A virtual element method for the solution of 2D time-harmonic elastic wave equations via scalar potentials

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.