Achieving robustness through coarse space enrichment in the two level Schwarz framework

As many domain decomposition methods the two level Additive Schwarz method may suffer from a lack of robustness with respect to coefficient variation in the underlying set of PDEs. This is the case in particular if the partition into subdomains is not aligned with all jumps in the coefficients. Thanks to the theoretical analysis of two level Schwarz methods (see [11] and references therein) this lack of robustness can be traced back to the so called stable splitting property (already in [4]). Following the same ideas as in the pioneering work [1] we propose to solve a generalized eigenvalue problem in each subdomain which identifies which vectors are responsible for slow convergence. The spectral problem is specifically chosen to separate components that violate the stable splitting property. These vectors are then used to span the coarse space which is taken care of by a direct solve while all remaining components can be resolved on the subdomains. The result is a preconditioned system with a condition number estimate that does not depend on the number of subdomains or any jumps in the coefficients. We refer to this method as GenEO for Generalized Eigenproblems in the Overlaps. It is closely related to the work of [2] where the same strategy leads to a different eigenproblem and different condition number estimate (which also does not depend on the jumps in the coefficients or on the number of subdomains). A full theoretical analysis of the two level Additive Schwarz method with the GenEO coarse space (first briefly introduced in [8]) is given in [7]. Here our purpose is to show the steps leading from the abstract Schwarz theory to the choice of our generalized eigenvalue problem (5). In the first section we introduce the rather wide range of problems to which the method applies and give the classical two-level Schwarz condition number estimate in the abstract framework (again, see [11] and references therein). In the second section we work to make this condition local (on each subdomain), identify the GenEO generalized eigenproblem and state our main result (Theorem 2). Finally in the third section we illustrate the result numerically.