradient method for discrete Stokes problems with lognormal random viscosity
Language: English
Abstract:

We study linear systems of equations arising from a stochastic Galerkin finite element discretization of saddle point problems with random data and its iterative solution. We consider the Stokes flow model with random viscosity described by the exponential of a correlated random process and shortly discuss the discretization framework and the representation of the emerging matrix equation. Due to the high dimensionality and the coupling of the associated symmetric, indefinite, linear system, we resort to iterative solvers and problem-specific preconditioners. As a standard iterative solver for this problem class, we consider the block diagonal preconditioned MINRES method and further introduce the Bramble-Pasciak conjugate gradient method as a promising alternative. This special conjugate gradient method is formulated in a non-standard inner product with a block triangular preconditioner. From a structural point of view, such a block triangular preconditioner enables a better approximation of the original problem than the block diagonal one. We derive eigenvalue estimates to assess the convergence behavior of the two solvers with respect to relevant physical and numerical parameters and verify our findings by the help of a numerical test case. We model Stokes flow in a cavity driven by a moving lid and describe the viscosity by the exponential of a truncated Karhunen-Lo{\`e}ve expansion. Regarding iteration numbers, the Bramble-Pasciak conjugate gradient method with block triangular preconditioner is superior to the MINRES method with block diagonal preconditioner in the considered example.

Title of Book: Recent Advances in Computational Engineering
Series Name: Lecture Notes in Computational Science and Engineering
Volume: 124
Place of Publication: Cham
Publisher: Springer International Publishin
ISBN: 978-3-319-93891-2
Divisions: Exzellenzinitiative
Exzellenzinitiative > Graduate Schools
Exzellenzinitiative > Graduate Schools > Graduate School of Computational Engineering (CE)
Exzellenzinitiative > Graduate Schools > Graduate School of Energy Science and Engineering (ESE)
04 Department of Mathematics
04 Department of Mathematics > Numerical Analysis and Scientific Computing
Date Deposited: 19 Dec 2017 08:49
DOI: 10.1007/978-3-319-93891-2_5
Official URL: https://link.springer.com/chapter/10.1007/978-3-319-93891-2_...
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