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    Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients

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    Type
    Article
    Authors
    Ayuso Dios, Blanca
    Holst, Michael
    Zhu, Yunrong
    Zikatanov, Ludmil
    KAUST Department
    Applied Mathematics and Computational Science Program
    Center for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2013-10-30
    Permanent link to this record
    http://hdl.handle.net/10754/594284
    
    Metadata
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    Abstract
    We introduce and analyze two-level and multilevel preconditioners for a family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with large jumps in the diffusion coefficient. Our approach to IPDG-type methods is based on a splitting of the DG space into two components that are orthogonal in the energy inner product naturally induced by the methods. As a result, the methods and their analysis depend in a crucial way on the diffusion coefficient of the problem. The analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes; dealing simultaneously with the jump in the diffusion coefficient and the non-nested character of the relevant discrete spaces presents additional difficulties in the analysis, which precludes a simple extension of existing results. However, we are able to establish robustness (with respect to the diffusion coefficient) and near-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners, by using a more refined Conjugate Gradient theory. Useful by-products of the analysis are the supporting results on the construction and analysis of simple, efficient and robust two-level and multilevel preconditioners for non-conforming Crouzeix-Raviart discretizations of elliptic problems with jump coefficients. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods. © 2013 American Mathematical Society.
    Citation
    Ayuso de Dios B, Holst M, Zhu Y, Zikatanov L (2013) Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients. Math Comp 83: 1083–1120. Available: http://dx.doi.org/10.1090/s0025-5718-2013-02760-3.
    Publisher
    American Mathematical Society (AMS)
    Journal
    Mathematics of Computation
    DOI
    10.1090/s0025-5718-2013-02760-3
    ae974a485f413a2113503eed53cd6c53
    10.1090/s0025-5718-2013-02760-3
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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