Additive average Schwarz with adaptive coarse spaces: scalable algorithms for multiscale problems

We present an analysis of the additive average Schwarz preconditioner with two newly proposed adaptively enriched coarse spaces, which were presented at the twenty-third international conference on domain decomposition methods in Korea, for solving second-order elliptic problems with highly varying and discontinuous coefficients. It is shown that the condition number of the preconditioned system is bounded independently of the variations and the jumps in the coefficient while depending only on a prescribed threshold for the eigenvalues of the coarse space, and it depends linearly on the mesh parameter ratio H/h that is the ratio between the subdomain size and the mesh size thereby retaining the same optimality and scalability of the original additive average Schwarz preconditioner.