Additive Schwarz Methods for DG Discretization of Elliptic Problems with Discontinuous Coefficient

In this paper we consider a second order elliptic problem defined on a polygonal region Ω, where the diffusion coefficient is a discontinuous function. The problem is discretized by a symmetric interior penalty discontinuous Galerkin (DG) finite element method with triangular elements and piecewise linear functions. Our goal is to design and analyze an additive Schwarz method (ASM), see the book by Toselli and Widlund [2005], for solving the resulting discrete problem with rate of convergence independent of the jumps of the coefficient. The method is two-level and without overlap of the substructures into which the original region Ω is partitioned. Usually, two level ASMs for discretizations on fine mesh of size h are being built by introducing a partitioning of the domain into subdomains of size H > h, where local solvers are applied in parallel. A global coarse problem is then typically based on the same partitioning. This approach has been generalized for nonoverlapping domain decomposition methods for DG discretizations by Feng and Karakashian [2001] and further extended by Antonietti and Ayuso [2007] by allowing the coarse grid with mesh size H to be a refinement of the original partitioning into subdomains where the local solvers are applied. The ASM discussed here is a generalization to non-constant diffusion coefficient and very small subdomains of methods mentioned above and of those presented in Dryja and Sarkis [2010] and Dryja et al. [2014]. Other recent works towards domain decomposition preconditioning of DG discretizations of problems with strongly varying coefficients include Ayuso de Dios et al. [2014], Brix et al. [2013] and Canuto et al. [2014]. In this paper, local solvers act on subdomains which are equal to single elements of the fine mesh. By allowing single element subdomains we substantially increase the level of paral-