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Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case

Published online by Cambridge University Press:  26 April 2007

Paola F. Antonietti  and
Blanca Ayuso
Paola F. Antonietti
Affiliation:
Dipartimento di Matematica, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy. paola.antonietti@unipv.it
Blanca Ayuso
Affiliation:
Istituto di Matematica Applicata e Tecnologie Informatiche, CNR, via Ferrata 1, 27100 Pavia, Italy. blanca@imati.cnr.it
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Abstract


We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain decomposition methods are presented in a unified framework. For symmetric schemes, it is shown that the condition number of the preconditioned system is of order O(H/h), where H and h are the mesh sizes of the coarse and fine grids respectively, which are assumed to be nested. For non-symmetric schemes, we show by numerical computations that the Eisenstat et al. [SIAM J. Numer. Anal.20 (1983) 345–357] GMRES convergence theory, generally used in the analysis of Schwarz methods for non-symmetric problems, cannot be applied even if the numerical results show that the GMRES applied to the preconditioned systems converges in a finite number of steps and the proposed preconditioners seem to be scalable. Extensive numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods are presented.


Keywords

Domain decomposition methodsdiscontinuous Galerkinelliptic problems.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 1 , January 2007 , pp. 21 - 54
Copyright
© EDP Sciences, SMAI, 2007

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