Optimally convergent mixed finite element methods for the time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise

Abstract

In this paper, a new time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise is developed and studied. This model considers heat transfer between the free flow in the pipe region and the porous media flow in the porous media region. Darcy’s law and stochastic Navier-Stokes equations are used to control the flows in the pipe and porous media regions, respectively. The heat equation is coupled with the flow equation to describe the heat transfer in these both regions. In order to avoid sub-optimal convergence, a new mixed finite element method is proposed by using the Helmholtz decomposition that drives the multiplicative noise. Then, the stability of the proposed method is proved, and we obtain the optimal convergence order $o(\Delta t^{\frac{1}{2}}+h)$ of global error estimation. Finally, numerical results indicate the efficiency of the proposed model and the accuracy of the numerical method.

A Appendix

Proof of Theorem 3.2. In (2.32) - (2.35), let $t=t_{n+1}$ and $t=t_{n}$, and then make the difference. Subtract (3.1) - (3.6) from the resulting formula and define $a_{f}^{n+1}\!=\!\textbf{u}_{f}(t_{n+1})\!-\textbf{u}_{f}^{n+1},b_{f}^{n+1}\!=\!R(t_{n+1})\!-r_{f}^{n+1}, c_{f}^{n+1}\!=\!\theta _{f}(t_{n+1})\!-\theta _{f}^{n+1}; a_{p}^{n+1}\!=\textbf{u}_{p}(t_{n+1})-\textbf{u}_{p}^{n+1},b_{p}^{n}\!=\phi _{p}(t_{n+1})\!-\phi _{p}^{n+1}, c_{p}^{n}\!=\!\theta _{p}(t_{n+1})-\theta _{p}^{n+1}$, we get

$$\begin{aligned}{} & {} (a_{f}^{n+1}-a_{f}^{n},\textbf{v}_{f})_{\Omega _{f}} +Pr\int _{t_{n}}^{t_{n+1}}\big (\nabla (\textbf{u}_{f}(s)-\textbf{u}_{f}^{n+1}),\nabla \textbf{v}_{f}\big )_{\Omega _{f}}ds\nonumber \\{} & {} +(c_{f}^{n+1}-c_{f}^{n},\varphi )_{\Omega _{f}}+k_{f}\int _{t_{n}}^{t_{n+1}}\big (\nabla (\theta _{f}(s)-\theta _{f}^{n+1}),\nabla \varphi \big )_{\Omega _{f}}ds\nonumber \\{} & {} +\int _{t_{n}}^{t_{n+1}}\big (c(\textbf{u}_{f}(s),\textbf{u}_{f}(s),\textbf{v}_{f})_{\Omega _{f}}-c(\textbf{u}_{f}^{n},\textbf{u}_{f}^{n+1},\textbf{v}_{f})_{\Omega _{f}}\big ]ds\nonumber \\{} & {} +\int _{t_{n}}^{t_{n+1}}\big (t_{f}(\textbf{u}_{f}(s),\theta _{f}(s),\varphi )_{\Omega _{f}}-t_{f}(\textbf{u}_{f}^{n},\theta _{f}^{n+1},\varphi )_{\Omega _{f}}\big ]ds\nonumber \\{} & {} -k_{f}\int _{t_{n}}^{t_{n+1}}\bigg [\int _{\uppercase {i}}\hat{n}_{f}\cdot \nabla (\theta _{f}(s)-\theta _{f}^{n})\varphi dl\bigg ]ds\nonumber \\{} & {} +\frac{k_{f}\gamma }{h}\int _{t_{n}}^{t_{n+1}}\bigg [\int _{\uppercase {i}}(\theta _{f}(s)\!-\!\theta _{p}(s)\!-\!\theta _{f}^{n+1}\!+\!\theta _{p}^{n})\varphi dl\bigg ]ds \!+\!C_{a}Da(a_{p}^{n+1}\!-\!a_{p}^{n},\textbf{v}_{p})_{\Omega _{p}}\nonumber \\{} & {} +Pr\int _{t_{n}}^{t_{n+1}}\big (\textbf{u}_{p}(s)-\textbf{u}_{p}^{n+1},\textbf{v}_{p}\big )_{\Omega _{p}}ds +(c_{p}^{n+1}-c_{p}^{n},\omega )_{\Omega _{p}}\nonumber \\{} & {} +k_{p}\int _{t_{n}}^{t_{n+1}}\big (\nabla (\theta _{p}(s)-\theta _{p}^{n+1}),\nabla \omega \big )_{\Omega _{p}}ds +\int _{t_{n}}^{t_{n+1}}\big (t_{p}(\textbf{u}_{p}(s),\theta _{p}(s),\omega )_{\Omega _{p}}\nonumber \\{} & {} -t_{p}(\textbf{u}_{p}^{n},\theta _{p}^{n+1},\omega )_{\Omega _{p}}\big ]ds +k_{f}\int _{t_{n}}^{t_{n+1}}\bigg [\int _{\uppercase {i}}\hat{n}_{f}\cdot \nabla (\theta _{f}(s)-\theta _{f}^{n})\omega dl\bigg ]ds\nonumber \\{} & {} -\frac{k_{f}\gamma }{h}\int _{t_{n}}^{t_{n+1}}\bigg [\int _{\uppercase {i}}(\theta _{f}(s)-\theta _{p}(s)-\theta _{f}^{n+1}+\theta _{p}^{n+1})\omega dl\bigg ]ds\nonumber \\= & {} PrRa\mathbf {\xi }\int _{t_{n}}^{t_{n+1}}(\theta _{f}(s)-\theta _{f}^{n},\textbf{v}_{f})_{\Omega _{f}}ds +PrDaRa\mathbf {\xi }\int _{t_{n}}^{t_{n+1}}(\theta _{p}(s)-\theta _{p}^{n},\textbf{v}_{p})_{\Omega _{p}}ds\nonumber \\{} & {} +\bigg (\int _{t_{n}}^{t_{n+1}}\big (\mathbf {\varepsilon }(s)-\mathbf {\varepsilon }^{n},\textbf{u}_{f}^{n})\big )dW(s),\textbf{v}_{f}\bigg )_{\Omega _{f}} +\int _{t_{n}}^{t_{n+1}}(\textbf{f}_{f}(s)-\textbf{f}_{f}(t_{n+1}),a_{f}^{n+1})ds.\nonumber \\ \end{aligned}$$

