A low-order discontinuous Petrov–Galerkin method for the Stokes equations

Abstract

This paper introduces a low-order discontinuous Petrov-Galerkin (dPG) finite element method (FEM) for the Stokes equations. The ultra-weak formulation utilizes piecewise constant and affine ansatz functions and piecewise affine and discontinuous lowest-order Raviart–Thomas test search functions. This low-order discretization for the Stokes equations allows for a direct proof of the discrete inf-sup condition with explicit constants. The general framework of Carstensen et al. (SIAM J Numer Anal 52(3):1335–1353, 2014) then implies a complete a priori and a posteriori error analysis of the dPG FEM in the natural norms. Numerical experiments investigate the performance of the method and underline its quasi-optimal convergence.

Appendix: Fortin interpolation in an extended test search space

Proof of Lemma 5.4

For ${\hat{Y}}_h:=({RT_0^\text {pw}(\mathcal {T};\mathbb {R}^{2})/\mathbb {R}\oplus \mathcal {B}_3(\mathcal {T})\mathbb {R}^{2\times 2}_{{{\mathrm{dev}}}}}) \times P_1(\mathcal {T};\mathbb {R}^{2})$ the Fortin interpolator ${\hat{\varPi }}: Y\rightarrow {\hat{Y}}_h$ maps $(\varvec{\tau },v)\mapsto (\hat{\varvec{\tau }}_{\text { RT}}, I_{\text { NC}}^\text {pw}v)$ such that (45)–(46) hold. Recall the edge-based Raviart–Thomas basis $\psi _{E}$ for $E\in \mathcal {E}(T)$, the opposite vertex $P_E\in \mathcal {N}(T)$, and the component $\kappa =1,\,2$, let $\varPsi _{E,\kappa }:=e_\kappa \otimes \psi _E :=e_\kappa \otimes \chi (T)(x-P_E)|E|/(2|T|)$. The Crouzeix-Raviart basis functions in two components $\varPhi _{E,\kappa }$ read $\varPhi _{E,\kappa }:=\phi _{E}\ e_\kappa :=\chi (T)(1-2\varphi _{P_E})\ e_\kappa$ with nodal basis functions $\varphi _{z}$ of $z\in \mathcal {N}$. Given $T\in \mathcal {T},\varvec{\tau }\in H({{\mathrm{div}}},T;\mathbb {R}^{2\times 2})$, define

$$\begin{aligned} I_F\varvec{\tau }:=\sum _{\kappa =1,\,2}\sum _{E\in \mathcal {E}(T)}\left( \frac{1}{|E|} \int _{\partial T} \varPhi _{E,\kappa }\cdot \varvec{\tau }\nu _T\,\mathrm{d}s\ \varPsi _{E,\kappa }\right) . \end{aligned}$$

(50)

Since $\left( \psi _E\cdot \nu _T\right) |_F=\delta _{EF}$ and $\int _F \phi _E \,\mathrm{d}s=\delta _{EF}|E|$ for $E,F\in \mathcal {E}(T),I_F=I_F^2$ is a projection. Moreover, $\left\langle I_F\varvec{\tau }\nu _T , \varPhi _{E,\kappa } \right\rangle _{\partial T}=\left\langle \varvec{\tau }\nu _T , \varPhi _{E,\kappa } \right\rangle _{\partial T}$ for any $E\in \mathcal {E}(T)$ and $\kappa =1,\,2$ implies (46). An integration by parts allows to rewrite $I_F\varvec{\tau }$. The projection property implies $I_F q=q$ for all $q\in P_0(T)$. This and $\sum _{E\in \mathcal {E}(T)} \varphi _E=1$ reveal

where . Given $b_T:=60 \varphi _1\varphi _2\varphi _3\in B_3(T)$, for $T\in \mathcal {T}$, set

$$\begin{aligned} \varPi _\tau \varvec{\tau }:= I_F\varvec{\tau }+ b_T {{\mathrm{dev}}}\varPi _0(\varvec{\tau }-I_F \varvec{\tau }) = I_F\varvec{\tau }- b_T {{\mathrm{dev}}}Q(T). \end{aligned}$$

(51)

Since $b_T|_{\partial T}=0$ and , this operator $\varPi _\tau$ satisfies (45)–(46). To compute $\Vert \varPi _\tau \Vert _{}$ with

$$\begin{aligned}\Vert \varPi _\tau (\varvec{\tau }) \Vert _{H({{\mathrm{div}}},T)}^2= & {} \Vert I_F\varvec{\tau }- b_T {{\mathrm{dev}}}Q(T) \Vert _{L^2(T)}^2\\&+\Vert {{\mathrm{div}}}I_F\varvec{\tau }-{{\mathrm{dev}}}Q(T) \nabla b_T \Vert _{L^2(T)}^2,\end{aligned}$$

estimate the Frobenius norm $\Vert Q(T) \Vert _{F}$ of Q(T) as follows. Since $\vert {{\mathrm{mid}}}(T)-P_E\vert \le 2h_T/3$ and $\Vert \phi _E \Vert _{L^2(T)}^2=|T|/3$, the Cauchy-Schwarz inequality allows

