Uniformly Convergent Cubic Nonconforming Element For Darcy–Stokes Problem

Abstract

In this paper we construct a cubic element named DSC33 for the Darcy---Stokes problem of three-dimensional space. The finite element space $${{\varvec{V}}}_{h}$$Vh for velocity is [InlineEquation not available: see fulltext.]-conforming, i.e., the normal component of a function in $${{\varvec{V}}}_{h}$$Vh is continuous across the element boundaries, meanwhile the tangential component of a function in $${{\varvec{V}}}_{h}$$Vh is averagely continuous across the element boundaries, hence $${{\varvec{V}}}_{h}$$Vh is $${{\varvec{H}}}^{1}$$H1-average conforming. We prove that this element is uniformly convergent with respect to the perturbation constant $$\varepsilon $$ź 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.