Error analysis for a non-conforming virtual element discretization of the acoustic problem

We introduce non conforming virtual elements to approximate the eigenvalues and eigenfunctions of the two dimensional acoustic vibration problem. We focus our attention on the pressure formulation of the acoustic vibration problem in order to discretize it with a suitable non conforming virtual space for $\mathrm{H}^1$. With the aid of the theory of non-compact operators we prove convergence and spectral correctness of the method. To illustrate the theoretical results, we report numerical tests on different polygonal meshes, in order to show the accuracy of the method on the approximation of the spectrum.