We design a Mixed Virtual Element Method for the approximated solution to the first-order form of the acoustic wave equation. In the absence of external loads, 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 an 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 applications of the method.

中文翻译：

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线性声波方程的混合虚拟元近似

我们设计了一种混合虚拟单元法来求解声波方程的一阶形式。在没有外部负载的情况下，半离散方法恰好守恒了系统能量。为了及时积分半离散问题，我们考虑经典的 $\theta $ 方法方案。我们对半离散问题的能量范数进行了稳定性和收敛性分析，显示了相对于网格尺寸的最佳收敛率。我们进一步研究了全离散系统的能量守恒性质。最后，我们介绍了该方法的一些验证测试以及工程应用。