非定常 Stokes/Darcy 模型一种新的time filter 算法的分析

图 1 全局域 $\Omega$ 由被交界面 $\Gamma$ 分隔开的自由流体区域 $\Omega_f$ 和多孔介质区域 $\Omega_p$ 构成

本文研究由 Stokes 方程和 Darcy 方程组成的 Stokes/Darcy 模型. 设 $T>0$ 是一个有限的时间. 自由流体区域 $\Omega_f$ 流动由 Stokes 方程控制: 求流体速度 ${\bf u}_f:\Omega_f\times[T]\rightarrow{\bf R}^d$ 和压力 $p_f:\Omega_f\times[T]\rightarrow{\bf R}$ 满足

$\begin{eqnarray*} \label{Stokes1}\frac{\partial {\bf u}_f}{\partial t}-\nabla\cdot({\bf T}_\nu({\bf u}_f,p_f))={\bf g}_f, \qquad &&\mbox{在}\;\Omega_f\times(0,T]\ \mbox{中}\;,\end{eqnarray*}$

(2.1)

$\begin{eqnarray*} \nabla\cdot {\bf u}_f=0, \qquad\qquad\qquad\qquad\qquad&&\mbox{在}\;\Omega_f\times(0,T] \ \mbox{中}\;, \end{eqnarray*}$

(2.2)

其中

${\bf T}_\nu({\bf u}_f,p_f)=-p_f{\bf I}+2\nu{\bf D}({\bf u}_f),\quad {\bf D}({\bf u}_f)=\frac12(\nabla {\bf u}_f+\nabla^Tu_f),$

分别是应力张量和应变张量, $\nu>0$ 是粘性系数, ${\bf g}_f({\bf x},t)$ 是外力项.

多孔介质区域 $\Omega_p$ 流动由以下方程控制

$\begin{matrix} \label{Darcy1}&&\frac{S\partial\phi_p}{\partial t}+\nabla\cdot {\bf u}_p={\bf g}_p, \qquad\qquad\qquad\ \mbox{在}\;\Omega_p\times(0,T]\ \mbox{中}\;,\end{matrix}$

(2.3)

$\begin{matrix} \label{Darcy2}&&{\bf u}_p=-{\bf K}\nabla\phi_p, \qquad\qquad\qquad\qquad\quad\mbox{在}\;\Omega_p\times(0,T]\ \mbox{中}\;, \end{matrix}$

(2.4)

其中, 第一个方程是饱和流动模型, $S$ 是比质量储能系数, $\phi_p=z+\frac{p_p}{\rho g}$ 是测压水头, $p_p$ 表示流体在 $\Omega_p$ 中的压力, $z$ 是从参考平面算起的高度, $\rho$ 和 $g$ 分别表示密度和重力加速度. 第二个方程是 Darcy 定律, ${\bf u}_p$ 是多孔介质区域 $\Omega_p$ 的流体速度, 与 $\phi_p$ 的梯度成正比关系. ${\bf K}$ 表示水利传导率张量, 它是一个最小特征值大于零的对称正定的矩阵, 且允许在空间中变化. ${\bf g}_p({\bf x},t)$ 是源项.

将方程(2.4)代入方程(2.3)可得 Darcy 方程: 求测压水头 $\phi_p:\Omega_p\times[T]\rightarrow{\bf R}^d$ 满足

$\begin{equation}\label{Darcy} \frac{S\partial\phi_p}{\partial t}-\nabla\cdot({\bf K}\nabla\phi_p)={\bf g}_p,\qquad\mbox{在}\;\Omega_p\times(0,T]\ \mbox{中}\;. \end{equation}$

(2.5)

下面考虑 Stokes/Darcy 模型满足以下初值条件

${\bf u}_f(x,0)={\bf u}_f^0(x),\qquad\mbox{在}\;\Omega_f\ \mbox{中}\;.$

(2.6)

$\phi_p(x,0)=\phi_p^0(x),\qquad\mbox{在}\;\Omega_p\ \mbox{中}\;.$

(2.7)

以及齐次 Dirichlet 边界条件

$\label{boundary} {\bf u}_f=0, \qquad\mbox{在}\;\Gamma_f\ \mbox{上}\;.$

(2.8)

$\phi_p=0,\qquad\mbox{在}\;\Gamma_p\ \mbox{上}\;.$

(2.9)

交界面条件是 Stokes/Darcy 模型的重要组成部分, 包括质量守恒条件, 法向力的平衡条件和 Beavers-Joseph-Saffman(BJS) 条件. 具体表示如下: 在 $\Gamma$ 上满足

