Journal Article

THE DIVERGENCE-FREE NONCONFORMING VIRTUAL ELEMENT FOR THE STOKES PROBLEM

ZHANG, BEI, ZHAO, JIKUN, CHEN, SHAOCHUN, MAO, SHIPENG

2019

Overview

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 H(div)-conforming virtual 2 element with some divergence-free functions from the C⁰-continuous H²-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 H¹-nonconforming and H²-nonconforming virtual elements. Finally, the numerical results are shown to confirm the convergence of the nonconforming virtual element.

• Publisher: Society for Industrial and Applied Mathematics