Abstract
In this article, we formulate and analyze a twolevel preconditioner for optimized Schwarz and 2Lagrange multiplier methods for PDEs with highly heterogeneous (multiscale) diffusion coefficients. The preconditioner is equipped with an automatic coarse space consisting of lowfrequency modes of approximate subdomain DirichlettoNeumann maps. Under a suitable change of basis, the preconditioner is a $2 \times 2$ block upper triangular matrix with the identity matrix in the upperleft block. We show that the spectrum of the preconditioned system is included in the disk having center $z=1/2$ and radius $r=1/2  \epsilon$, where $0 < \epsilon < 1/2$ is a parameter that we can choose. We further show that the GMRES algorithm applied to our heterogeneous system converges in $O(1/\epsilon)$ iterations (neglecting certain polylogarithmic terms). The number $\epsilon$ can be made arbitrarily large by automatically enriching the coarse space. Our theoretical results are confirmed by numerical experiments.
Original language  English 

Pages (fromto)  A2896–A2923 
Number of pages  28 
Journal  SIAM Journal on Scientific Computing 
Volume  37 
Issue number  6 
Early online date  10 Dec 2015 
DOIs  
Publication status  Published  2015 
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Profiles

Sebastien Loisel
 School of Mathematical & Computer Sciences  Assistant Professor
 School of Mathematical & Computer Sciences, Mathematics  Assistant Professor
Person: Academic (Research & Teaching)