Stabilized Finite Element Methods for the Oberbeck–Boussinesq Model

Abstract

We consider conforming finite element approximations for the time-dependent Oberbeck–Boussinesq model with inf-sup stable pairs for velocity and pressure and use a stabilization of the incompressibility constraint. In case of dominant convection, a local projection stabilization method in streamline direction is considered both for velocity and temperature. For the arising nonlinear semi-discrete problem, a stability and convergence analysis is given that does not rely on a mesh width restriction. Numerical experiments validate a suitable parameter choice within the bounds of the theoretical results.

Appendix

Lemma 1

Let $\epsilon >0$ and $({\varvec{u}},p,\theta )\in {\varvec{V}}^{div} \times Q\times {\varTheta }$, $({\varvec{u}}_h,p_h,\theta _h) \in {\varvec{V}}{}_h^{\,div} \times Q_h\times {\varTheta }_h$ be solutions of (2), (3) and (7), (8) satisfying ${\varvec{u}}\in [W^{1,\infty }({\varOmega })]^d$, $\theta \in W^{1,\infty }({\varOmega })$ and ${\varvec{u}}_h\in [L^{\infty }({\varOmega })]^d$. If Assumptions 1 and 2 hold, we can estimate the difference of the convective terms in the momentum equation

$$\begin{aligned}&c_u({\varvec{u}};{\varvec{u}},{\varvec{e}}_{u,h}) - c_u({\varvec{u}}_h;{\varvec{u}}_h,{\varvec{e}}_{u,h}) \\&\quad \le \frac{C}{\epsilon } \sum _{M\in {\mathcal {M}}_h} \frac{1}{h_M^2} \Vert \varvec{\eta }_{u,h}\Vert ^2_{0,M} + 3 \epsilon ||| \varvec{\eta }_{u,h}|||^2_{\textit{LPS}} + 3 \epsilon ||| {\varvec{e}}_{u,h}|||^2_{\textit{LPS}} \\&\qquad + \left[ |{\varvec{u}}|_{W^{1,\infty }({\varOmega })} + \epsilon \max _{M\in {\mathcal {M}}_h} \{ h_M^2 |{\varvec{u}}|_{W^{1,\infty }(M)}^2 \} + \frac{C}{\epsilon }\max _{M\in {\mathcal {M}}_h} \left\{ \frac{h_M^2}{\gamma _M} | {\varvec{u}}|^2_{W^{1,\infty }(M)} \right\} \right. \\&\left. \qquad +\, \frac{C}{\epsilon } \max _{M\in {\mathcal {M}}_h} \{ \gamma _M^{-1}\Vert {\varvec{u}}\Vert _{\infty ,M}^2 \} + \epsilon \Vert {\varvec{u}}_h\Vert _{\infty }^2 \right] \Vert {\varvec{e}}_{u,h}\Vert ^2_{0} \end{aligned}$$

with C independent of $h_M$, $h_L$, $\epsilon$, the problem parameters and the solutions. The difference of the convective terms in the Fourier equation can be bounded as

$$\begin{aligned}&c_\theta ({\varvec{u}};\theta ,e_{\theta ,h}) - c_\theta ({\varvec{u}}_h;\theta _h,e_{\theta ,h}) \\&\quad \le \frac{C}{\epsilon } \sum _{M\in {\mathcal {M}}_h} h_M^{-2} \Vert \varvec{\eta }_{u,h}\Vert ^2_{0,M} + 3 \epsilon ||| \varvec{\eta }_{u,h}|||^2_{\textit{LPS}} + 3 \epsilon ||| {\varvec{e}}_{u,h}|||^2_{\textit{LPS}}\\&\qquad + \frac{1}{2}|\theta |_{W^{1,\infty }({\varOmega })} \Vert {\varvec{e}}_{u,h}\Vert ^2_{0} + \frac{C}{\epsilon } \sum _{L\in {\mathcal {L}}_h} h_L^{-2}\Vert \eta _{\theta ,h}\Vert _{0,L}^2\\&\qquad + \Vert e_{\theta ,h}\Vert ^2_{0} \left( \frac{1}{2} |\theta |_{W^{1,\infty }({\varOmega })} +\epsilon \Vert {\varvec{u}}_h\Vert _{\infty }^2 + \epsilon \max _{M\in {\mathcal {M}}_h} \{ h_M^2 |\theta |^2_{W^{1,\infty }(M)} \}\right. \\&\left. \qquad +\, \frac{C}{\epsilon } \max _{M\in {\mathcal {M}}_h} \left\{ \frac{h_M^2}{\gamma _M} | \theta |^2_{W^{1,\infty }(M)} \right\} +\frac{C}{\epsilon } \max _{M\in {\mathcal {M}}_h} \{ \gamma _M^{-1} \Vert \theta \Vert _{\infty ,M}^2\} \right) \end{aligned}$$

with $C>0$ independent of the problem parameters, $h_M$, $h_L$ and the solutions.

