The weak Galerkin finite element method for the transport-reaction equation

Abstract

We present and analyze a weak Galerkin finite element method for solving the transportreaction equation in d space dimensions. This method is highly flexible by allowing the use of discontinuous finite element on general meshes consisting of arbitrary polygon/polyhedra. We derive the L 2-error estimate of O(h k+ 1 2)-order for the discrete solution when the kth-order polynomials are used for k ≥ 0. Moreover, for a special class of meshes, we also obtain the optimal error estimate of O(h k+1)-order in the L 2-norm. A derivative recovery formula is presented to approximate the convection directional derivative and the corresponding superconvergence estimate is given. Numerical examples on compatible and non-compatible meshes are provided to show the effectiveness of this weak Galerkin method.