The Weak Galerkin Finite Element Method for the Symmetric Hyperbolic Systems

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.