A Domain Decomposition Method for Discretization of Multiscale Elliptic Problems by Discontinuous Galerkin Method

In this paper boundary value problems for second order elliptic equations with highly discontinuous coefficients are considered on a 2D polygonal region. The problems are discretized by a discontinuous Galerkin (DG) with finite element method (FEM) on triangular elements using piecewise linear functions. The goal is to design and analyze a parallel algorithm for solving the discrete problem whose rate of convergence is independent of the jumps of the coefficients. The method discussed is an additive Schwarz method (ASM) which belongs to a class of domain decomposition methods and is one of the most efficient parallel algorithm for solving discretizations of PDEs. It turns out that the convergence of the method presented here is almost optimal and only weakly depends on the jumps of coefficients. The suggested method is very well suited for parallel computations.