A New Splitting Mixed Finite Element Analysis of the Viscoelastic Wave Equation

This paper aims to propose a new splitting mixed finite element method (MFE) for solving viscoelastic wave equations and give convergence analysis. First, by introducing two new variables q=u_t and σ=A(x)∇ u+B(x)∇ u_t , a new system of first-order differential-integral equations is derived from the original second-order viscoelastic wave equation. Then, the semi-discrete and fully-discrete splitting MFE schemes are proposed by using the MFE spaces and the second-order time discetization. By the two schemes the approximate solutions for the unknowns u, u_t and σ are obtained simultaneously. It is proved that the semi-discrete and fully-discrete schemes have the optimal error estimates in L^2 -norm. Meanwhile, it is proved that the fully-discrete SMFE scheme based on the Raviart-Thomas mixed finite element spaces and the uniform rectangular mesh partitions is super convergent. Finally, numerical experiments to compute the L^2 errors for approximating u, q and σ and their convergence rates are presented, and the theoretical analysis on error estimates and convergence is then confirmed.