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

Authors: Blanca Ayuso de Dios, Michael Holst, Yunrong Zhu and Ludmil Zikatanov
Journal: Math. Comp. 83 (2014), 1083-1120
MSC (2010): Primary 65N30, 65N55
Published electronically: October 30, 2013
MathSciNet review: 3167451
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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.

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Blanca Ayuso de Dios
Affiliation: Centre de Recerca Matematica, Campus de Bellaterra, Bellaterra, 08193, Spain
Address at time of publication: Center for Uncertainty Quantification in Computational Science & Engineering, Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE), King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia

Michael Holst
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093

Yunrong Zhu
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093
Address at time of publication: Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085

Ludmil Zikatanov
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

Keywords: Multilevel preconditioner, discontinuous Galerkin methods, Crouzeix-Raviart finite elements, space decomposition
Received by editor(s): January 14, 2011
Received by editor(s) in revised form: February 1, 2012, June 15, 2012, and October 19, 2012
Published electronically: October 30, 2013
Article copyright: © Copyright 2013 American Mathematical Society

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