(A.1)

Taking $\textbf{v}_{f}=2a_{f}^{n+1},\varphi =2c_{f}^{n+1};\textbf{v}_{p}=2a_{p}^{n+1},\omega =2c_{p}^{n+1}$, and using the identity $2(a-b,a)=a^{2}-b^{2}+(a-b)^{2}$, we find that

$$\begin{aligned}{} & {} \parallel a_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2}-\parallel a_{f}^{n}\parallel _{L^{2}(\Omega _{f})}^{2} +\parallel a_{f}^{n+1}-a_{f}^{n}\parallel _{L^{2}(\Omega _{f})}^{2}+2\Delta tPr\parallel \nabla a_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2}\nonumber \\{} & {} +\parallel c_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2}-\parallel c_{f}^{n}\parallel _{L^{2}(\Omega _{f})}^{2} +\parallel c_{f}^{n+1}-c_{f}^{n}\parallel _{L^{2}(\Omega _{f})}^{2}+2\Delta tk_{f}\parallel \nabla c_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2}\nonumber \\{} & {} +C_{a}Da\parallel a_{p}^{n+1}\parallel _{L^{2}(\Omega _{p})}^{2}-C_{a}Da\parallel a_{p}^{n}\parallel _{L^{2}(\Omega _{p})}^{2} +C_{a}Da\parallel a_{p}^{n+1}-a_{p}^{n}\parallel _{L^{2}(\Omega _{p})}^{2}\nonumber \\{} & {} +2\Delta tPr\parallel a_{p}^{n+1}\parallel _{L^{2}(\Omega _{p})}^{2} +\parallel c_{p}^{n+1}\parallel _{L^{2}(\Omega _{p})}^{2}-\parallel c_{p}^{n}\parallel _{L^{2}(\Omega _{p})}^{2} +\parallel c_{p}^{n+1}-c_{p}^{n}\parallel _{L^{2}(\Omega _{p})}^{2}\nonumber \\{} & {} +2\Delta tk_{p}\parallel \nabla c_{p}^{n+1}\parallel _{L^{2}(\Omega _{p})}^{2}\nonumber \\\le & {} \!-2Pr\int _{t_{n}}^{t_{n+1}}\!\!\big (\nabla (\textbf{u}_{f}(s)\!-\!\textbf{u}_{f}(t_{n+1})),\nabla a_{f}^{n+1}\big )_{\Omega _{f}}ds\!-\!2k_{f}\int _{t_{n}}^{t_{n+1}}\!\big (\nabla (\theta _{f}(s)\!-\!\theta _{f}(t_{n+1})),\nabla c_{f}^{n+1}\big )_{\Omega _{f}}ds\nonumber \\{} & {} -2Pr\int _{t_{n}}^{t_{n+1}}\big (\textbf{u}_{p}(s)-\textbf{u}_{p}(t_{n+1}),a_{p}^{n+1}\big )_{\Omega _{p}}ds -2k_{p}\int _{t_{n}}^{t_{n+1}}\big (\nabla (\theta _{p}(s)-\theta _{p}(t_{n+1})),\nabla c_{p}^{n+1}\big )_{\Omega _{p}}ds\nonumber \\{} & {} +2k_{f}\int _{t_{n}}^{t_{n+1}}\bigg [\int _{\uppercase {i}}\hat{n}_{f} \cdot \nabla (\theta _{f}(s)-\theta _{f}^{n})(c_{f}^{n+1}-c_{p}^{n+1})dl\bigg ]ds-\frac{2k_{f}\gamma }{h}\int _{t_{n}}^{t_{n+1}}\bigg [\int _{\uppercase {i}} (\theta _{f}(s)-\theta _{p}(s)\nonumber \\{} & {} -\theta _{f}^{n+1}+\theta _{p}^{n})c_{f}^{n+1}dl\bigg ]ds +\frac{2k_{f}\gamma }{h}\int _{t_{n}}^{t_{n+1}}\bigg [\int _{\uppercase {i}} (\theta _{f}(s)-\theta _{p}(s)-\theta _{f}^{n+1}+\theta _{p}^{n+1})c_{p}^{n+1}dl\bigg ]ds\nonumber \\{} & {} +2PrRa\mathbf {\xi }\int _{t_{n}}^{t_{n+1}}\big (\theta _{f}(s)-\theta _{f}^{n},a_{f}^{n+1}\big )_{\Omega _{f}}ds+2PrDaRa\mathbf {\xi }\int _{t_{n}}^{t_{n+1}}\big (\theta _{p}(s)-\theta _{p}^{n},a_{p}^{n+1}\big )_{\Omega _{p}}ds\nonumber \\{} & {} +2\bigg (\int _{t_{n}}^{t_{n+1}}\big (\mathbf {\varepsilon }(s)-\mathbf {\varepsilon }^{n}\big )dW(s),a_{f}^{n+1}\bigg )_{\Omega _{p}}+\int _{t_{n}}^{t_{n+1}}(\textbf{f}_{f}(s)-\textbf{f}_{f}(t_{n+1}),a_{f}^{n+1})ds\nonumber \\{} & {} \!+2\int _{t_{n}}^{t_{n+1}}\!\big [c(\textbf{u}_{f}(s),\textbf{u}_{f}(s),a_{f}^{n+1})_{\Omega _{f}} \!-\!c(\textbf{u}_{f}^{n},\textbf{u}_{f}^{n+1},a_{f}^{n+1})_{\Omega _{f}}\!\big ]ds \!+\!2\int _{t_{n}}^{t_{n+1}}\!\big [t_{f}(\textbf{u}_{f}(s),\!\theta _{f}(s),\!c_{f}^{n+1})_{\Omega _{f}}\nonumber \\{} & {} -t_{f}(\textbf{u}_{f}^{n},\theta _{f}^{n+1},c_{f}^{n+1})_{\Omega _{f}}\big ]ds+2\int _{t_{n}}^{t_{n+1}}\big [t_{p}(\textbf{u}_{p}(s),\theta _{p}(s),c_{p}^{n+1})_{\Omega _{p}} -t_{p}(\textbf{u}_{p}^{n},\theta _{p}^{n+1},c_{p}^{n+1})_{\Omega _{p}}\big ]ds.\nonumber \\ \end{aligned}$$

