Domain decomposition for multiscale PDEs |
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Authors: | I G Graham P O Lechner R Scheichl |
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Institution: | (1) Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK;(2) d-fine GmbH, Opernplatz 2, 60313 Frankfurt am Main, Germany |
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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, when the classical method fails
to be robust. In particular our estimates prove very precisely the previously made empirical observation that the use of low-energy
coarse spaces can lead to robust preconditioners. We go on to consider coarse spaces constructed from multiscale finite elements
and prove that preconditioners using this type of coarsening lead to robust preconditioners for a variety of binary (i.e.,
two-scale) media model problems. Moreover numerical experiments show that the new preconditioner has greatly improved performance
over standard preconditioners even in the random coefficient case. We show also how the analysis extends in a straightforward
way to multiplicative versions of the Schwarz method.
We would like to thank Bill McLean for very useful discussions concerning this work. We would also like to thank Maksymilian
Dryja for helping us to improve the result in Theorem 4.3. |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 65F10 65N22 65N55 |
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