A Simplified Weak Galerkin Finite Element Method: Algorithm and Error Estimates

TL;DR

This paper introduces a simplified weak Galerkin finite element method for convection-diffusion-reaction equations, reducing computational complexity while maintaining stability and accuracy, supported by theoretical error estimates and numerical verification.

Contribution

It develops a simplified weak Galerkin method that uses boundary degrees of freedom, offering computational efficiency and proven error bounds.

Findings

Reduced computational complexity compared to regular weak Galerkin methods

Established stability and optimal error estimates in $H^1$ and $L^2$ norms

Numerical results confirm theoretical error estimates and reveal superconvergence phenomena

Abstract

In this article a simplified weak Galerkin finite element method is developed for the Dirichlet boundary value problem of convection-diffusion-reaction equations. The simplified weak Galerkin method utilizes only the degrees of freedom on the boundary of each element and, hence, has significantly reduced computational complexity over the regular weak Galerkin finite element method. A stability and some optimal order error estimates in the $H^1$ and $L^2$ norms are established for the corresponding numerical solutions. Numerical results are presented to verify the theory error estimates and a superconvergence phenomena on rectangular partitions.