(A.2)

Using the Cauchy-Schwarz inequality, the Young inequality and (2.36), yields

$$\begin{aligned}{} & {} \!-2Pr\int _{t_{n}}^{t_{n+1}}\!\big (\nabla (\textbf{u}_{f}(s)\!-\!\textbf{u}_{f}(t_{n+1})),\nabla a_{f}^{n+1}\big )_{\Omega _{f}}ds \!\le \!\frac{\Delta tPr}{4}\!\parallel \nabla a_{f}^{n+1}\!\parallel _{L^{2}(\Omega _{f})}^{2}+C\Delta t,\nonumber \\ \end{aligned}$$

(A.3)

$$\begin{aligned}{} & {} \!-2k_{f}\int _{t_{n}}^{t_{n+1}}\!\big (\nabla (\theta _{f}(s)\!-\!\theta _{f}(t_{n+1})),\nabla c_{f}^{n+1}\big )_{\Omega _{f}}ds\! \le \!\frac{\Delta tk_{f}}{2}\!\parallel \nabla c_{f}^{n+1}\!\parallel _{L^{2}(\Omega _{f})}^{2}+C\Delta t,\nonumber \\\end{aligned}$$

(A.4)

$$\begin{aligned}{} & {} \!-2Pr\int _{t_{n}}^{t_{n+1}}\!\big (\textbf{u}_{p}(s)\!-\!\textbf{u}_{p}(t_{n+1}),a_{p}^{n+1}\big )_{\Omega _{p}}ds\! \le \!\frac{\Delta tPr}{2}\!\parallel a_{p}^{n+1}\!\parallel _{L^{2}(\Omega _{p})}^{2}+C\Delta t, \end{aligned}$$

(A.5)

$$\begin{aligned}{} & {} \!-2k_{p}\int _{t_{n}}^{t_{n+1}}\!\big (\nabla (\theta _{p}(s)\!-\theta _{p}(t_{n+1})),\nabla c_{p}^{n+1}\big )_{\Omega _{p}}ds \!\le \!\frac{\Delta tk_{p}}{2}\!\parallel \nabla c_{p}^{n+1}\!\parallel _{L^{2}(\Omega _{p})}^{2}+C\Delta t.\nonumber \\\end{aligned}$$

