An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems

We consider a scalar advection-diffusion problem and a recently proposed discontinuous Galerkin approximation, which employs discontinuous finite element spaces and suitable bilinear forms containing interface terms that ensure consistency. For the corresponding sparse, non-symmetric linear system, we propose and study an additive, two--level overlapping Schwarz preconditioner, consisting of a coarse problem on a coarse triangulation and local solvers associated to suitable problems defined on a family of subdomains. This is a generalization of the corresponding overlapping method for approximations on continuous finite element spaces. Related to the lack of continuity of our approximation spaces, some interesting new features arise in our generalization, which have no analog in the conforming case. We prove an upper bound for the number of iterations obtained by using this preconditioner with GMRES, which is independent of the number of degrees of freedom of the original problem and the number of subdomains. The performance of the method is illustrated by several numerical experiments for different test problems, using linear finite elements in two dimensions.