A divergence-free reconstruction of the nonconforming virtual element method for the Stokes problem

Abstract

In this paper, we propose and investigate a divergence-free reconstruction of the nonconforming virtual element for the Stokes problem. By constructing the computable Raviart–Thomas-like interpolation operator, we guarantee the independence between the velocity error estimation |u−uh|1,h and the continuous pressure p, as it happens for the divergence-free flow solver. Moreover, this modified scheme can also inherit the advantages of the classical nonconforming virtual element method, such as, very general meshes including non-convex and degenerate elements, a unified scheme for an arbitrary-order approximation accuracy k, etc. Then, we provide the optimal L2-error estimates for the velocity gradient and the pressure by taking advantage of the Raviart–Thomas-like interpolation operator and avoiding the use of a trace inequality. Finally, three numerical experiments are presented to conform the theoretical analysis.