A Simple Preconditioner for a Discontinuous Galerkin Method for the Stokes Problem

Abstract

In this paper we construct Discontinuous Galerkin approximations of the Stokes problem where the velocity field is $H(\mathrm{div},\Omega )$-conforming. This implies that the velocity solution is divergence-free in the whole domain. This property can be exploited to design a simple and effective preconditioner for the final linear system.

Appendix: Proof of Proposition 4.3

We now state and prove a result, Proposition 8.1 given below, used in Sect. 4 to show Korn inequality (cf. Lemma 4.2). After giving its proof, we comment briefly on how the result can be applied to show the corresponding Korn inequality (4.13) (cf. Lemma 4.2) for $d=3$.

Proposition 8.1

Let $T$ be a triangle (or a tetrahedron for $d=3$) with minimum angle $\theta >0$, and let $e$ be an edge (resp. face) of $T$. Then for every $p>2$ and for every integer $k_{max}$ there exists a constant $C_{p,\theta ,kmax}$ such that

$$\begin{aligned} \int \limits _e\varvec{v}\cdot ({\varvec{\tau }}\cdot \mathbf{n})\,\text{ d }s\le C_{p,\theta ,k_{max}}\, h_{T}^{-1/2}\Vert \varvec{v}\Vert _{0,e}\, \Big (h_{T}\Vert \mathrm{\varvec{div}}{\varvec{\tau }}\Vert _{0,T}\,+\,h_{T}^{\frac{d(p-2)}{2p}}\Vert {\varvec{\tau }}\Vert _{0,p,T}\Big ) \end{aligned}$$

(8.1)

for every ${\varvec{\tau }}\in (L^p(\Omega ))^{d\times d}_{sym}$ having divergence in $\varvec{L}^2$ and for every $\varvec{v} \in {\varvec{P}}^{k_{max}}(T)$.

Proof

First we go to the reference element $\hat{T}$:

$$\begin{aligned} \Big |\int \limits _e\varvec{{v}}\cdot ({{\varvec{\tau }}}\cdot \mathbf{n})\,\,\text{ d }s\le C_{\theta }|e|\Big |\int \limits _{\hat{e}}\varvec{\hat{v}}\cdot (\hat{{\varvec{\tau }}}\cdot \hat{\mathbf{n}})\,d\hat{s}\Big | \le C_{\theta }h_e^{d-1}\Big |\int \limits _{\hat{e}}\varvec{\hat{v}}\cdot (\hat{{\varvec{\tau }}}\cdot \hat{\mathbf{n}})\,d\hat{s}\Big | \end{aligned}$$

(8.2)

where $\hat{\varvec{v}}$ and $\hat{{\varvec{\tau }}}$ are the usual covariant and contra-variant images of $\varvec{v}$ and ${\varvec{\tau }}$, respectively. And, here and throughout his proof, the constants $C_{\theta }$ and $C_{\theta ,k_{max}}$ may assume different values at different occurrences. Note that $\hat{\varvec{v}}$ will still be a vector-valued polynomial of degree $\le k_{max}$ and the space $H(\mathrm{div}, T)$ is effectively mapped into $H(\mathrm{div}, \hat{T})$ by means of the contra-variant mapping. Then for every component $\hat{v}$ of $\hat{\varvec{v}}$, we construct the auxiliary function $\varphi _v$ as follows. First we define $\varphi _v$ on $\partial \hat{T}$ by setting it as equal to $\hat{v}$ on $\hat{e}$ and zero on the rest of $\partial \hat{T}$. Then we define $\varphi _v$ in the interior using the harmonic extension. It is clear that $\varphi _v$ will belong to $W^{1,p'}(\hat{T})$ (remember that $p>2$ so that its conjugate index $p'$ will be smaller than $2$). Using the fact that $\hat{\varvec{v}}$ is a polynomial of degree $\le k_{max}$, it is not difficult to see that

$$\begin{aligned} \Vert \varvec{\varphi }_v\Vert _{W^{1,p'}(\hat{T})} \le \,\hat{C}_{\theta ,k_{max}}\Vert \hat{\varvec{v}}\Vert _{0,\hat{e}}. \end{aligned}$$

