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**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: 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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