4.5 Article

Nested BDDC for a saddle-point problem

Journal

NUMERISCHE MATHEMATIK
Volume 125, Issue 4, Pages 761-783

Publisher

SPRINGER HEIDELBERG
DOI: 10.1007/s00211-013-0548-2

Keywords

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Funding

  1. National Science Foundation [DMS-0713876]
  2. Grant Agency of the Czech Republic GACR [106/08/0403]
  3. DOE/ASCR

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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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