We propose and study some new additive, two-level non-overlapping Schwarz preconditioned 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
Antonietti, P., Ayuso De Dios, B. (2007). Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case. MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE, 41(1), 21-54 [10.1051/m2an:2007006].
Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case
Ayuso De Dios, BP
2007
Abstract
We propose and study some new additive, two-level non-overlapping Schwarz preconditioned 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 presentedFile | Dimensione | Formato | |
---|---|---|---|
m2an0568Schwarz.pdf
accesso aperto
Tipologia di allegato:
Publisher’s Version (Version of Record, VoR)
Dimensione
494.56 kB
Formato
Adobe PDF
|
494.56 kB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.