$\begin{equation}\label{BJ} \left\{\begin{array}{l} {\bf u}_f\cdot {\bf n}_f-{\bf K}\nabla\phi_p\cdot {\bf n}_p=0,\\ -[{\bf T}_\nu({\bf u}_f,p_f)\cdot {\bf n}_f]\cdot {\bf n}_f=g\phi_p,\\ -[{\bf T}_\nu({\bf u}_f,p_f)\cdot {\bf n}_f]\cdot {\boldsymbol{\tau}}_i=\frac{\alpha\nu\sqrt{d}}{\sqrt{\mbox{trace}({\bf \Pi})}}{\bf u}_f\cdot {\boldsymbol{\tau}}_i. \end{array}\right. \end{equation}$

(2.10)

其中 ${\boldsymbol{\tau}}_i$ , $i=1,2,\cdots,d-1$ , 是沿着 $\Gamma$ 的正交切向单位向量, $\alpha$ 是一个确定的实验参数, ${\bf \Pi}$ 表示渗透率, 与水利传导率张量存在关系: ${\bf K}=\frac{{\bf \Pi}g}{\nu}$ .

首先定义 Hilbert 空间

$\begin{eqnarray*} &&{\bf H}_f=\{{\bf v}_f \in ({\bf H}^1(\Omega_f)^d): {\bf v}_f|_{\Gamma_f}=0\},\\ &&{\bf H}_p=\{\psi_p\in ({\bf H}^1(\Omega_p)^d): \psi_p|_{\Gamma_p}=0\},\\ &&Q_f=L^2(\Omega_f),\\ &&{\bf U}={\bf H}_f\times {\bf H}_p. \end{eqnarray*}$

区域 $D$ 中, $(\cdot,\cdot)_D$ 表示 $D=\Omega_f$ 或 $\Omega_p$ 中的内积. 并对空间 ${\bf U}$ 配备以下范数: $\forall$ $\underline{\bf u}=({\bf u}_f,\phi_p)\in {\bf U}$ ,有

$\begin{eqnarray*} &&\|\underline{\bf u}\|_0=\sqrt{({\bf u}_f,{\bf u}_f)_{\Omega_f}+gS(\phi_p,\phi_p)_{\Omega_p}}\,\\ &&\|\underline{\bf u}\|_{\bf U}=\sqrt{\nu(\nabla {\bf u}_f,\nabla {\bf u}_f)_{\Omega_f}+g{\bf K}(\nabla\phi_p,\nabla\phi_p)_{\Omega_p}}. \end{eqnarray*}$

用 $\Vert\cdot\Vert_{{\bf H}_f/{\bf H}_p}$ , $\Vert\cdot\Vert_{f/p}$ 和 $\Vert\cdot\Vert_{\Gamma}$ 分别表示范数 ${\bf H}^1(\Omega_{f/p})$ , $L^2(\Omega_{f/p})$ , $L^2(\Gamma)$ . 同时定义下列范数

$\begin{eqnarray*} &&\Vert {\bf u}_f\Vert_f=\Vert {\bf u}_f\Vert_{L^2(\Omega_f)},\ \ \Vert {\bf u}_f\Vert_{{\bf H}_f}=\Vert\nu^{\frac{1}{2}}\nabla u_f\Vert_{L^2(\Omega_f)},\\ &&\Vert\phi_p\Vert_p=\Vert\phi_p\Vert_{L^2(\Omega_p)}, \Vert\phi_p\Vert_{{\bf H}_p}=\Vert{\bf K}^{\frac{1}{2}}\nabla\phi_p\Vert_{L^2(\Omega_p)}. \end{eqnarray*}$

对函数 $v(x,t)$ 定义范数

$\|v\|_{L^2(0,T;L^2)}:=(\int_{0}^{T}\|v(\cdot,t)\|_{L^2}^2{\rm d}t)^{\frac{1}{2}},\quad \|v\|_{L^2(0,T;L^{\infty})}:=ess\sup_{[T]}\|v(\cdot,t)\|_{L^2}.$

