Uniformly Convergent Cubic Nonconforming Element For Darcy-Stokes Problem

In this paper we construct a cubic element named DSC33 for the Darcy-Stokes problem of three-dimensional space. The finite element space for velocity is -conforming, i.e., the normal component of a function in is continuous across the element boundaries, meanwhile the tangential component of a function in is averagely continuous across the element boundaries, hence is -average conforming. We prove that this element is uniformly convergent with respect to the perturbation constant for the Darcy-Stokes problem. In addition, we construct a discrete de Rham complex corresponding to DSC33 element. The finite element spaces in the discrete de Rham complex can be applied to some singular perturbation problems.