A divergence-free weak virtual element method for the Navier-Stokes equation on polygonal meshes

In this paper, we present a divergence-free weak virtual element method for the Navier-Stokes equation on polygonal meshes. The velocity and the pressure are discretized by the H(div) virtual element and discontinuous piecewise polynomials, respectively. An additional polynomial space that lives on the element edges is introduced to approximate the tangential trace of the velocity. The velocity at the discrete level is point-wise divergence-free and thus the exact mass conservation is preserved in the discretization. Given suitable data conditions, the well-posedness of the discrete problem is proved and a rigorous error analysis of the method is derived. The error with respect to a mesh dependent norm for the velocity depends on the smoothness of the velocity and the approximation of the load term. A series of numerical experiments are reported to validate the performance o f the method.