Uniformly Convergent Nonconforming Tetrahedral Element for Darcy-Stokes Problem (Journal of Computational Mathematics, 2018)
In this paper, we construct a tetrahedral element named DST20 for the three dimensionalDarcy-Stokes problem, which reduces the degrees of velocity in [30]. The finite elementspace $\boldsymbol{V}_h$ for velocity is $\boldsymbol{H}$(div)-conforming, i.e., the normal component of a functionin $\boldsymbol{V}_h$ is continuous across the element boundaries, meanwhile the tangential componentof 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 respectto the perturbation constant ε for the Darcy-Stokes problem. At the same time, we give adiscrete de Rham complex corresponding to DST20 element.
Keywords: Darcy-Stokes problem; Mixed finite elements; Tetrahedral element; Uniformly convergent.