Proof

Similar estimates can be performed for velocity and temperature. We present the steps for the velocity; for details for the temperature terms, we refer the reader to [21].

We choose the same interpolation operators $j_u:{\varvec{V}}^{div}\rightarrow {\varvec{V}}{}_h^{\,div}$ and $j_\theta :{\varTheta }\rightarrow {\varTheta }_h$ as in Theorem 2. With the splitting $\varvec{\eta }_{u,h}+{\varvec{e}}_{u,h}= ({\varvec{u}}-j_u {\varvec{u}}) + (j_u {\varvec{u}}-{\varvec{u}}_h)$ from (13) and integration by parts, we have

$$\begin{aligned}&c_u({\varvec{u}};{\varvec{u}},{\varvec{e}}_{u,h}) - c_u({\varvec{u}}_h;{\varvec{u}}_h,{\varvec{e}}_{u,h}) \\&\quad = \underbrace{(({\varvec{u}}-{\varvec{u}}_h) \cdot \nabla {\varvec{u}},{\varvec{e}}_{u,h})}_{=: T^u_1} + \underbrace{({\varvec{u}}_h \cdot \nabla ({\varvec{u}}-j_u{\varvec{u}}),{\varvec{e}}_{u,h})}_{=: T^u_2} - \frac{1}{2} \underbrace{((\nabla \cdot {\varvec{u}}_h)j_u{\varvec{u}}, {\varvec{e}}_{u,h})}_{=: T^u_3}. \end{aligned}$$

Now, we bound each term separately. Using Young’s inequality with $\epsilon >0$, we calculate:

$$\begin{aligned} T^u_1&\le \sum _{M\in {\mathcal {M}}_h} \Vert \nabla {\varvec{u}}\Vert _{\infty ,M} \left( \Vert {\varvec{e}}_{u,h}\Vert ^2_{0,M} + \Vert \varvec{\eta }_{u,h}\Vert _{0,M} \Vert {\varvec{e}}_{u,h}\Vert _{0,M} \right) \nonumber \\&\le |{\varvec{u}}|_{W^{1,\infty }({\varOmega })} \Vert {\varvec{e}}_{u,h}\Vert ^2_{0} + \sum _{M\in {\mathcal {M}}_h} \frac{1}{h_M} | {\varvec{u}}|_{W^{1,\infty }(M)} \Vert \varvec{\eta }_{u,h}\Vert _{0,M} h_M \Vert {\varvec{e}}_{u,h}\Vert _{0,M} \nonumber \\&\le \frac{1}{4\epsilon } \sum _{M\in {\mathcal {M}}_h} \frac{1}{h_M^2} \Vert \varvec{\eta }_{u,h} \Vert ^2_{0,M} + \left( |{\varvec{u}}|_{W^{1,\infty }({\varOmega })} + \epsilon \max _{M\in {\mathcal {M}}_h} \{ h_M^2 |{\varvec{u}}|_{W^{1,\infty }(M)}^2 \} \right) \Vert {\varvec{e}}_{u,h}\Vert ^2_{0}. \end{aligned}$$

(33)

For the term $T^u_2$, we have via integration by parts

$$\begin{aligned} T^u_2 = ({\varvec{u}}_h \cdot \nabla \varvec{\eta }_{u,h},{\varvec{e}}_{u,h}) = - ({\varvec{u}}_h \cdot \nabla {\varvec{e}}_{u,h},\varvec{\eta }_{u,h}) - ((\nabla \cdot {\varvec{u}}_h){\varvec{e}}_{u,h}, \varvec{\eta }_{u,h}) =: T^u_{21}+T^u_{22}. \end{aligned}$$

Term $T^u_{21}$ is the most critical one. We calculate using Assumption 2 and Young’s inequality:

$$\begin{aligned} T^u_{21}&= - ({\varvec{u}}_h \cdot \nabla {\varvec{e}}_{u,h},\varvec{\eta }_{u,h}) \le \sum _{M\in {\mathcal {M}}_h} \Vert {\varvec{u}}_h\Vert _{\infty ,M} \Vert \nabla {\varvec{e}}_{u,h}\Vert _{0,M} \Vert \varvec{\eta }_{u,h}\Vert _{0,M} \nonumber \\&\le C \sum _{M\in {\mathcal {M}}_h} \Vert {\varvec{u}}_h\Vert _{\infty ,M} \Vert {\varvec{e}}_{u,h}\Vert _{0,M}h_M^{-1}\Vert \varvec{\eta }_{u,h}\Vert _{0,M} \nonumber \\&\le \epsilon \Vert {\varvec{u}}_h\Vert _{\infty }^2 \Vert {\varvec{e}}_{u,h}\Vert _{0}^2 + \frac{C}{\epsilon } \sum _{M\in {\mathcal {M}}_h} h_M^{-2}\Vert \varvec{\eta }_{u,h}\Vert _{0,M}^2. \end{aligned}$$

