A compact divergence-free H(div)-conforming finite element method for Stokes flows

In this paper, we construct a $P_{1}^{c}\oplus RT0-P0$ discretization of the Stokes equations for general simplicial meshes in two/three dimensions (2D/3D), which yields a exactly divergence-free and pressure-independent velocity approximation with optimal order. Some interesting properties of $RT0$ are fully utilized. Unlike the popular H(div)-conforming methods in a discontinuous Galerkin (DG) framework, our scheme only consists of integrals on each element, and does not involve any trace operators such as jumps and averages on the element interfaces. We propose a simple stabilization on $RT0$ which only acts on the diagonal of the coefficient matrix. The stencil of our method is the same as the lowest order Bernardi and Raugel (B-R) element method (see C. Bernardi and G. Raugel, Math. Comp., 44 (1985), pp. 71-79). Finally, by a diagonal perturbation of the bilinear form, we further decouple the unknowns corresponding to $P_{1}^{c}$ and $RT0$, solving a stabilized $P_{1}^{c}-P0$ discretization first. Then the $RT0$ part can be obtained locally and explicitly. Numerical experiments illustrating the robustness of our methods are also provided.