A General Degree Divergence-Free Finite Element Method for the Two-Dimensional Stokes Problem on Smooth Domains

Abstract

In this paper, we construct and analyze divergence-free finite element methods for the Stokes problem on smooth domains. The discrete spaces are based on the Scott-Vogelius finite element pair of arbitrary polynomial degree greater than 2. By combining the Piola transform with the classical isoparametric framework, and with a judicious choice of degrees of freedom, we prove that the method converges with optimal order in the energy norm. We also show that the discrete velocity error converges with optimal order in the $L^2$-norm. Numerical experiments are presented, which support the theoretical results.

Appendices

Appendix A. Proof of Lemma 3.2

Proof

Write $\varvec{v}(x) = A_T {\hat{\varvec{v}}} (\hat{x})$ for some ${\hat{\varvec{v}}}\in \hat{\varvec{V}}_k$. We then use Lemma 2.3, (2.5), and equivalence of norms to obtain

$$\begin{aligned} \begin{aligned} \Vert \varvec{v}\Vert _{W^{\ell ,p}(K)}&\le C h_T^{2/p-\ell } \Vert A_T {\hat{\varvec{v}}}\Vert _{W^{\ell ,p}({\hat{K}})}\\&\le C h_T^{2/p-\ell } \Vert A_T\Vert _{W^{j,\infty }({\hat{K}})} \Vert {\hat{\varvec{v}}}\Vert _{W^{\ell ,p}({\hat{K}})}\\&\le C h_T^{2/p-\ell -1} \Vert {\hat{\varvec{v}}}\Vert _{W^{\ell ,p}({\hat{K}})}\le C h_T^{2/p-\ell -1} \Vert {\hat{\varvec{v}}}\Vert _{L^{q}({\hat{K}})}. \end{aligned} \end{aligned}$$

(A.1)

Likewise, we have

$$\begin{aligned} \begin{aligned} \Vert {\hat{\varvec{v}}}\Vert _{L^q({\hat{K}})}\le \Vert A_T^{-1}\Vert _{L^\infty ({\hat{K}})} \Vert A_T {\hat{\varvec{v}}}\Vert _{L^q({\hat{K}})} \le C h_T^{1-2/q} \Vert \varvec{v}\Vert _{L^q(K)}. \end{aligned} \end{aligned}$$

(A.2)

Combining (A.1)–(A.2) yields (3.1) for the case $m=0$. The estimate (3.1) for general m then follows by standard arguments (cf. [9, Lemma 4.5.3]).

To prove (3.2), we first use (2.5):

$$\begin{aligned} |\varvec{v}|_{W^{\ell ,p}(K)}&\le C \Big [\underbrace{h_T^{2/p+\ell } \sum _{r=0}^k h_T^{-2r} |A_T {\hat{\varvec{v}}}|_{W^{r,p}({\hat{K}})}}_{=:I} +\underbrace{h_T^{2/p+\ell } \sum _{r=k+1}^\ell h_T^{-2r} |A_T \hat{\varvec{v}}|_{W^{r,p}({\hat{K}})}}_{=:II}\Big ]. \end{aligned}$$

To bound I, we use (2.5) once again to obtain

$$\begin{aligned} I\le h_T^{2/p+\ell }\sum _{r=0}^k h_T^{-2r} \cdot h_T^{r-2/p} \Vert \varvec{v}\Vert _{W^{r,p}(K)}\le C h_T^{\ell -k}\Vert \varvec{v}\Vert _{W^{k,p}(K)}. \end{aligned}$$

