published by the press syndicate of the university of cambridgeThe Pitt

No abstract

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 hp-version discontinuous Galerkin finite element method for a class of these problems that satisfy the Cordès condition. It is shown that the method exhibits a convergence rate that is optimal with respect to the mesh size h and suboptimal with respect to the polynomial degree p by only half an order. Numerical experiments demonstrate the accuracy of the method and illustrate the potential of exponential convergence under hp-refinement for problems with discontinuous coefficients and nonsmooth solutions.

Abstract. We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of ∆x only. For example, when polynomials of degree k are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order k + 1/2 in the L 2 -norm, whereas the post-processed approximation is of order 2k + 1; if the exact solution is in L 2 only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order k + 1/2 in L 2 (Ω 0 ), where Ω 0 is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

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 hp-version discontinuous Galerkin finite element method for a class of these problems that satisfy the Cordès condition. It is shown that the method exhibits a convergence rate that is optimal with respect to the mesh size h and suboptimal with respect to the polynomial degree p by only half an order. Numerical experiments demonstrate the accuracy of the method and illustrate the potential of exponential convergence under hp-refinement for problems with discontinuous coefficients and nonsmooth solutions.