The nonconforming virtual element method for the Darcy–Stokes problem

We present the nonconforming virtual element method for the Darcy–Stokes problem. By using a gradient projection operator and a special polynomial subspace orthogonal to the gradient of some polynomial space, we construct a H1-nonconforming but H(div)-conforming virtual element that allows us to compute the L2-projection. The optimal convergence is proved under the assumption of sufficient regularity, and the uniform convergence is also obtained for the lowest-order case. Besides, we establish a discrete exact sequence of de Rham complex related to the nonconforming virtual element. Finally, we carry out some numerical tests to make a comparison of the convergence between different nonconforming virtual elements and confirm the optimal and uniform convergence of the nonconforming virtual element.