(8.3)

Integration by parts then gives

$$\begin{aligned} \begin{aligned} \int \limits _{\hat{e}}\hat{\varvec{v}}\cdot (\hat{{\varvec{\tau }}}\cdot \hat{\mathbf{n}})\,d\hat{s}&= \int \limits _{\partial \hat{T}}\varvec{\varphi }_{v}\cdot (\hat{{\varvec{\tau }}}\cdot \hat{\mathbf{n}})\,d\hat{s}\\&= \int \limits _{\hat{T}}\varvec{\nabla }\varvec{\varphi }_{v}:\hat{{\varvec{\tau }}}\,d\hat{x} -\int \limits _{\hat{T}}\varvec{\varphi }_{v}\cdot \mathrm{\varvec{div}}{\hat{{\varvec{\tau }}}}\,d\hat{x}\\&\le |\varvec{\varphi }_v|_{W^{1,p'}(\hat{T})}\Vert \hat{{\varvec{\tau }}}\Vert _{(L^p(\hat{T}))^{d\times d}_{sym}}+ \Vert \varvec{\varphi }_v\Vert _{0,\hat{T}}\Vert \mathrm{\varvec{div}}{\hat{{\varvec{\tau }}}}\Vert _{0,\hat{T}}\\&\le \hat{C}\Big (\Vert \hat{\varvec{v}}\Vert _{0,\hat{e}}\,\Vert \hat{{\varvec{\tau }}}\Vert _{(L^p(\hat{T}))^{d\times d}_{sym}}\,+\, \Vert \varvec{\varphi }_v\Vert _{0,\hat{e}}\Vert \mathrm{\varvec{div}}{\hat{{\varvec{\tau }}}}\Vert _{0,\hat{T}}\Big )\\&\le \hat{C}\,\Vert \hat{\varvec{v}}\Vert _{0,\hat{e}}\,\Big (\Vert \hat{{\varvec{\tau }}}\Vert _{(L^p(\hat{T}))^{d\times d}_{sym}}\,+\Vert \mathrm{\varvec{div}}{\hat{{\varvec{\tau }}}}\Vert _{0,\hat{T}}\Big ). \end{aligned} \end{aligned}$$

(8.4)

Then we recall the inverse transformations (from $\hat{T}$ to $T$):

$$\begin{aligned}&\Vert \hat{\varvec{v}}\Vert _{0,\hat{e}}\le C_{\theta }h_e^{-\frac{d-1}{2}}\Vert \varvec{v}\Vert _{0,{e}},\quad \Vert \hat{{\varvec{\tau }}}\Vert _{(L^p(\hat{T}))^{d\times d}_{sym}}\le C_{\theta } h_{T}^{-\frac{d}{p}}\Vert {{\varvec{\tau }}}\Vert _{(L^p({T}))^{d\times d}_{sym}},\\&\Vert \mathrm{\varvec{div}}\hat{{\varvec{\tau }}}\Vert _{0,\hat{T}}\le C_{\theta }h_{T}^{\frac{2-d}{2}} \Vert \mathrm{\varvec{div}{\varvec{\tau }}}\Vert _{0,T}. \end{aligned}$$

Inserting this into (8.4) and then in (8.2) we have then

$$\begin{aligned} \int \limits _e\varvec{v}\cdot ({\varvec{\tau }}\cdot \mathbf{n})\,\text{ d }s\le C_{p,\theta ,k_{max}}\, h_e^{d-1}\,h_e^{-\frac{d-1}{2}}\Vert \varvec{v}\Vert _{0,{e}}\Big ( h_{T}^{-\frac{d}{p}}\Vert {{\varvec{\tau }}}\Vert _{(L^p({T}))^{d\times d}_{sym}}+h_{T}^{\frac{2-d}{2}} \Vert \mathrm{\varvec{div}{\varvec{\tau }}}\Vert _{0,T} \Big ). \end{aligned}$$

