This article deals with the development of the virtual element method for the approximation of semilinear hyperbolic problems. For the space discretization, two different operators are used: the energy projection operator ${\mathrm{\Pi }}_{k,K}^{\mathrm{\nabla }}$ and an internal $L^2$-projection operator ${\mathrm{\Pi }}_{k,K}^0$. In order to deal with the time derivative, a Newmark scheme is employed; and the resulted fully discrete scheme is analysed. Moreover, with the help of projection operators, optimal error estimates are derived for both semi- and fully discrete schemes in $L^2$-norm and $H^1$-norm. We have conducted numerical experiments on polygonal meshes to illustrate the performance of the proposed scheme and validate the theoretical findings.

In this work, a new Virtual Element Method (VEM) of arbitrary order k ≥ 2 for the time dependent Navier–Stokes equations in stream-function form is proposed and analyzed. Using suitable projection operators, the bilinear and trilinear terms are discretized by only using the proposed degrees of freedom associated with the virtual space. Under certain assumptions on the computational domain, error estimations are derived and shown that the method is optimally convergent in both space and time variables. Finally, to justify the theoretical analysis, four benchmark examples are examined numerically.