BDDC algorithms for advection-diffusion problems with HDG discretizations

In this paper, a preconditioned GMRES method is developed and analyzed for solving the linear system from advection-diffusion equations with the hybridizable discontinuous Galerkin (HDG) discretization. The preconditioner is the balancing domain decomposition methods (BDDC), one of the most popular nonoverlapping domain decomposition methods. 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 subdomain problem size. The convergence deteriorates when the viscosity decreases. These results are similar to those with the standard finite element discretizations. Numerical results of two examples in two dimensions are provided to confirm the theory.

Automated summary

A preconditioned GMRES method is developed and analyzed for solving the linear system from advection-diffusion equations with HDG discretization. The use of BDDC as a preconditioner showed promising results, with the number of iterations being independent of subdomain quantity for large viscosity. However, convergence deteriorates with decreasing viscosity, similar to standard finite element discretizations.