  We present the non-conforming Virtual Element Method (VEM) for the numerical
approximation of velocity and pressure in the steady Stokes problem. The
pressure is approximated using discontinuous piecewise polynomials, while each
component of the velocity is approximated using the nonconforming virtual
element space. On each mesh element the local virtual space contains the space
of polynomials of up to a given degree, plus suitable non-polynomial functions.
The virtual element functions are implicitly defined as the solution of local
Poisson problems with polynomial Neumann boundary conditions. As typical in VEM
approaches, the explicit evaluation of the non-polynomial functions is not
required. This approach makes it possible to construct nonconforming (virtual)
spaces for any polynomial degree regardless of the parity, for two-and
three-dimensional problems, and for meshes with very general polygonal and
polyhedral elements. We show that the non-conforming VEM is inf-sup stable and
establish optimal a priori error estimates for the velocity and pressure
approximations. Numerical examples confirm the convergence analysis and the
effectiveness of the method in providing high-order accurate approximations.
