In this paper, we propose and analyse a numerical method to solve 2D Dirichlet time-harmonic elastic wave equations. The procedure is based on the decoupling of the elastic vector field into scalar Pressure ($P$-) and Shear ($S$-) waves via a suitable Helmholtz-Hodge decomposition. For the approximation of the two scalar potentials we apply a virtual element method associated with different mesh sizes and degrees of accuracy. We provide for the stability of the method and a convergence error estimate in the $L^2$-norm for the displacement field, in which the contributions to the error associated with the $P$- and $S$- waves are separated. In contrast to standard approaches that are directly applied to the vector formulation, this procedure allows for keeping track of the two different wave numbers, that depend on the $P$- and $S$- speeds of propagation and, therefore, for using a high-order method for the approximation of the wave associated with the higher wave number. Some numerical tests, validating the theoretical results and showing the good performance of the proposed approach, are presented.

翻译：本文提出并分析了一种数值方法，用于求解二维Dirichlet时谐弹性波动方程。该方法基于通过适当的Helmholtz-Hodge分解将弹性矢量场解耦为标量压缩波（P-波）和剪切波（S-波）。为逼近这两个标量势，我们应用了关联不同网格尺寸和精度的虚拟元方法。我们提供了方法的稳定性分析以及位移场$L^2$-范数下的收敛误差估计，其中与P-波和S-波相关的误差贡献被分离。与直接应用于矢量形式的传统方法相比，该过程能够追踪依赖于P-波和S-波传播速度的两个不同波数，从而可采用高阶方法逼近与较高波数相关的波。文中还给出了验证理论结果并展示所提方法良好性能的数值试验。