On conditioning of Schur complements of H-TFETI clusters for 2D problems governed by Laplacian

Abstract

Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements of the clusters arising in the discretization of problems governed by 2D Laplacian. The estimates depend on the decomposition and discretization parameters and gluing conditions. We also show how to plug the results into the analysis of H-TFETI methods and compare the estimates with numerical experiments. The results are useful for the analysis and implementation of powerful massively parallel scalable algorithms for the solution of variational inequalities.