A mass-lumped mixed finite element method for acoustic wave propagation

Abstract

We consider the numerical approximation of acoustic wave propagation in time domain by a mixed finite element method based on the $$\text {BDM}_1$$–$$\text {P}_0$$ spaces. A mass-lumping strategy for the $$\text {BDM}_1$$ element, originally proposed by Wheeler and Yotov in the context of subsurface flow, is utilized to enable an efficient integration in time. By this mass-lumping strategy, the accuracy of the mixed method is formally reduced to first order. We will show however that the numerical approximation still carries global second order information which is expressed in the super-convergence of the numerical approximation to certain projections of the true solution. Based on this fact, we propose post-processing strategies for the pressure and the velocity leading to piecewise linear approximations with second order accuracy. A complete convergence analysis is provided for the semi-discrete and corresponding fully-discrete approximations, which result from time discretization by the leapfrog method. The efficiency of the proposed strategy is illustrated also in numerical tests.