BDDC preconditioners for continuous and discontinuous Galerkin methods using spectral/hp elements with variable local polynomial degree

Abstract

Locally adapted meshes and polynomial degrees can greatly improve spectral element accuracy and applicability. A balancing domain decomposition by constraints (BDDC) preconditioner is constructed and analysed for both continuous (CG) and discontinuous (DG) Galerkin discretizations of scalar elliptic problems, built by nodal spectral elements with variable polynomial degrees. The DG case is reduced to the CG case via the auxiliary space method. The proposed BDDC preconditioner is proved to be scalable in the number of subdomains and quasi-optimal in both the ratio of local polynomial degrees and element sizes and the ratio of subdomain and element sizes. Several numerical experiments in the plane confirm the obtained theoretical convergence rate estimates, and illustrate the preconditioner performance for both CG and DG discretizations. Different configurations with locally adapted polynomial degrees are studied, as well as the preconditioner robustness with respect to discontinuities of the elliptic coefficients across subdomain boundaries. These results apply also to other dual-primal preconditioners defined by the same set of primal constraints, such as FETI-DP preconditioners.