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    Multilevel Balancing Domain Decomposition by Constraints Deluxe Algorithms with Adaptive Coarse Spaces for Flow in Porous Media

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    Type
    Article
    Authors
    Zampini, Stefano cc
    Tu, Xuemin
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Extreme Computing Research Center
    Date
    2017-08-03
    Online Publication Date
    2017-08-03
    Print Publication Date
    2017-01
    Permanent link to this record
    http://hdl.handle.net/10754/625318
    
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    Abstract
    Multilevel balancing domain decomposition by constraints (BDDC) deluxe algorithms are developed for the saddle point problems arising from mixed formulations of Darcy flow in porous media. In addition to the standard no-net-flux constraints on each face, adaptive primal constraints obtained from the solutions of local generalized eigenvalue problems are included to control the condition number. Special deluxe scaling and local generalized eigenvalue problems are designed in order to make sure that these additional primal variables lie in a benign subspace in which the preconditioned operator is positive definite. The current multilevel theory for BDDC methods for porous media flow is complemented with an efficient algorithm for the computation of the so-called malign part of the solution, which is needed to make sure the rest of the solution can be obtained using the conjugate gradient iterates lying in the benign subspace. We also propose a new technique, based on the Sherman--Morrison formula, that lets us preserve the complexity of the subdomain local solvers. Condition number estimates are provided under certain standard assumptions. Extensive numerical experiments confirm the theoretical estimates; additional numerical results prove the effectiveness of the method with higher order elements and high-contrast problems from real-world applications.
    Citation
    Zampini S, Tu X (2017) Multilevel Balancing Domain Decomposition by Constraints Deluxe Algorithms with Adaptive Coarse Spaces for Flow in Porous Media. SIAM Journal on Scientific Computing 39: A1389–A1415. Available: http://dx.doi.org/10.1137/16m1080653.
    Sponsors
    The authors would like to thank the two anonymous referees for their comments and suggestions that helped improve the quality of the manuscript. For computer time, this research used the resources of the Supercomputing Laboratory at King Abdullah University of Science & Technology (KAUST) in Thuwal, Saudi Arabia.
    Publisher
    Society for Industrial & Applied Mathematics (SIAM)
    Journal
    SIAM Journal on Scientific Computing
    DOI
    10.1137/16m1080653
    Additional Links
    http://epubs.siam.org/doi/10.1137/16M1080653
    ae974a485f413a2113503eed53cd6c53
    10.1137/16m1080653
    Scopus Count
    Collections
    Articles; Extreme Computing Research Center; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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