Moreover, $\Vert b_T \Vert _{L^2(T)}^2=10|T|/7$ and $\Vert \nabla b_T \Vert _{L^2(T)}^2=20 |T|\Vert {{\mathrm{G}}}(T) \Vert _{F}^2$, with

$$\begin{aligned} G(T):=\begin{pmatrix} \nabla \varphi _1^\top \\ \nabla \varphi _2^\top \\ \nabla \varphi _3^\top \end{pmatrix} =\begin{pmatrix} 1&{}\quad 1&{}\quad 1\\ P_1&{}\quad P_2&{}\quad P_3\end{pmatrix}^{-1}\begin{pmatrix} 0&{}\quad 0\\ 1&{}\quad 0\\ 0&{}\quad 1\end{pmatrix} \end{aligned}$$

and $\Vert G(T) \Vert _{F}^2=(2|T|)^{-2}\sum _{E\in \mathcal {E}(T)}|E|^2\le 3 h_T^2(2|T|)^{-2}.$ Let $\alpha _T$ denote the smallest angle in T and recall $h_T^2\le 4|T|\cot (\alpha _T)$. Hence,

$$\begin{aligned} \Vert b_T {{\mathrm{dev}}}Q(T) \Vert _{L^2(T)}^2&\le \Vert Q(T) \Vert _{F}^2 \Vert b_T \Vert _{L^2(T)}^2 = {20h_T^2}/{21}\ \Vert {{\mathrm{div}}}\varvec{\tau } \Vert _{L^2(T)}^2, \\ \Vert {{\mathrm{dev}}}Q(T)\ \nabla b_T \Vert _{L^2(T)}^2&\le \Vert Q(T) \Vert _{F}^2\Vert \nabla b_T \Vert _{L^2(T)}^2={15h_T^4}/{|T|}\ \Vert {{\mathrm{div}}}\varvec{\tau } \Vert _{L^2(T)}^2 \\&\le {60 h_T^2\cot (\alpha _T)}\Vert {{\mathrm{div}}}\varvec{\tau } \Vert _{L^2(T)}^2. \end{aligned}$$

Alltogether,

$$\begin{aligned} \Vert \varPi _\tau (\varvec{\tau }) \Vert _{H({{\mathrm{div}}},T)}^2&\le \left( \Vert \varPi _0\varvec{\tau } \Vert _{L^2(T)}+\Vert Q(T) \Vert _{L^2(T)}+\Vert b_T {{\mathrm{dev}}}Q(T) \Vert _{L^2(T)}\right) ^2 \\&\quad +\left( \Vert {{\mathrm{div}}}\varPi _0\varvec{\tau } \Vert _{L^2(T)}+\Vert {{\mathrm{dev}}}Q(T) \nabla b_T \Vert _{L^2(T)}\right) ^2 \\&\le \left( \Vert \varvec{\tau } \Vert _{L^2(T)}+\left( \sqrt{2/3}+\sqrt{20/21}\right) h_T\Vert {{\mathrm{div}}}\varvec{\tau } \Vert _{L^2(T)}\right) ^2 \\&\quad +\left( 1+2\sqrt{15\cot (\alpha _T)}h_T\right) ^2\Vert {{\mathrm{div}}}\varvec{\tau } \Vert _{L^2(T)}^2 \\&\le \left( 1+3.22h_T^2+(1+7.75\sqrt{\cot (\alpha )}h_T)^2\right) \Vert \varvec{\tau } \Vert _{H({{\mathrm{div}}},T)}^2 \\&\le (2+15.5 \sqrt{\cot (\alpha _T)}h_T+(3.22+60\cot (\alpha _T)) h_T^2)\Vert \varvec{\tau } \Vert _{H({{\mathrm{div}}},T)}^2. \end{aligned}$$

Set ${\hat{\varPi }}(\varvec{\tau },v)|_T:=(\varPi _\tau (\varvec{\tau }|_T),I_{\text { NC}}^\text {pw}v|_T)$ as in (51). Given $y=(\varvec{\tau },v)\in Y$, the above computation, the abbreviation $\alpha _{\min }$ for the smallest angle of the triangulation (which is bounded in a regular triangulation) and [12, Thm. 4] prove

$\square$