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Nested BDDC for a saddle-point problem

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Abstract

We propose a Nested BDDC for a class of saddle-point problems. The method solves for both flux and pressure variables. The fluxes are resolved in three-steps: the coarse solve is followed by subdomain solves, and last we look for a divergence-free flux correction and pressure variables using conjugate gradients with a Multilevel BDDC preconditioner. Because the coarse solve in the first step has the same structure as the original problem, we can use this procedure recursively and solve (a hierarchy of) coarse problems only approximately, utilizing the coarse problems known from the BDDC. The resulting algorithm thus first performs several upscaling steps, and then solves a hierarchy of problems that have the same structure but increase in size while sweeping down the levels, using the same components in the first and in the third step on each level, and also reusing the components from the higher levels. Because the coarsening can be quite aggressive, the number of levels can be kept small and the additional computational cost is significantly reduced due to the reuse of the components. We also provide the condition number bound and numerical experiments confirming the theory.

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Acknowledgments

I would like to thank Dr. Christopher Harder and Prof. Jan Mandel for many discussions over the paper, and the referees for useful comments and suggestions.

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Authors and Affiliations

  1. Department of Aerospace and Mechanical Engineering, Viterbi School of Engineering, University of Southern California, Los Angeles, CA, 90089-2531, USA

    Bedřich Sousedík

  2. Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Dolejškova 1402/5, Prague 8, Czech Republic

    Bedřich Sousedík

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  1. Bedřich Sousedík
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Correspondence to Bedřich Sousedík.

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Part of the work has been completed while the author was a Research Assistant Professor at the Department of Mathematical and Statistical Sciences, University of Colorado Denver.

Supported in part by the National Science Foundation under grant DMS-0713876, and by the Grant Agency of the Czech Republic GAČR 106/08/0403. Support from DOE/ASCR is also gratefully acknowledged.

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Sousedík, B. Nested BDDC for a saddle-point problem. Numer. Math. 125, 761–783 (2013). https://doi.org/10.1007/s00211-013-0548-2

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  • DOI: https://doi.org/10.1007/s00211-013-0548-2

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