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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 (MonteCarlo 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 lowenergy 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., twoscale) 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.
Original language  English 

Pages (fromto)  589626 
Number of pages  38 
Journal  Numerische Mathematik 
Volume  106 
Issue number  4 
DOIs  
Publication status  Published  Jun 2007 
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 1 Finished

ESTABLISHMENT OF THE UNIVERSITY OF BATH CENTRE FOR COMPLEX S YSTEMS  O/HEAD SPLIT SM75% EN25%
Budd, C., Almond, D., Britton, N., Hunt, G., Hurn, M. & Smyshlyaev, V. P.
Engineering and Physical Sciences Research Council
29/11/04 → 28/11/09
Project: Research council