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.