BDDC Algorithms with deluxe scaling and adaptive selection of primal constraints for Raviart-Thomas vector fields

Abstract

A BDDC domain decomposition preconditioner is defined by a coarse component, expressed in terms of primal constraints, a weighted average across the interface between the subdomains, and local components given in terms of solvers of local subdomain problems. BDDC methods for vector field problems discretized with Raviart-Thomas finite elements are introduced. The methods are based on a new type of weighted average and an adaptive selection of primal constraints developed to deal with coefficients with high contrast even inside individual subdomains. For problems with very many subdomains, a third level of the preconditioner is introduced. Under the assumption that the subdomains are all built from elements of a coarse triangulation of the given domain, and that the material parameters are constant in each subdomain, a bound is obtained for the condition number of the preconditioned linear system which is independent of the values and the jumps of these parameters across the interface between the subdomains. Numerical experiments, using the PETSc library, are also presented which support the theory and show the effectiveness of the algorithms even for problems not covered by the theory. Included are also experiments with Brezzi-Douglas-Marini finite element approximations.