For II, we use the fact that ${\hat{\varvec{v}}}$ is a polynomial of degree $\le k$ on K to obtain

$$\begin{aligned} |A_T {\hat{\varvec{v}}}|_{W^{r,p}({\hat{K}})}&\le C \sum _{j=0}^k |A_T|_{W^{r-j,\infty }({\hat{K}})} |{\hat{\varvec{v}}}|_{W^{j,p}({\hat{K}})}\\&\le C\sum _{j=0}^k h_T^{r-j-1} |A_T^{-1} A_T {\hat{\varvec{v}}}|_{W^{j,p}({\hat{K}})}\\&\le C\sum _{j=0}^k \sum _{i=0}^j h_T^{r-j-1} |A_T^{-1}|_{W^{j-i,\infty }({\hat{K}})} |A_T {\hat{\varvec{v}}}|_{W^{i,p}({\hat{K}})}\\&\le C\sum _{j=0}^k \sum _{i=0}^j h_T^{r-i} |A_T {\hat{\varvec{v}}}|_{W^{i,p}({\hat{K}})}\\&\le C\sum _{j=0}^k \sum _{i=0}^j h_T^{r-i}\cdot h_T^{i-2/p} \Vert \varvec{v}\Vert _{W^{i,p}(K)}\\&\le C h_T^{r-2/p} \Vert \varvec{v}\Vert _{W^{k,p}(K)}. \end{aligned}$$

Thus,

$$\begin{aligned} II \le C h_T^{2/p+\ell } \sum _{r=k+1}^\ell h_T^{-r-2/p} \Vert \varvec{v}\Vert _{W^{k,p}(K)}\le C \Vert \varvec{v}\Vert _{W^{k,p}(K)}. \end{aligned}$$

Combining the bounds for I and II completes the proof of (3.2). $\square$

Appendix B. Proof of Lemma 4.5

Proof

Define $\varvec{E}_h: \varvec{V}^h \rightarrow \varvec{H}_0^1(\Omega _h)$ such that, for $\varvec{v}\in \varvec{V}^h$,

$$\begin{aligned} \varvec{E}_h \varvec{v}|_{T} = (\tilde{\varvec{v}} \circ F_{\tilde{T}}\circ F_{T}^{-1})|_{T}, \end{aligned}$$

where $\tilde{\varvec{v}}$ is the function in $\tilde{\varvec{V}}$ uniquely defined by

$$\begin{aligned} \varvec{v}|_T(a) = \tilde{\varvec{v}}|_{\tilde{T}}(\tilde{a}) \quad \forall {a} \in \mathcal {N}_T, \quad \forall T \in \mathcal {T}_h, \end{aligned}$$

where $T= G_h(\tilde{T})$. In other words, in a standard isoparametric, kth degree Lagrange finite element method, $\varvec{E}_h \varvec{v}$ would be the function on the isoparametric element associated with $\tilde{\varvec{v}}$ on $\tilde{T}$. Thus, $\varvec{E}_h\varvec{v}\in \varvec{H}_0^1(\Omega _h)$.

As shown in [6], $\tilde{\varvec{v}} = \varvec{E}_h \varvec{v}$ on affine triangles, and we may conclude

$$\begin{aligned} \varvec{E}_h\varvec{v}|_T(a) = \varvec{v}|_T(a) \quad \forall {a} \in \mathcal {N}_T, \quad \forall T \in \mathcal {T}_h. \end{aligned}$$

Our goal is to estimate $\varvec{v}- \varvec{E}_h \varvec{v}$, and our proof follows closely with the proof of Lemma 4.5 in [6]. However, here we provide a more general result.

As $\varvec{v}= \varvec{E}_h \varvec{v}$ on affine triangles, we only consider $T \in \mathcal {T}_h$ with curved boundaries. Additionally, we know $\varvec{v}|_{\partial T \cap \partial \Omega _h} = 0$. We may write $\varvec{v}|_T (x) = A_T (\hat{x})\hat{\varvec{v}}(\hat{x})$, for some $\hat{\varvec{v}}\in \hat{\varvec{V}}$, where $A_T = DF_T/\det {(DF_T)}$. Furthermore, there exists $\hat{\varvec{w}} \in \hat{\varvec{V}}$ such that $\hat{\varvec{w}}(\hat{x}) = \varvec{E}_h \varvec{v}|_{T}(x)$. Consequently,

$$\begin{aligned} A_T(\hat{a})\hat{\varvec{v}}(\hat{a}) = \hat{\varvec{w}}(\hat{a}) \quad \forall \hat{a} \in \mathcal {N}_{\hat{T}}, \end{aligned}$$

so $\hat{\varvec{w}}$ is the piecewise kth degree Lagrange interpolant of $A_T\hat{\varvec{v}}$ on $\hat{T}^{CT}$.

