An Analysis of the Weak Finite Element Method for Convection-Diffusion Equations

TL;DR

This paper analyzes the weak finite element method for convection-diffusion equations, establishing error estimates and superconvergence results, supported by numerical examples.

Contribution

It introduces a weak finite element scheme based on a new variational form and proves optimal error estimates and superconvergence properties.

Findings

Optimal order error estimates in multiple norms

H^1-superconvergence of order k+2 under certain conditions

Numerical examples confirming theoretical results

Abstract

We study the weak finite element method solving convection-diffusion equations. A weak finite element scheme is presented based on a spacial variational form. We established a weak embedding inequality that is very useful in the weak finite element analysis. The optimal order error estimates are derived in the discrete $H^1$-norm, the $L_2$-norm and the $L_\infty$-norm, respectively. In particular, the $H^1$-superconvergence of order $k+2$ is given under certain condition. Finally, numerical examples are provided to illustrate our theoretical analysis