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

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 Zhao et al. (SIAM J. Numer. Anal. 57(6):2730-2759, 2019). The proposed construction can be seen as a generalization of the divergence-free basis in Crouzeix-Raviart finite element space (Brenner, Math. Comp. 55(192):411-437, 1990; Thomasset, 1981) 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.

Automated summary

In this work, a formal construction of a pointwise divergence-free basis in the nonconforming virtual element method for the Stokes problem is developed, extending the concept from the traditional finite element space. The elimination of the pressure variable from the mixed system and the achievement of a symmetric positive definite system are demonstrated through numerical tests, confirming the efficiency and accuracy of the proposed construction.