Mixed Virtual Element approximation of linear acoustic wave equation

TL;DR

This paper introduces a Mixed Virtual Element Method for the linear acoustic wave equation that conserves energy, is stable, and converges optimally, with practical verification and engineering applications.

Contribution

It develops a novel mixed virtual element approach for acoustic waves, ensuring energy conservation and optimal convergence in both semi- and fully-discrete forms.

Findings

Exact energy conservation in the semi-discrete system

Optimal convergence rate with respect to mesh size

Successful verification tests and engineering applications

Abstract

We design a Mixed Virtual Element Method for the approximated solution to the first-order form of the acoustic wave equation. In absence of external load, the semi-discrete method exactly conserves the system energy. To integrate in time the semi-discrete problem we consider a classical theta-method scheme. We carry out the stability and convergence analysis in the energy norm for the semi-discrete problem showing optimal rate of convergence with respect to the mesh size. We further study the property of energy conservation for the fully-discrete system. Finally, we present some verification tests as well as engineering application of the method.