By the Bramble-Hilbert lemma, we have

$$\begin{aligned} \Vert A_T \hat{\varvec{v}} - \hat{\varvec{w}}\Vert _{H^i(\hat{K})} \le C \vert A_T \hat{\varvec{v}}\vert _{H^{k+1}(\hat{K})} \quad \forall \hat{K} \in \hat{T}^{CT}, \quad i = 0,1,\ldots ,k. \end{aligned}$$

(B.1)

We may then bound the right-hand side using Lemma 2.3 and recognizing that $\hat{\varvec{v}}$ is a polynomial of degree k. Thus we have

$$\begin{aligned} \begin{aligned} \vert A_T \hat{\varvec{v}}\vert _{H^{k+1}(\hat{K})} \le&C \sum _{j=0}^{k+1} \vert A_T\vert _{W^{k+1-j}({\hat{K}})} \vert {\hat{\varvec{v}}} \vert _{H^j({\hat{K}})} = C \sum _{j=0}^{k} \vert A_T\vert _{W^{k+1-j}({\hat{K}})} \vert {\hat{\varvec{v}}} \vert _{H^j({\hat{K}})}\\ \le&C \sum _{j=0}^{k} h_T^{k-j} \vert {\hat{\varvec{v}}} \vert _{H^j({\hat{K}})}. \end{aligned} \end{aligned}$$

(B.2)

Using Lemmas 2.3 and 2.4, we have

$$\begin{aligned} \begin{aligned} \vert {\hat{\varvec{v}}} \vert _{H^j({\hat{K}})}&= \vert A_T^{-1} A_T \hat{\varvec{v}} \vert _{H^j({\hat{K}})} \le C \sum _{\ell =0}^j |A_T^{-1}|_{W^{j-\ell }({\hat{K}})} |A_T {\hat{\varvec{v}}}|_{H^\ell ({\hat{K}})}\\&\le C \sum _{\ell =0}^j h_T^{1+j-\ell } |A_T {\hat{\varvec{v}}}|_{H^\ell ({\hat{K}})}\\&\le C\sum _{\ell =0}^j h_T^{1+j-\ell } h_T^{\ell -1}\Vert \varvec{v}\Vert _{H^\ell ( K)}\le C h_T^j \Vert \varvec{v}\Vert _{H^j(K)}. \end{aligned} \end{aligned}$$

(B.3)

Inserting this estimate into (B.2) yields

$$\begin{aligned} |A_T {\hat{\varvec{v}}}|_{H^{k+1}({\hat{K}})}\le C h_T^k \Vert \varvec{v}\Vert _{H^k(K)}, \end{aligned}$$

and therefore by (B.1) and Lemma 2.4,

$$\begin{aligned} |\varvec{v}-\varvec{E}_h \varvec{v}|_{H^i(K)}\le C h_T^{1-i} \Vert A_T {\hat{\varvec{v}}}-\hat{\varvec{w}}\Vert _{H^i({\hat{K}})}\le C h_T^{1-i} |A_T {\hat{\varvec{v}}}|_{H^{k+1}({\hat{K}})}\le C h_T^{k+1-i}\Vert \varvec{v}\Vert _{H^k(K)}. \end{aligned}$$

(B.4)

An applicaiton of the inverse inequality (3.1) then yields the desired estimate (4.3). $\square$