(A.6)

$$\begin{aligned}{} & {} \!\int _{t_{n}}^{t_{n+1}}(\textbf{f}_{f}(s)\!-\textbf{f}_{f}(t_{n+1}),a_{f}^{n+1})_{\Omega _{f}}ds \le \parallel a_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2}+C\Delta t. \end{aligned}$$

(A.7)

As for the trilinear term, we can bound

$$\begin{aligned}{} & {} 2\int _{t_{n}}^{t_{n+1}}\big [c(\textbf{u}_{f}(s),\textbf{u}_{f}(s),a_{f}^{n+1})_{\Omega _{f}} -c(\textbf{u}_{f}^{n},\textbf{u}_{f}^{n+1},a_{f}^{n+1})_{\Omega _{f}}\big ]ds\\= & {} 2\int _{t_{n}}^{t_{n+1}}\big [c(\textbf{u}_{f}(s),\textbf{u}_{f}(s),a_{f}^{n+1})_{\Omega _{f}} -c(\textbf{u}_{f}(t_{n}),\textbf{u}_{f}(t_{n+1)},a_{f}^{n+1})_{\Omega _{f}}\big ]ds\\{} & {} +2\int _{t_{n}}^{t_{n+1}}\big [c(\textbf{u}_{f}(t_{n}),\textbf{u}_{f}(t_{n+1)},a_{f}^{n+1})_{\Omega _{f}} -c(\textbf{u}_{f}^{n},\textbf{u}_{f}^{n+1},a_{f}^{n+1})_{\Omega _{f}}\big ]ds\\ \end{aligned}$$

$$\begin{aligned}= & {} 2\int _{t_{n}}^{t_{n+1}}\!\big [c(\textbf{u}_{f}(s)\!-\textbf{u}_{f}(t_{n}),\textbf{u}_{f}(s),a_{f}^{n+1})_{\Omega _{f}} \!+c(\textbf{u}_{f}(t_{n}),\textbf{u}_{f}(s)\!-\textbf{u}_{f}(t_{n+1)},a_{f}^{n+1})_{\Omega _{f}}\big ]ds\nonumber \\{} & {} +2\int _{t_{n}}^{t_{n+1}}c(a_{f}^{n},\textbf{u}_{f}(t_{n+1)},a_{f}^{n+1})_{\Omega _{f}}ds\nonumber \\\le & {} Pr\Delta t\parallel \nabla a_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2}+C\Delta t, \end{aligned}$$

(A.8)

Similarly, we can get

$$\begin{aligned}{} & {} 2\int _{t_{n}}^{t_{n+1}}\big [t_{f}(\textbf{u}_{f}(s),\theta _{f}(s),c_{f}^{n+1})_{\Omega _{f}} -t_{f}(\textbf{u}_{f}^{n},\theta _{f}^{n+1},c_{f}^{n+1})_{\Omega _{f}}\big ]ds \nonumber \\\le & {} \frac{k_{f}\Delta t}{2}\parallel \nabla c_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2} +C\Delta t\Vert a_{f}^{n}\Vert _{L^{2}(\Omega _{f})}^{2} +C\Delta t,\end{aligned}$$

(A.9)

$$\begin{aligned}{} & {} 2\int _{t_{n}}^{t_{n+1}}\big [t_{p}(\textbf{u}_{p}(s),\theta _{p}(s),c_{p}^{n+1})_{\Omega _{p}} -t_{p}(\textbf{u}_{p}^{n},\theta _{p}^{n+1},c_{p}^{n+1})_{\Omega _{p}}\big ]ds\nonumber \\\le & {} \frac{k_{p}\Delta t}{2}\parallel \nabla c_{p}^{n+1}\parallel _{L^{2}(\Omega _{p})}^{2} +C\Delta t\Vert a_{p}^{n}\Vert _{L^{2}(\Omega _{p})}^{2} +C\Delta t. \end{aligned}$$

(A.10)

Applying the trace inequality, the inverse inequality and (2.36), we obtain

$$\begin{aligned}{} & {} 2k_{f}\int _{t_{n}}^{t_{n+1}}\bigg [\int _{\uppercase {i}}\hat{n}_{f} \cdot \nabla (\theta _{f}(s)-\theta _{f}^{n})(c_{f}^{n+1}-c_{p}^{n+1})dl\bigg ]ds\nonumber \\= & {} 2k_{f}\int _{t_{n}}^{t_{n+1}}\!\bigg [\int _{\uppercase {i}}\hat{n}_{f} \cdot \nabla (\theta _{f}(s)\!-\!\theta _{f}(t_{n}))(c_{f}^{n+1}\!-\!c_{p}^{n+1})dl\bigg ]ds \!+\!2k_{f}\int _{t_{n}}^{t_{n+1}}\!\bigg [\!\int _{\uppercase {i}}\hat{n}_{f} \cdot \nabla c_{f}^{n}(c_{f}^{n+1}-c_{p}^{n+1})dl\bigg ]ds\nonumber \\\le & {} \frac{4k_{f}C_{inv}\Delta t}{\gamma }\parallel \nabla c_{f}^{n}\parallel _{L^{2}(\Omega _{f})}^{2} +\frac{k_{f}\gamma \Delta t}{2h}\parallel c_{f}^{n+1}-c_{p}^{n+1}\parallel _{L^{2}(\uppercase {i})}^{2} +C\Delta t, \end{aligned}$$