(34)

Using $(\nabla \cdot {\varvec{u}},q)=0$ for all $q\in L^2({\varOmega })$, Assumption 1 and Young’s inequality with $\epsilon >0$, we obtain

$$\begin{aligned} T^u_{22}&= - ((\nabla \cdot {\varvec{u}}_h)\varvec{\eta }_{u,h}, {\varvec{e}}_{u,h})~ = ((\nabla \cdot (\varvec{\eta }_{u,h} + {\varvec{e}}_{u,h}))\varvec{\eta }_{u,h}, {\varvec{e}}_{u,h}) \nonumber \\&\le \sum _{M\in {\mathcal {M}}_h} \Vert \varvec{\eta }_{u,h}\Vert _{\infty ,M} \left( \Vert \nabla \cdot {\varvec{e}}_{u,h}\Vert _{0,M} + \Vert \nabla \cdot \varvec{\eta }_{u,h} \Vert _{0,M} \right) \Vert {\varvec{e}}_{u,h}\Vert _{0,M} \nonumber \\&\le \sum _{M\in {\mathcal {M}}_h} \frac{Ch_M}{\sqrt{\gamma _M}}|{\varvec{u}}|_{W^{1,\infty }(M)}\sqrt{\gamma _M} \left( \Vert \nabla \cdot {\varvec{e}}_{u,h}\Vert _{0,M} + \Vert \nabla \cdot \varvec{\eta }_{u,h}\Vert _{0,M} \right) \Vert {\varvec{e}}_{u,h}\Vert _{0,M} \nonumber \\&\le \epsilon ||| \varvec{\eta }_{u,h}|||^2_{\textit{LPS}} + \epsilon ||| {\varvec{e}}_{u,h}|||^2_{\textit{LPS}} + \frac{C}{\epsilon } \max _{M\in {\mathcal {M}}_h} \left\{ \frac{h_M^2}{\gamma _M} | {\varvec{u}}|^2_{W^{1,\infty }(M)} \right\} \Vert {\varvec{e}}_{u,h}\Vert ^2_{0}. \end{aligned}$$

(35)

Utilizing the splitting according to (13), we have

$$\begin{aligned} T^u_3 = ((\nabla \cdot {\varvec{u}}_h)j_u{\varvec{u}}, {\varvec{e}}_{u,h}) = -((\nabla \cdot {\varvec{u}}_h)\varvec{\eta }_{u,h}, {\varvec{e}}_{u,h}) +((\nabla \cdot {\varvec{u}}_h) {\varvec{u}}, {\varvec{e}}_{u,h}) = T^u_{22} + T^u_{32}. \end{aligned}$$

and use the same estimate as in (35). For the term $T^u_{32}$, we use that $(\nabla \cdot {\varvec{u}},q)=0$ for all $q\in L^2({\varOmega })$ and Young’s inequality:

$$\begin{aligned} |T^u_{32}|&= |(\nabla \cdot {\varvec{u}}_h, {\varvec{u}} \cdot {\varvec{e}}_{u,h})| = | (\nabla \cdot (-\varvec{\eta }_{u,h} - {\varvec{e}}_{u,h} + {\varvec{u}}), {\varvec{u}} \cdot {\varvec{e}}_{u,h} )|\nonumber \\&\le | (\nabla \cdot \varvec{\eta }_{u,h},{\varvec{u}} \cdot {\varvec{e}}_{u,h} ) | +|(\nabla \cdot {\varvec{e}}_{u,h}, {\varvec{u}} \cdot {\varvec{e}}_{u,h} )|\nonumber \\&\le \sum _{M\in {\mathcal {M}}_h} \left( \Vert {\varvec{u}}\Vert _{\infty ,M} \sqrt{\gamma _M}\Vert \nabla \cdot \varvec{\eta }_{u,h}\Vert _{0,M} \frac{1}{\sqrt{\gamma _M}} \Vert {\varvec{e}}_{u,h}\Vert _{0,M} \right. \nonumber \\&\quad \left. + \Vert {\varvec{u}}\Vert _{\infty ,M} \sqrt{\gamma _M}\Vert \nabla \cdot {\varvec{e}}_{u,h}\Vert _{0,M} \frac{1}{\sqrt{\gamma _M}}\Vert {\varvec{e}}_{u,h}\Vert _{0,M} \right) \nonumber \\&\le \epsilon ||| \varvec{\eta }_{u,h}|||_{\textit{LPS}}^2 + \epsilon ||| {\varvec{e}}_{u,h}|||_{\textit{LPS}}^2 + \frac{C}{\epsilon } \max _{M\in {\mathcal {M}}_h} \{ \gamma _M^{-1}\Vert {\varvec{u}}\Vert _{\infty ,M}^2 \} \Vert {\varvec{e}}_{u,h}\Vert _0^2. \end{aligned}$$

(36)

Combining the above bounds (33)–(36) yields the claim. $\square$