Bounds on the spectra of Schur complements of large H-TFETI-DP clusters for 2D Laplacian

Abstract

Bounds on the spectrum of Schur complements of subdomain stiffness matrices of the discretized Laplacian with respect to interior variables are important in the convergence analysis of finite element tearing and interconnecting (FETI)-based domain decomposition methods. Here, we are interested in bounds on the regular condition number of Schur complements of "floating" clusters, that is, of matrices comprising the Schur complements of subdomains with prescribed zero Neumann conditions that are joined on the primal level by edge averages. Using some known results, angles of subspaces, and known bounds on the spectrum of Schur complements associated with square domains, we give bounds on the regular condition number of the Schur complement of some "floating" clusters arising from the discretization and decomposition of 2D Laplacian on domains comprising square subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m x m square subdomains joined by edge averages increases proportionally to m. The estimates are compared with numerical values and used in the analysis of H-FETI-DP methods. Though the research has been motivated by an effort to extend the scope of scalability of FETI-based solvers to variational inequalities, the experiments indicate that H-TFETI-DP with large clusters can be useful for the solution of huge linear elliptic problems discretized by sufficiently regular grids.