Multilevel Preconditioners for Discontinuous Galerkin Approximations of Elliptic Problems with Jump Coefficients

Details Multilevel Preconditioners for Discontinuous Galerkin Approximations of Elliptic Problems with Jump Coefficients Multilevel Preconditioners for Discontinuous Galerkin Approximations of Elliptic Problems with Jump Coefficients

2010-12-06

arxiv.org/abs/1012.1287v2

Mathematics

Numerical Analysis

32 pages, 4 figures, 7 tables

Scientific paper

We introduce and analyze two-level and multi-level 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 extra 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 nearly-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods. The paper includes an Appendix with a collection of proofs of several technical results required for the analysis.

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