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High-Performance Scientific Computing pp 219–250Cite as

A Preconditioned Scheme for Nonsymmetric Saddle-Point Problems

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Abstract

In this paper, we present an effective preconditioning technique for solving nonsymmetric saddle-point problems. In particular, we consider those saddle-point problems that arise in the numerical simulation of particulate flows—flow of solid particles in incompressible fluids, using mixed finite element discretization of the Navier–Stokes equations.

These indefinite linear systems are solved using a preconditioned Krylov subspace method with an indefinite preconditioner. This creates an inner–outer iteration, in which the inner iteration is handled via a preconditioned Richardson scheme. We provide an analysis of our approach that relates the convergence properties of the inner to the outer iterations. Also “optimal” approaches are proposed for the implicit construction of the Richardson’s iteration preconditioner. The analysis is validated by numerical experiments that demonstrate the robustness of our scheme, its lack of sensitivity to changes in the fluid–particle system, and its “scalability”.

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Acknowledgements

This work has been done in collaboration with Prof. Ahmed Sameh, and the author would like to acknowledge him for his continuous support.

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Authors and Affiliations

  1. College of Science and Engineering, Université Laval, Quebec City, Canada

    Abdelkader Baggag

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  1. Abdelkader Baggag
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Correspondence to Abdelkader Baggag .

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Editors and Affiliations

  1. Dept. Electrical Eng. & Computer Science, University of Tennessee, Volunteer Boulevard 1122, Knoxville, 37996-3450, Tennessee, USA

    Michael W. Berry

  2. Department of Mathematics, Florida State University, Academic Way 1017, Tallahassee, 32306, Florida, USA

    Kyle A. Gallivan

  3. Dept. Computer Engineering & Informatics, University of Patras, Patras, 26500, Greece

    Efstratios Gallopoulos

  4. Department of Computer Science, Purdue University, 305 North University Street, West Lafayette, 47907-2107, Indiana, USA

    Ananth Grama

  5. IRISA, INRIA Rennes - Bretagne Atlantique, Campus de Beaulieu, Rennes, 35042, France

    Bernard Philippe

  6. Dept. of Computer Science & Engineering, University of Minnesota, Union Street SE 200, Minneapolis, 55455, Minnesota, USA

    Yousef Saad

  7. Department of Computer Science, Purdue University, N. University Street 305, West Lafayette, 47907-2107, Indiana, USA

    Faisal Saied

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Baggag, A. (2012). A Preconditioned Scheme for Nonsymmetric Saddle-Point Problems. In: Berry, M., et al. High-Performance Scientific Computing. Springer, London. https://doi.org/10.1007/978-1-4471-2437-5_11

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