BDDC Algorithms for Advection-diffusion problems with HDG Discretizations

TL;DR

This paper analyzes the convergence of BDDC algorithms preconditioned GMRES for advection-diffusion equations discretized with HDG, highlighting effects of viscosity and discretization degree on performance.

Contribution

It extends BDDC methods to HDG discretizations for advection-diffusion problems and analyzes convergence behavior with respect to viscosity and primal constraints.

Findings

Iteration count is independent of the number of subdomains for large viscosity.

Convergence deteriorates as viscosity decreases.

Higher degree HDG discretizations are more affected by primal constraints.

Abstract

The balancing domain decomposition methods (BDDC) are originally introduced for symmetric positive definite systems and have been extended to the nonsymmetric positive definite system from the linear finite element discretization of advection-diffusion equations. In this paper, the convergence of the GMRES method is analyzed for the BDDC preconditioned linear system from advection-diffusion equations with the hybridizable discontinuous Galerkin (HDG) discretization. Compared to the finite element discretizations, several additional norms for the numerical trace have to be used and the equivalence between the bilinear forms and norms needs to be established. For large viscosity, if the subdomain size is small enough, the number of iterations is independent of the number of subdomains and depends only slightly on the sudomain problem size. The convergence deteriorates when the viscosity decreases. These results are similar to those with the finite element discretizations. Moreover, the effects of the additional primal constraints used in the BDDC algorithms are more significant with the higher degree HDG discretizations. The results of two two-dimensional examples are provided to confirm our theory.