(A.11)

$$\begin{aligned}{} & {} -\frac{2k_{f}\gamma }{h}\int _{t_{n}}^{t_{n+1}}\bigg [\int _{\uppercase {i}} (\theta _{f}(s)-\theta _{p}(s)-\theta _{f}^{n+1}+\theta _{p}^{n})c_{f}^{n+1}dl\bigg ]ds\nonumber \\{} & {} +\frac{2k_{f}\gamma }{h}\int _{t_{n}}^{t_{n+1}}\bigg [\int _{\uppercase {i}} (\theta _{f}(s)-\theta _{p}(s)-\theta _{f}^{n+1}+\theta _{p}^{n+1})c_{p}^{n+1}dl\bigg ]ds\nonumber \\= & {} -\frac{2k_{f}\gamma \Delta t}{h}\int _{\uppercase {i}}(c_{f}^{n+1}-c_{p}^{n})c_{f}^{n+1}dl +\frac{2k_{f}\gamma \Delta t}{h}\int _{\uppercase {i}}(c_{f}^{n+1}-c_{p}^{n+1})c_{p}^{n+1}dl -\frac{2k_{f}\gamma \Delta t}{h}\int _{\uppercase {i}}(\theta _{p}(t_{n+1})\nonumber \\{} & {} -\theta _{p}(t_{n}))c_{f}^{n+1}dl\nonumber \\\le & {} -\frac{2k_{f}\gamma \Delta t}{h}\parallel c_{f}^{n+1}-c_{p}^{n+1}\parallel _{L^{2}(\uppercase {i})}^{2} -\frac{k_{f}\gamma \Delta t}{h}\big [\parallel c_{p}^{n+1}\parallel _{L^{2}(\uppercase {i})}^{2}-\parallel c_{p}^{n}\parallel _{L^{2}(\uppercase {i})}^{2} -\parallel c_{f}^{n+1}\nonumber \\{} & {} -c_{p}^{n+1}\parallel _{L^{2}(\uppercase {i})}^{2}\big ]+\frac{4k_{f}C_{inv}\Delta t}{\gamma }\Vert \nabla c_{f}^{n+1}\Vert _{L^{2}(\Omega _{f})}^{2}+C\Delta t. \end{aligned}$$

(A.12)

Thanks to the Cauchy-Schwarz inequality, the Young’s inequality and (2.36), yield the following inequalities

$$\begin{aligned}{} & {} 2PrRa\mathbf {\xi }\int _{t_{n}}^{t_{n+1}}\big (\theta _{f}(s)-\theta _{f}^{n},a_{f}^{n+1}\big )_{\Omega _{f}}ds\nonumber \\= & {} 2PrRa\mathbf {\xi }\int _{t_{n}}^{t_{n+1}}\big (\theta _{f}(s)-\theta _{f}(t_{n}),a_{f}^{n+1}\big )_{\Omega _{f}}ds +2PrRa\mathbf {\xi }\int _{t_{n}}^{t_{n+1}}\big (\theta _{f}(t_{n})-\theta _{f}^{n},a_{f}^{n+1}\big )_{\Omega _{f}}ds\nonumber \\\le & {} 4PrRa^{2}\Delta t\parallel c_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2} +\frac{1}{4}Pr\Delta t\parallel \nabla a_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2} +2PrRa\Delta t\parallel a_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2}+C\Delta t,\nonumber \\ \end{aligned}$$

(A.13)

$$\begin{aligned}{} & {} 2PrDaRa\mathbf {\xi }\int _{t_{n}}^{t_{n+1}}\big (\theta _{p}(s)-\theta _{p}^{n},a_{p}^{n+1}\big )_{\Omega _{p}}ds\\\le & {} 2PrRa^{2}Da^{2}\Delta t\parallel c_{p}^{n+1}\parallel _{L^{2}(\Omega _{p})}^{2} \!+\frac{1}{2}Pr\Delta t\parallel a_{p}^{n+1}\parallel _{L^{2}(\Omega _{p})}^{2} \!+2PrDaRa\Delta t\parallel a_{p}^{n+1}\parallel _{L^{2}(\Omega _{p})}^{2}+C\Delta t.\nonumber \end{aligned}$$

(A.14)

