ASM-BDDC Preconditioners with Variable Polynomial Degree for CG- and DG-SEM

Abstract

Discontinuous Galerkin (DG) solvers for partial differential equations are well suited to treat nonconforming meshes and inhomogeneous polynomial orders required by hp-adaptivity. In this paper, we present a new preconditioner for spectral-DG based on the so-called Auxiliary Space Method (ASM). First, Balancing Domain Decomposition by Constraints (BDDC) preconditioners for continuous spectral elements with homogeneous polynomial degree are extended to inhomogeneous polynomial distributions. Second, a BDDC-based preconditioner for DG is drawn from the ASM to solve second order elliptic problems. Such a preconditioner is proved to be quasi-optimal with respect to the maximal polynomial jump between two spectral elements and the ratio between subdomain size and element size. Numerical results confirm the theoretical bounds. In particular, it is shown that for acceptable variations of the polynomial degree, the preconditioner is scalable and quasi-optimal.