A second order finite element method with mass lumping for wave equations in $H(\mathrm{div})$

TL;DR

This paper develops an efficient second order finite element method with mass lumping for wave equations in $H( ext{div})$, enabling accurate and computationally efficient simulations of acoustic wave propagation with internal damping.

Contribution

It introduces a novel mass-lumping strategy for $H( ext{div})$ finite elements that preserves second order accuracy despite quadrature approximations.

Findings

Mass-lumping leads to a block-diagonal mass matrix.

Second order accuracy is maintained despite quadrature errors.

Numerical tests confirm theoretical error estimates.

Abstract

We consider the efficient numerical approximation of acoustic wave propagation in time domain by a finite element method with mass lumping. In the presence of internal damping, the problem can be reduced to a second order formulation in time for the velocity field alone. For the spatial approximation we consider $H(\mathrm{div})$--conforming finite elements of second order. In order to allow for an efficient time integration, we propose a mass-lumping strategy based on approximation of the $L^2$-scalar product by inexact numerical integration which leads to a block-diagonal mass matrix. A careful error analysis allows to show that second order accuracy is not reduced by the quadrature errors which is illustrated also by numerical tests.