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

TL;DR

This paper introduces a novel low-order divergence-free finite element method for Stokes flows that is pressure-independent, optimally accurate, and computationally efficient on general meshes.

Contribution

The paper proposes a new ${ P_{1}^{c}}igoplus {RT0}-P0$ discretization for Stokes equations that is divergence-free, pressure-independent, and reduces degrees of freedom compared to existing methods.

Findings

Exact divergence-free velocity approximation achieved.

Method has fewer degrees of freedom than comparable finite element methods.

Numerical experiments confirm robustness and efficiency.

Abstract

In this paper, we propose a ${ P_{1}^{c}}\oplus {RT0}-P0$ discretization of the Stokes equations on general simplicial meshes in two/three dimensions (2D/3D), which yields an exactly divergence-free and pressure-independent velocity approximation with optimal order. Our method has the following features. Firstly, the global number of the degrees of freedom of our method is the same as the low order Bernardi and Raugel ($B$-$R$) finite element method (Bernardi and Raugel, 1985), while the number of {the non-zero entries} of the former is about half of the latter in the velocity-velocity region of the coefficient matrix. Secondly, the ${ P_{1}^{c}}$ component of the velocity, the $RT0$ component of the velocity and the pressure seem to solve a popular ${ P_{1}^{c}}-{RT0}-P0$ discretization of a poroelastic-type system formally. Finally, our method can be easily transformed into a pressure-robust and stabilized ${ P_{1}^{c}}-P0$ discretization for the Stokes problem via the static condensation of the $RT0$ component, which has a much smaller number of global degrees of freedom. Numerical experiments illustrating the robustness of our method are also provided.