A new splitting mixed finite element analysis of the viscoelastic wave equation

Abstract

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 \(\varvec{\sigma }=A(x)\nabla u+B(x)\nabla u_t\), 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 \(\sigma \) 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 \(\varvec{\sigma }\) and their convergence rates are presented, and the theoretical analysis on error estimates and convergence is then confirmed.