The Divergence-Free Nonconforming Virtual Element for the Stokes Problem

Abstract

We present the divergence-free nonconforming virtual element method for the Stokes problems. We first construct a nonconforming virtual element with continuous normal component and weak continuous tangential component by enriching the previous $\bm H$(div)-conforming virtual element with some divergence-free functions from the $C^0$-continuous $H^2$-nonconforming virtual element. By imposing a restriction on each edge for the resulting nonconforming virtual element, we obtain the desired nonconforming virtual element with the less space dimension. The nonconforming virtual element provides the exact divergence-free approximation to the velocity and is proved to be convergent with the optimal convergence rate. Further, we present two exact sequences of differential complex between the $\bm H^1$-nonconforming and $H^2$-nonconforming virtual elements. Finally, the numerical results are shown to confirm the convergence of the nonconforming virtual element.