Uniformly Convergent Nonconforming Tetrahedral Element for Darcy-Stokes Problem

Abstract

In this paper, we construct a tetrahedral element named DST20 for the three dimensional Darcy-Stokes problem, which reduces the degrees of velocity in [30]. The finite element space $\boldsymbol{V}_h$ for velocity is $\boldsymbol{H}$(div)-conforming, i.e., the normal component of a function in $\boldsymbol{V}_h$ is continuous across the element boundaries, meanwhile the tangential component of a function in $\boldsymbol{V}_h$ is average continuous across the element boundaries, hence $\boldsymbol{V}_h$ is $\boldsymbol{H}^1$-average conforming. We prove that this element is uniformly convergent with respect to the perturbation constant ε for the Darcy-Stokes problem. At the same time, we give a discrete de Rham complex corresponding to DST20 element.