Now we note that

$$\begin{aligned} -\frac{1}{2}+\frac{d(p-2)}{2p}=d-1-\frac{d-1}{2}-\frac{d}{p}, \end{aligned}$$

and that

$$\begin{aligned} -\frac{1}{2}+1=d-1-\frac{d-1}{2}+\frac{2-d}{2}, \end{aligned}$$

and the proof then follows immediately. $\square$

With this result in hand, we can show the Korn inequality (4.13) given in Lemma 4.2 for $d=3$. It is necessary to modify the proof only at two places: the definition of the space of rigid motions on $\Omega , \varvec{RM}(\Omega )$, and the application of Proposition 4.21. The space $\varvec{RM}(\Omega )$ is now defined by:

$$\begin{aligned} \varvec{RM}(\Omega )=\left\{ \, \varvec{a}+\varvec{b} \varvec{x}\,\,:\,\, \varvec{a}\in \mathbb R ^{d} \quad \varvec{b} \in so(d) \,\,\right\} \end{aligned}$$

with $so(d)$ denoting the space of the skew-symmetric $d\times d$ matrices.

To prove (4.16) (and so conclude the proof of (4.13)), estimate (4.23) is replaced by estimate (8.5) below, which is obtained as follows: first, by applying (8.1) (instead of (4.21)) from Proposition 8.1 to each $e$ in the last term in (4.22) and then by using the generalized Hölder inequality with the same exponents as for $d=2$ (with $q=1/2$ and $r=2p/(p-2)$, so that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$)

$$\begin{aligned} \sum _{e\in {\mathcal{E }^{o}_h}}\int \limits _e [\![\,\varvec{v}_t\,]\!] : \{ \varvec{\tau }\}&\le C_{p,\theta ,kmax}\, \sum _{T\in \mathcal{T }_h}\sum _{e\in \partial T} h_{T}^{-1/2}\Vert [\![\,\varvec{v}_t\,]\!]\Vert _{0,e}\,h_{T}\Vert \mathbf{div}\varvec{\tau }\Vert _{0,T} \nonumber \\&+ C_{p,\theta ,kmax}\, \sum _{T\in \mathcal{T }_h}\sum _{e\in \partial T} h_{T}^{-1/2}\Vert [\![\,\varvec{v}_t\,]\!]\Vert _{0,e}\Vert \,h_{T}^{\frac{d(p-2)}{2p}}\Vert {\varvec{\tau }}\Vert _{0,p,T} \nonumber \\&\le C h\,|[\![\,\varvec{v}_t\,]\!]|_{*}\,\Vert \mathbf{div}{\varvec{\tau }}\Vert _{0,\Omega } \\&+ C\Big (\sum _{e\in {\mathcal{E }^{o}_h}}h_e^{-1}|[\![\,\varvec{v}_t\,]\!]|^2_{0,e}\Big )^{1/2} \Big (\sum _{e\in {\mathcal{E }^{o}_h}}\Vert {\varvec{\tau }}\Vert _{0,p,T(e)}^p\Big )^{1/p} \Big (\sum _{e\in {\mathcal{E }^{o}_h}}h_e^{\frac{d(p-2)}{2p} r}\Big )^{1/r}\nonumber \\&\le C |[\![\,\varvec{v}_t\,]\!]|_{*}\,h\,\Vert \mathbf{div}{\varvec{\tau }}\Vert _{0,\Omega }+C\,|[\![\,\varvec{v}_t\,]\!]|_{*}\,\Vert {\varvec{\tau }}\Vert _{0,p,\Omega }\,\mu ({\Omega })^{1/r}\nonumber \end{aligned}$$

(8.5)

Here, as in estimate (4.23), $\mu (\Omega )$ denotes the measure of the domain $\Omega$, and the constant $C$ still depends on $p, k_{max}$, and on the maximum angle in the decomposition $\mathcal{T }_h$. The rest of the proof of Lemma 4.2 proceeds as for $d=2$.