Analysis of divergence free conforming virtual elements for the Brinkman problem

In this paper, we develop stability analysis, including inverse inequality, L-2 norm equivalence and interpolation error estimates, for divergence free conforming virtual elements in arbitrary dimension. A local energy projector based on the local Stokes problem is suggested, which commutes with the divergence operator. After defining a discrete bilinear form and a stabilization involving only the boundary degrees of freedom (DoF) and parts of the interior DoF, a new divergence free conforming virtual element method is advanced for the Brinkman problem, which can be reduced to a simpler method due to the divergence free discrete velocity. An optimal convergence rate is derived for the discrete method. Furthermore, we achieve a uniform half convergence rate of the discrete method in consideration of the boundary layer phenomenon. Finally, some numerical results are provided to validate the convergence of the discrete method.