A formal construction of a divergence-free basis in the nonconforming virtual element method for the Stokes problem

TL;DR

This paper presents a new divergence-free basis construction for the nonconforming virtual element method applied to the Stokes problem, enabling pressure elimination and resulting in a symmetric positive definite system.

Contribution

It generalizes divergence-free basis construction from Crouzeix-Raviart finite elements to virtual element spaces for arbitrary order.

Findings

Eliminates pressure variable, simplifying the system.

Produces a symmetric positive definite system.

Numerical tests confirm efficiency and accuracy.

Abstract

We develop a formal construction of a pointwise divergence-free basis in the nonconforming virtual element method of arbitrary order for the Stokes problem introduced in [19]. The proposed construction can be seen as a generalization of the divergence-free basis in Crouzeix-Raviart finite element space [10, 17] to the virtual element space. Using the divergence-free basis obtained from our construction, we can eliminate the pressure variable from the mixed system and obtain a symmetric positive definite system. Several numerical tests are presented to confirm the efficiency and the accuracy of our construction.