GenEO spectral coarse spaces in SPD domain decomposition
Résumé
Two-level domain decomposition methods are preconditioned Krylov solvers. What sep- arates one and two- level domain decomposition method is the presence of a coarse space in the latter. The abstract Schwarz framework is a formalism that allows to define and study a large variety of two-level methods. The objective of this article is to define, in the abstract Schwarz framework, a family of coarse spaces called the GenEO coarse spaces (for General- ized Eigenvalues in the Overlaps). This is a generalization of methods that exist already for particular choices of domain decomposition methods. Bounds for the condition numbers of the preconditioned operators are proved that are independent of the parameters in the prob- lem (e.g., any coefficients in an underlying PDE or the number of subdomains). The coarse spaces are computed by finding low or high frequency spaces of some well chosen generalized eigenvalue problems in each subdomain. The abstract framework is illustrated by defining two-level Additive Schwarz, Neumann-Neumann and Inexact Schwarz preconditioners for a two dimensional linear elasticity problem. Explicit theoretical bounds as well as numerical results are provided for this example.
Domaines
Analyse numérique [math.NA]Origine | Fichiers produits par l'(les) auteur(s) |
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