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Parallel algebraic multilevel Schwarz preconditioners for a class of elliptic PDE systems

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Computing and Visualization in Science

Abstract

Algebraic multilevel preconditioners for algebraic problems arising from the discretization of a class of systems of coupled elliptic partial differential equations (PDEs) are presented. These preconditioners are based on modifications of Schwarz methods and of the smoothed aggregation technique, where the coarsening strategy and the restriction and prolongation operators are defined using a point-based approach with a primary matrix corresponding to a single PDE. The preconditioners are implemented in a parallel computing framework and are tested on two representative PDE systems. The results of the numerical experiments show the effectiveness and the scalability of the proposed methods. A convergence theory for the twolevel case is presented.

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Acknowledgments

The authors wish to thank the anonymous referees for their careful reading of this paper and their valuable comments.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Str. 30, 97074, Würzburg, Germany

    Alfio Borzì

  2. Dipartimento di Matematica e Fisica, Seconda Università degli Studi di Napoli, viale A. Lincoln 5, 81100, Caserta, Italy

    Valentina De Simone & Daniela di Serafino

  3. Istituto di Calcolo e Reti ad Alte Prestazioni (ICAR), CNR, via P. Castellino 111, 80131, Naples, Italy

    Daniela di Serafino

Authors
  1. Alfio Borzì
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  2. Valentina De Simone
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  3. Daniela di Serafino
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Correspondence to Alfio Borzì.

Additional information

Communicated by Gabriel Wittum.

Work partially supported by INdAM-GNCS Projects ‘Advanced numerical methods for large-scale nonlinear constrained optimization problems’ (2011) and ‘Numerical methods and software for preconditioning linear systems arising in PDE and optimization problems’ (2012), and by BMBF Verbundproject 05M2013 ‘ROENOBIO: Robust energy optimization of fermentation processes for the production of biogas and wine’.

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Borzì, A., De Simone, V. & di Serafino, D. Parallel algebraic multilevel Schwarz preconditioners for a class of elliptic PDE systems. Comput. Visual Sci. 16, 1–14 (2013). https://doi.org/10.1007/s00791-014-0220-0

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  • DOI: https://doi.org/10.1007/s00791-014-0220-0

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