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

Abstract

Abstract In this paper, we propose a ${\textbf{P}_{1}^{c}}\oplus{RT0}-P0$ discretization of the Stokes equations on general simplicial meshes in two/three dimensions, which yields an exactly divergence-free and pressure-independent velocity approximation with optimal order. Our method has the following features. First, the global number of the degrees of freedom (DOFs) of our method is the same as the low-order Bernardi and Raugel ($B$-$R$) finite element method (Bernardi & Raugel, 1985, Analysis of some finite elements for the Stokes problem. Math. Comp., 44, 71–79), while the number of the nonzero entries of the former is about half of the latter in the velocity–velocity region of the coefficient matrix. Secondly, the ${\textbf{P}_{1}^{c}}$ component of the velocity, the $RT0$ component of the velocity and the pressure seem to solve a popular ${\textbf{P}_{1}^{c}}-{RT0}-P0$ discretization of a poroelastic-type system formally. Finally, our method can be easily transformed into a pressure-robust and stabilized ${\textbf{P}_{1}^{c}}-P0$ discretization for the Stokes problem via the static condensation of the $RT0$ component, which has a much smaller number of global DOFs. Numerical experiments illustrating the robustness of our method are also provided.