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In this paper, we propose and analyse a numerical method to solve 2DDirichlet time-harmonic elastic wave equations. The procedure is based on thedecoupling of the elastic vector field into scalar Pressure ($P$-) and Shear($S$-) waves via a suitable Helmholtz-Hodge decomposition. For theapproximation of the two scalar potentials we apply a virtual element methodassociated with different mesh sizes and degrees of accuracy. We provide forthe stability of the method and a convergence error estimate in the $L^2$-normfor the displacement field, in which the contributions to the error associatedwith the $P$- and $S$- waves are separated. In contrast to standard approachesthat are directly applied to the vector formulation, this procedure allows forkeeping 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 forthe approximation of the wave associated with the higher wave number. Somenumerical tests, validating the theoretical results and showing the goodperformance of the proposed approach, are presented.

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