Discontinuous Galerkin finite element approximation of non-divergence form elliptic equations with Cordes coefficients

Abstract

Non-divergence form elliptic equations with discontinuous coefficients do not generally posses a weak formulation, thus presenting an obstacle to their numerical solution by classical finite element methods. We propose a new -version discontinuous Galerkin finite element method for a class of these problems that satisfy the Cordes condition. It is shown that the method exhibits a convergence rate that is optimal with respect to the mesh size and suboptimal with respect to the polynomial degree by only half an order. Numerical experiments demonstrate the accuracy of the method and illustrate the potential of exponential convergence under -refinement for problems with discontinuous coefficients and nonsmooth solutions.