A virtual element discretization for the time dependent Navier-Stokes equations in stream-function formulation

In this work, a new Virtual Element Method (VEM) of arbitrary order $k \geq 2$ 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.