The stabilized nonconforming virtual element method for the Darcy–Stokes problem

A stabilized nonconforming virtual element method is designed in order to solve the Darcy–Stokes problem, which preserves a divergence-free approximation to the velocity. The same degrees of freedom as the usual Crouzeix–Raviart-type virtual element is used, but a different virtual element space is obtained by modifying the conforming Stokes virtual element with the $H^1$-projection operator. The proposed stabilized scheme contains two jump penalty terms over edges. One is the penalty for jumps of velocity approximation and the other one is the penalty for jumps of its normal component. We analyze this method’s well-posedness and prove its uniform convergence in a discrete energy norm. Finally, we verify the validity of this stabilized scheme by some numerical experiments.