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

TL;DR

This paper introduces a mass-lumped mixed finite element method for acoustic wave simulation that achieves second order accuracy through super-convergence and post-processing, despite a formal first order reduction.

Contribution

It develops a mass-lumped mixed finite element approach for acoustic waves, with super-convergence and post-processing techniques to recover second order accuracy.

Findings

The method achieves super-convergence of second order for pressure and velocity.

Post-processing improves the accuracy to second order.

Numerical tests confirm the efficiency and accuracy of the approach.

Abstract

We consider the numerical approximation of acoustic wave propagation in the time domain by a mixed finite element method based on the BDM1-P0 spaces. A mass-lumping strategy for the BDM1 element, originally proposed by Wheeler and Yotov in the context of subsurface flow problems, 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 as super-convergence of the numerical approximation to certain projections of the true solution. Based on this fact, we propose post-processing strategies for both variables, the pressure and the velocity, which yield piecewise linear approximations of 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. In addition, some numerical tests are presented to illustrate the efficiency of the proposed approach.