A Weak Galerkin Mixed Finite Element Method for Acoustic Wave Equation

Abstract

This paper is concerned with the weak Galerkin mixed finite element method (WG-MFEM) for the second-order hyperbolic acoustic wave equation in velocity-pressure formulation. In this formulation, the original second-order differential equation in time and space is reduced to first-order differential equations by introducing the velocity and pressure variables. We employ the usual discontinuous piecewise-polynomials of degree $k\geq 0$ for the pressure and $k+1$ for the velocity. Furthermore, the normal component of the pressure on the interface of elements is enhanced by polynomials of degree $k+1$. The time derivative is approximated by the backward Euler difference. We show the stability of the semi-discrete and fully-discrete schemes, and obtain the suboptimal order error estimates for the velocity and pressure variables. Numerical experiment confirms our theoretical analysis.