A nonoverlapping spectral additive Schwarz method for interior penalty discontinuous Galerkin discretizations of Anisotropic Elliptic Problems

We design and analyze a nonoverlapping additive Schwarz preconditioner for interior penalty discontinuous Galerkin discretizations of anisotropic elliptic problems. The preconditioned method coupled with a Krylov subspace iteration is shown to be independent of the highly discontinuous (and anisotropic) jump coefficients as well as the subdomain size. To increase efficacy, various auxiliary spaces are considered to reduce the size of the coarse grid operator. We demonstrate how to modify the additive Schwarz preconditioner such that it is applicable to the nonsymmetric IPDG schemes. Several numerical experiments verify the theory and validate the robustness of the preconditioner.