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International Conference on Computational Engineering

ICCE 2017: Recent Advances in Computational Engineering pp 63–87Cite as

A Bramble-Pasciak Conjugate Gradient Method for Discrete Stokes Problems with Lognormal Random Viscosity

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 124))

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-Loève 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.

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Acknowledgements

This work is supported by the ‘Excellence Initiative’ of the German federal and state governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt.

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Authors and Affiliations

  1. Graduate School of Computational Engineering, Technische Universität Darmstadt, Darmstadt, Germany

    Christopher Müller, Sebastian Ullmann & Jens Lang

  2. Department of Mathematics, Technische Universität Darmstadt, Darmstadt, Germany

    Christopher Müller, Sebastian Ullmann & Jens Lang

Authors
  1. Christopher Müller
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  2. Sebastian Ullmann
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  3. Jens Lang
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Corresponding author

Correspondence to Christopher Müller .

Editor information

Editors and Affiliations

  1. Graduate School of Computational Engineering, Technische Universität Darmstadt, Darmstadt, Germany

    Michael Schäfer

  2. Lehrstuhl für CATS, RWTH Aachen University, Aachen, Germany

    Marek Behr

  3. Institut für Parallele und Verteilte Systeme (IPVS), Universität Stuttgart, Stuttgart, Germany

    Miriam Mehl

  4. Lehrstuhl für Numerische Mathematik, Technische Universität München, Garching bei München, Germany

    Barbara Wohlmuth

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Müller, C., Ullmann, S., Lang, J. (2018). A Bramble-Pasciak Conjugate Gradient Method for Discrete Stokes Problems with Lognormal Random Viscosity. In: Schäfer, M., Behr, M., Mehl, M., Wohlmuth, B. (eds) Recent Advances in Computational Engineering. ICCE 2017. Lecture Notes in Computational Science and Engineering, vol 124. Springer, Cham. https://doi.org/10.1007/978-3-319-93891-2_5

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