By the definition of the martingale, (2.27), (2.20) and (2.36), we can get

$$\begin{aligned}{} & {} 2\bigg (\int _{t_{n}}^{t_{n+1}}\big (\mathbf {\varepsilon }(s)-\mathbf {\varepsilon }^{n}\big )dW(s),a_{f}^{n+1}\bigg )_{\Omega _{p}}\nonumber \\= & {} 2\bigg (\!\int _{t_{n}}^{t_{n+1}}\big (B(s,\textbf{u}_{f}(s))\!-\!B(t_{n},\textbf{u}_{f}^{n})\big )dW(s),a_{f}^{n+1}\!-\!a_{f}^{n}\bigg )_{\Omega _{p}} \!+\!2\bigg (\!\int _{t_{n}}^{t_{n+1}}\big (b(s)\!-\!b^{n}\big )dW(s),a_{f}^{n+1}\!-\!a_{f}^{n}\!\bigg )_{\Omega _{p}}\nonumber \\\le & {} \frac{1}{2}\Vert a_{f}^{n+1}-a_{f}^{n}\Vert _{L^{2}(\Omega _{f})}^{2} +C\Delta t\Vert a_{f}^{n}\Vert _{L^{2}(\Omega _{f})}^{2} +C\Delta t^{2}\parallel a_{f}^{n}\parallel _{L^{2}(\Omega _{f})}^{2}+C\Delta t^{3}. \end{aligned}$$

(A.15)

Substitute (A.3) - (A.15) into (A.2), taking the sum from $n=0$ to $l-1$, using Gronwall’s lemma, and taking the expectation, we obtain

$$\begin{aligned}{} & {} E\big [\parallel a_{f}^{l}\parallel _{L^{2}(\Omega _{f})}^{2}\big ] +\frac{1}{2}E\bigg [\sum _{n=0}^{l-1}\parallel a_{f}^{n+1}-a_{f}^{n}\parallel _{L^{2}(\Omega _{f})}^{2}\bigg ] +\frac{\Delta tPr}{2}E\bigg [\sum _{n=0}^{l-1}\parallel \nabla a_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2}\bigg ]\nonumber \\{} & {} +E\big [\!\parallel c_{f}^{l}\!\parallel _{L^{2}(\Omega _{f})}^{2}\!\big ] \!+E\bigg [\!\sum _{n=0}^{l-1}\!\parallel c_{f}^{n+1}\!-c_{f}^{n}\parallel _{L^{2}(\Omega _{f})}^{2}\!\bigg ] \!+\Delta tk_{f}\Big (1\!-\frac{8C_{inv}}{\gamma }\Big )E\bigg [\sum _{n=0}^{l-1}\!\parallel \nabla c_{f}^{n+1}\!\parallel _{L^{2}(\Omega _{f})}^{2}\!\bigg ]\nonumber \\{} & {} +C_{a}DaE\big [\!\parallel a_{p}^{l}\!\parallel _{L^{2}(\Omega _{p})}^{2}\big ] \!+C_{a}DaE\bigg [\!\sum _{n=0}^{l-1}\parallel a_{p}^{n+1}\!-\!a_{p}^{n}\parallel _{L^{2}(\Omega _{p})}^{2}\!\bigg ] \!+\Delta tPrE\bigg [\!\sum _{n=0}^{l-1}\!\parallel a_{p}^{n+1}\!\parallel _{L^{2}(\Omega _{p})}^{2}\!\bigg ]\nonumber \\{} & {} +E\big [\parallel c_{p}^{l}\parallel _{L^{2}(\Omega _{p})}^{2}\big ] +E\bigg [\sum _{n=0}^{l-1}\parallel c_{p}^{n+1}-c_{p}^{n}\parallel _{L^{2}(\Omega _{p})}^{2}\bigg ] +\Delta tk_{p}E\bigg [\sum _{n=0}^{l-1}\parallel \nabla c_{p}^{n+1}\parallel _{L^{2}(\Omega _{p})}^{2}\bigg ]\nonumber \\{} & {} +\frac{k_{f}\gamma \Delta t}{h}E\big [\parallel c_{p}^{l}\parallel _{L^{2}(\Omega _{p})}^{2}\big ] +\frac{k_{f}\gamma \Delta t}{4h}E\bigg [\sum _{n=0}^{l-1}\parallel c_{f}^{n+1}-c_{p}^{n+1}\parallel _{L^{2}(\uppercase {i})}^{2}\bigg ]\nonumber \\\le & {} C\Delta t, \end{aligned}$$

(A.16)

where $\gamma >8C_{inv}$. Thus we prove that (3.11).

