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

Ayuso De Dios, B., Holst, M., Zhu, Y., Zikatanov, L. (2014). Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients. MATHEMATICS OF COMPUTATION, 83(287), 1083-1120 [10.1090/S0025-5718-2013-02760-3].

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

Ayuso De Dios, B
;
2014

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
Articolo in rivista - Articolo scientifico
BPX preconditioner; DISCONTINUOUS GALERKIN; jump coefficients
English
2014
83
287
1083
1120
none
Ayuso De Dios, B., Holst, M., Zhu, Y., Zikatanov, L. (2014). Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients. MATHEMATICS OF COMPUTATION, 83(287), 1083-1120 [10.1090/S0025-5718-2013-02760-3].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/218128
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