Complete, Optimal and Optimized Coarse Spaces for Additive Schwarz
The additive Schwarz method does not converge in general when used as a stationary iterative method, it must be used as a preconditioner for a Krylov method. In the two level variant of the additive Schwarz method, a coarse grid correction is added to make the method scalable. We introduce a new coarse space for the additive Schwarz method which makes it convergent when used as a stationary iterative method, and show that an optimal choice even makes the method nilpotent, i.e. it converges in one iteration, independently of the overlap and the number of subdomains. We then show how this optimal choice can be approximated leading to a spectral harmonically enriched coarse space, based on interface eigenvalue problems. We present a convergence analysis of our new coarse corrections in the two subdomain case in two spatial dimensions, and also compare them with GenEO recently proposed by Spillane, Dolean, Hauret, Nataf, Pechstein, and Scheichl, and the local spectral multiscale coarse space proposed by Galvis and Efendiev.
The work is supported by University of Geneva. The second author is partially supported by the Fundamental Research Funds for the Central Universities under grant G2018KY0306. We appreciate the comments of the reviewers that led to a better presentation.
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