Decoupling (2.32), let $t=t_{n+1}$ and $t=t_{n}$, and then make the difference. Subtract (3.1) from the resulting formula to get

$$\begin{aligned}{} & {} \int _{t_{n}}^{t^{n+1}}(R(s)-r_{f}^{n+1},\nabla \cdot \textbf{v}_{f})_{\Omega _{f}}ds\nonumber \\= & {} (a_{f}^{n+1}\!-\!a_{f}^{n},\textbf{v}_{f})_{\Omega _{f}} \!+\!Pr\!\int _{t_{n}}^{t^{n+1}}\!\!\big (\nabla (\textbf{u}_{f}(s)\!-\textbf{u}_{f}^{n+1}),\nabla \textbf{v}_{f}\big )_{\Omega _{f}}ds \!+\!\int _{t_{n}}^{t^{n+1}}\!\!\big [c(\textbf{u}_{f}(s),\textbf{u}_{f}(s),a_{f}^{n+1})_{\Omega _{f}}\nonumber \\{} & {} \!-c(\textbf{u}_{f}^{n},\textbf{u}_{f}^{n+1},a_{f}^{n+1})_{\Omega _{f}}\big ]ds \!-PrRa\int _{t_{n}}^{t^{n+1}}\!\!(\theta _{f}(s)\!-\theta _{f}^{n},\textbf{v}_{f})_{\Omega _{f}}ds\nonumber \\{} & {} \!+\bigg (\!\int _{t_{n}}^{t^{n+1}}\!\!(\textbf{f}_{f}(s)\!-\textbf{f}_{f}(t_{n+1}))ds,\textbf{v}_{f}\bigg )_{\Omega _{f}}-\bigg (\int _{t_{n}}^{t^{n+1}}\big (\mathbf {\varepsilon }(s)-\mathbf {\varepsilon }^{n}\big )dW(s),\textbf{v}_{f}\!\bigg )_{\Omega _{f}}. \end{aligned}$$

(A.17)

Thus, sum from n=0 to $l-1$ of the above equations, using the Cauchy-Schwarz inequality, (A.3), (A.8) and (A.15), we gain

$$\begin{aligned}{} & {} \bigg (\sum _{n=0}^{l-1}\int _{t_{n}}^{t^{n+1}}(R(s)-r_{f}^{n+1})ds,\nabla \cdot \textbf{v}_{f}\bigg )_{\Omega _{f}}\nonumber \\= & {} \sum _{n=0}^{l-1}(a_{f}^{n+1}-a_{f}^{n},\textbf{v}_{f})_{\Omega _{f}} +Pr\sum _{n=0}^{l-1}\int _{t_{n}}^{t^{n+1}}\big (\nabla (\textbf{u}_{f}(s)-\textbf{u}_{f}^{n+1}),\nabla \textbf{v}_{f}\big )_{\Omega _{f}}ds\nonumber \\{} & {} +\sum _{n=0}^{l-1}\int _{t_{n}}^{t^{n+1}}\big [c(\textbf{u}_{f}(s),\textbf{u}_{f}(s),a_{f}^{n+1})_{\Omega _{f}} -c(\textbf{u}_{f}^{n},\textbf{u}_{f}^{n+1},a_{f}^{n+1})_{\Omega _{f}}\big ]ds\nonumber \\{} & {} -PrRa\mathbf {\xi }\sum _{n=0}^{l-1}\int _{t_{n}}^{t^{n+1}}(\theta _{f}(s)-\theta _{f}^{n},\textbf{v}_{f})_{\Omega _{f}}ds -\sum _{n=0}^{l-1}\bigg (\int _{t_{n}}^{t^{n+1}}(\textbf{f}_{f}(s)-\textbf{f}_{f}(t_{n+1}))ds,\textbf{v}_{f}\bigg )_{\Omega _{f}}\nonumber \\{} & {} -\sum _{n=0}^{l-1}\bigg (\int _{t_{n}}^{t^{n+1}}\big (\mathbf {\varepsilon }(s)-\mathbf {\varepsilon }^{n})\big )dW(s),\textbf{v}_{f}\bigg )_{\Omega _{f}}\nonumber \\\le & {} \bigg [\Big (\sum _{n=0}^{l-1}\parallel a_{f}^{n+1}-a_{f}^{n}\parallel _{L^{2}(\Omega _{f})}^{2}\Big )^{\frac{1}{2}} +Pr\Big (\sum _{n=0}^{l-1}\Delta t\parallel \nabla (\textbf{u}_{f}(s)-\textbf{u}_{f}(t_{n+1}))\parallel _{L^{2}(\Omega _{f})}^{2}\Big )^{\frac{1}{2}}\nonumber \\{} & {} +Pr\Big (\sum _{n=0}^{l-1}\Delta t\parallel \nabla a_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2}\Big )^{\frac{1}{2}}+C\bigg (\Delta t\sum _{n=0}^{l-1}\Vert \nabla \textbf{u}_{f}(t_{n+1})\Vert _{L^{2}(\Omega _{f})}^{2}\Vert \nabla a_{f}^{n+1}\Vert _{L^{2}(\Omega _{f})}^{2}\bigg )^{\frac{1}{2}}\nonumber \\{} & {} +PrRa\Big (\sum _{n=0}^{l-1}\Delta t\parallel \theta _{f}(s)-\theta _{f}(t_{n})\parallel _{L^{2}(\Omega _{f})}^{2}\Big )^{\frac{1}{2}}+PrRa\Big (\sum _{n=0}^{l-1}\Delta t\parallel c_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2}\Big )^{\frac{1}{2}}\nonumber \\{} & {} +\Big (\sum _{n=0}^{l-1}\Delta t\parallel \textbf{f}_{f}(s)-\textbf{f}_{f}(t_{n+1})\parallel _{L^{2}(\Omega _{f})}^{2}\Big )^{\frac{1}{2}}\bigg ] \bigg (\sum _{n=0}^{l-1}\parallel \nabla \textbf{v}_{f}^{n+1}\parallel _{L^{2}(\Omega _{f})}^{2}\bigg )^{\frac{1}{2}}. \end{aligned}$$

