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

TL;DR

This paper introduces a flexible weak Galerkin finite element method for the transport-reaction equation, achieving optimal error estimates and superconvergence, applicable to general polygonal/polyhedral meshes.

Contribution

The paper develops a weak Galerkin finite element approach with error analysis and superconvergence results for the transport-reaction equation on arbitrary meshes.

Findings

Achieves $O(h^{k+1})$-order error estimate on special meshes.

Provides a derivative recovery formula for directional derivatives.

Demonstrates effectiveness through numerical examples.

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.