DMCA
Domain decomposition for multiscale PDEs
| Venue: | Numer. Math |
| Citations: | 44 - 16 self |
BibTeX
@ARTICLE{Graham_domaindecomposition,
author = {I. G. Graham and P. Lechner and R. Scheichl},
title = {Domain decomposition for multiscale PDEs},
journal = {Numer. Math},
year = {}
}
Share
 |
 |
 |
 |
||
OpenURL
Abstract
We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises often in practice, for example in the computation of flows in heterogeneous porous media, in both the deterministic and (Monte-Carlo simulated) stochastic cases. We consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid. We perform a new analysis of the preconditioned matrix, which shows rather explicitly how its condition number depends on the variable coefficient in the PDE as well as on the coarse mesh and overlap parameters. The classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements. By contrast, our new results show that, with a good choice of subdomains and coarse space basis functions, the preconditioner can still be robust even for large coefficient variation inside domains,
Keyphrases
domain decomposition multiscale pdes variable coefficient coarse mesh classical estimate coefficient varies additive schwarz domain decomposition preconditioners stochastic case guarantee good conditioning coarse grid new analysis piecewise linear finite element approximation condition number new result global solve elliptic pdes coarse grid element overlap parameter general coarse space large coefficient variation coarse space basis function heterogeneous porous medium good choice local solves preconditioned matrix general overlapping
