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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.

In this paper, we propose and analyze a $C^1$-virtual element method of high order to solve the Brinkman problem formulated in terms of the stream function. The velocity is obtained as a simple post-process from stream function and a novel strategy is written to recover the fluid pressure. We establish optimal a priori error estimates for the stream function, velocity and pressure with constants independent of the viscosity. Finally, we report some numerical test illustrating the behavior of the virtual scheme and supporting our theoretical results on different families of polygonal meshes.

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