(A.18)

Taking the expectation of the above inequality and using the inf-sup condition, we can obtain

$$\begin{aligned} E\bigg [\int _{0}^{T}R(s)ds-\Delta t\sum _{n=0}^{l-1}r_{f}^{n+1}\bigg ]\le C\Delta t^{\frac{1}{2}}. \end{aligned}$$

(A.19)

which implies (3.11).

Decoupling (2.32), let $t=t_{n+1}$ and $t=t_{n}$, and then make the difference. Subtract (3.4) from the resulting formula to get

$$\begin{aligned}{} & {} C_{a}Da(a_{p}^{n+1}\!-a_{p}^{n},\textbf{v}_{p})_{\Omega _{p}}\!+Pr\int _{t_{n}}^{t^{n+1}}(\textbf{u}_{p}(s)\!-\textbf{u}_{p}^{n+1},\textbf{v}_{p})_{\Omega _{p}} \!-\!Da\int _{t_{n}}^{t^{n+1}}(\phi _{p}(s)-\phi _{p}^{n+1},\nabla \cdot \textbf{v}_{p})_{\Omega _{p}}\nonumber \\= & {} PrDaRa\mathbf {\xi }\int _{t_{n}}^{t^{n+1}}(\theta _{p}(s)-\theta _{p}^{n},\textbf{v}_{p})_{\Omega _{p}}. \end{aligned}$$

(A.20)

Thus, sum from $n=0$ to $l-1$ of the above equations, using the Cauchy-Schwarz inequality and (A.5), we gain

$$\begin{aligned}{} & {} Da\bigg (\sum _{n=0}^{l-1}\int _{t_{n}}^{t^{n+1}}(\phi _{p}(s)-\phi _{p}^{n+1})ds,\nabla \cdot \textbf{v}_{p}\bigg )_{\Omega _{p}}\nonumber \\= & {} C_{a}Da\sum _{n=0}^{l-1}(a_{p}^{n+1}-a_{p}^{n},\textbf{v}_{p})_{\Omega _{p}} +Pr\sum _{n=0}^{l-1}\int _{t_{n}}^{t^{n+1}}(\textbf{u}_{p}(s)-\textbf{u}_{p}^{n+1},\textbf{v}_{p})_{\Omega _{p}}ds\nonumber \\{} & {} +PrDaRa\mathbf {\xi }\sum _{n=0}^{l-1}\int _{t_{n}}^{t^{n+1}}(\theta _{p}^{n}-\theta _{p}(s),\textbf{v}_{p})_{\Omega _{p}}ds\nonumber \\\le & {} \bigg [\!C_{a}Da\Big (\sum _{n=0}^{l-1}\parallel a_{p}^{n+1}\!-\!a_{p}^{n}\parallel _{L^{2}(\Omega _{p})}^{2}\Big )^{\frac{1}{2}} \!+\!Pr\Big (\sum _{n=0}^{l-1}\Delta t\parallel \textbf{u}_{p}(s)\!-\!\textbf{u}_{p}(t_{n+1})\parallel _{L^{2}(\Omega _{p})}^{2}\!\Big )^{\frac{1}{2}}\nonumber \\{} & {} +Pr\Big (\sum _{n=0}^{l-1}\Delta t\parallel a_{p}^{n+1}\parallel _{L^{2}(\Omega _{p})}^{2}\Big )^{\frac{1}{2}} \!+\!PrDaRa\Big (\sum _{n=0}^{l-1}\Delta t\!\parallel \theta _{p}(s)\!-\!\theta _{p}(t_{n})\parallel _{L^{2}(\Omega _{p})}^{2}\Big )^{\frac{1}{2}}\nonumber \\{} & {} +PrDaRa\Big (\sum _{n=0}^{l-1}\Delta t\parallel c_{p}^{n}\parallel _{L^{2}(\Omega _{p})}^{2}\!\Big )^{\frac{1}{2}}\bigg ] \Big (\sum _{n=0}^{l-1}\Delta t\parallel \textbf{v}_{p}\parallel _{L^{2}(\Omega _{p})}^{2}\Big )^{\frac{1}{2}}. \end{aligned}$$

(A.21)

Taking the expectation of the above inequality and using the inf-sup condition, we can obtain

$$\begin{aligned} E\bigg [\int _{0}^{T}\phi _{p}(s)ds-\Delta t\sum _{n=0}^{l-1}\phi _{p}^{n+1}\bigg ]\le C\Delta t^{\frac{1}{2}}. \end{aligned}$$

(A.22)

At this point, the proof of 3.12 is complete.