The Weak Galerkin Finite Element Method for the Symmetric Hyperbolic Systems

TL;DR

This paper introduces a flexible weak Galerkin finite element method for symmetric hyperbolic systems, providing stability, optimal error estimates, and applications to singularly perturbed equations with numerical validation.

Contribution

It develops a novel weak Galerkin scheme with stability and optimal error bounds for symmetric hyperbolic systems, including singular perturbation problems.

Findings

Optimal $L_2$-error estimate of $O(h^{k+1/2})$ for the scheme

Effective application to singularly perturbed convection-diffusion-reaction equations

Numerical results confirm the method's accuracy and stability.

Abstract

In this paper, we present and analyze a weak Galerkin finite element (WG) method for solving the symmetric hyperbolic systems. This method is highly flexible by allowing the use of discontinuous finite elements on element and its boundary independently of each other. By introducing special weak derivative, we construct a stable weak Galerkin scheme and derive the optimal $L_2$-error estimate of $O(h^{k+\frac{1}{2}})$-order for the discrete solution when the $k$-order polynomials are used for $k\geq 0$. As application, we discuss this WG method for solving the singularly perturbed convection-diffusion-reaction equation and derive an $\varepsilon$-uniform error estimate of order $k+1/2$. Numerical examples are provided to show the effectiveness of the proposed WG method.