Stokes elements on cubic meshes yielding divergence-free approximations

Abstract

Conforming piecewise polynomial spaces with respect to cubic meshes are constructed for the Stokes problem in arbitrary dimensions yielding exactly divergence-free velocity approximations. The derivation of the finite element pair is motivated by a smooth de Rham complex that is well-suited for the Stokes problem. We derive the stability and convergence properties of the new elements as well as the construction of reduced elements with less global unknowns.

Appendix: A calculus identity

Lemma 8

Let $\{\varvec{a}_i\}_{i=1}^n \subset {\mathbb {R}}^n$ be a set of constant orthonormal (column) vectors.

Then there holds

$$\begin{aligned} {\mathrm{div}}\,{\varvec{v}}= \sum _{i=1}^n \frac{{\partial }{\varvec{v}}}{{\partial }\varvec{a}_i} \cdot \varvec{a}_i. \end{aligned}$$

Proof

Let A be the orthonormal matrix $A = [\varvec{a}_1| \varvec{a}_2|\cdots |\varvec{a}_n]\in {\mathbb {R}}^{n\times n}$, and define $\hat{{\varvec{v}}}(\hat{x}) = A^{-1} {\varvec{v}}(x) = A^T {\varvec{v}}(x)$, where $x = A \hat{x}$. We then have $D {\varvec{v}}(x) = A \hat{D} \hat{{\varvec{v}}}(\hat{x}) A^{T}$ by the chain rule [5].

Therefore, since the trace is invariant under similarity transforms, and since A is orthonormal, we have

$$\begin{aligned} \sum _{i=1}^n \frac{{\partial }{\varvec{v}}}{{\partial }\varvec{a}_i} \cdot \varvec{a}_i&= \sum _{i=1}^n \varvec{a}^T_i (D{\varvec{v}})\varvec{a}_i = \sum _{i=1}^n (\varvec{a}^T_i A) \hat{D} \hat{{\varvec{v}}} (\varvec{a}^T_i A)^T =\sum _{i=1}^n \frac{{\partial }\hat{v}^{(i)}}{{\partial }\hat{x}_i}\\&= \widehat{{\mathrm{div}}\,}\hat{{\varvec{v}}}= \mathrm{tr}(\hat{D}\hat{{\varvec{v}}}) = \mathrm{tr}(D{\varvec{v}}) = {\mathrm{div}}\,{\varvec{v}}. \end{aligned}$$

$\square$