The Weak Galerkin Finite Element Method for the Transport-Reaction Equation

Tie Zhang; Shangyou Zhang

doi:10.1016/j.jcp.2020.109399

paper

The Weak Galerkin Finite Element Method for the Transport-Reaction Equation

arXiv:2011.11861 · doi:10.1016/j.jcp.2020.109399

Abstract

We present and analyze a weak Galerkin finite element method for solving the transport-reaction 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 \textcolor[rgb]{0.00,0.00,1.00}{ $L_{2}$ -error estimate} of $O (h^{k + \frac{1}{2}})$ -order for the discrete solution when the $k$ th-order polynomials are used for $k \geq 0$ . Moreover, for a special class of meshes, we also obtain the \textcolor[rgb]{0.00,0.00,1.00}{optimal error} estimate of $O (h^{k + 1})$ -order in the $L_{2}$ -norm. A derivative recovery formula is presented to approximate the convection \textcolor[rgb]{1.00,0.00,0.00}{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.