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 -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.

中文翻译：

---

达西-斯托克斯问题的稳定非协调虚元法


为了解决达西-斯托克斯问题，设计了一种稳定的非一致性虚拟单元方法，该方法保留了速度的无散近似。使用与通常的Crouzeix-Raviart型虚拟单元相同的自由度，但通过使用-投影算子修改一致的Stokes虚拟单元来获得不同的虚拟单元空间。所提出的稳定方案包含两个边缘跳跃惩罚项。一种是对速度近似的跳跃的惩罚，另一种是对其法向分量的跳跃的惩罚。我们分析了该方法的适定性并证明了其在离散能量范数下的一致收敛性。最后，我们通过数值实验验证了该稳定方案的有效性。