其次, 非定常Stokes/Darcy 耦合模型(2.1)-(2.10)式的弱形式描述如下: 对于 ${\bf g}_f\in L^2(0,T; {\bf L}^2(\Omega_f))$ , ${\bf g}_p\in L^2(0,T; {\bf L}^2(\Omega_p))$ , 求 $\underline{\bf u}=(u_f, \phi_p) \in (L^2(0,T; {\bf H}_f)\cap L^\infty(0,T; L^2(\Omega_f)) \times L^2(0,T; {\bf H}_p)\cap L^\infty(0,T; L^2(\Omega_p)) )$ 和 $p_f\in L^2(0,T; Q_f)$ 使得 $\forall$ $(\underline{\bf v},q_f) \in{\bf U}\times Q_f$ 满足

$\begin{matrix}\label{coupled} &&(\frac{\partial\underline{\bf u}}{\partial t},\underline{\bf v})+a(\underline{\bf u},\underline{\bf v})+b(\underline{\bf v},p_f)=\langle {\bf F},\underline{\bf v} \rangle_{\bf U'},\nonumber\\ &&b(\underline{\bf u},q_f)=0,\\ &&\underline{\bf u}(\bf x,0)=\underline{\bf u}^0,\nonumber \end{matrix}$

(2.11)

其中

$\begin{eqnarray*} &&(\frac{\partial\underline{\bf u}}{\partial t},\underline{\bf v})=(\frac{\partial {\bf u}_f}{\partial t},{\bf v}_f)+g(\frac{S\partial\phi_p}{\partial t},\psi_p),\\ &&a(\underline{\bf u},\underline{\bf v})=a_{\Omega}(\underline{\bf u},\underline{\bf v})+a_\Gamma(\underline{\bf u},\underline{\bf v}),\\ &&a_{\Omega}(\underline{\bf u},\underline{\bf v})=a_{\Omega_f}({\bf u}_f,{\bf v}_f)+a_{\Omega_p}(\phi_p,\psi_p),\\ &&a_{\Omega_f}({\bf u}_f,{\bf v}_f)=\nu({\bf D}({\bf u}_f),{\bf D}({\bf v}_f))_{\Omega_f} +(\displaystyle{\frac{\alpha\nu\sqrt{d}}{\sqrt{\mbox{trace}({\bf \Pi})}}}P_{\tau}({\bf u}_f,{\bf v}_f))_\Gamma,\\ &&a_{\Omega_p}(\phi_p,\psi_p)=g({\bf K}\nabla\phi_p,\nabla\psi_p)_{\Omega_p},\\ &&a_\Gamma(\underline{\bf u},\underline{\bf v})=c_{\Gamma}({\bf v}_f,\phi_p)-c_{\Gamma}({\bf u}_f,\psi_p)= g(\phi_p,{\bf v}_f\cdot {\bf n}_f)_\Gamma-g(\psi_p,{\bf u}_f\cdot {\bf n}_f)_\Gamma,\\ &&b(\underline{\bf v},p_f)=-(p_f,\nabla\cdot {\bf v}_f)_{\Omega_f},\\ &&\langle {\bf F},\underline{\bf v}\rangle_{{\bf U}'}=({\bf g}_f,{\bf v}_f)_{\Omega_f}+g({\bf g}_p,\psi_p)_{\Omega_p}. \end{eqnarray*}$

${\bf U}'$ 是 ${\bf U}$ 的对偶空间, $P_{\tau}(\cdot)$ 是局部切平面上的投影, 表示为 $P_{\tau}({\bf v}_f)={\bf v}_f-({\bf v}_f\cdot{\bf n}_f){\bf n}_f$ .

双线性形式 $a(\cdot,\cdot)$ 满足连续性和强制性: $\forall$ $\underline{\bf u}, \underline{\bf v}\in {\bf U},$ ${\bf u}_f\in {\bf H}_f,$ $\phi_p \in {\bf H}_p,$ 有

$\begin{eqnarray*} &&a(\underline{\bf u},\underline{\bf v})\leq C_{con} \|\underline{\bf u}\|_{\bf U}\|\underline{\bf v}\|_{\bf U},\\ && a(\underline{\bf u},\underline{\bf u})\geq C_{coe} \|\underline{\bf u}\|_{\bf U}^2, \\ &&a_{\Omega_f}({\bf u}_f,{\bf u}_f)\geq \hat{C}_{coe} \|{\bf u}_f\|_{{\bf H}_f}^2,\\ &&a_{\Omega_p}(\phi_p,\phi_p)\geq \check{C}_{coe} \|\phi_p\|_{{\bf H}_p}^2. \end{eqnarray*}$

$a_\Gamma(\cdot,\cdot)$ 满足反对称性

$\begin{matrix}\label{c} a_\Gamma(\underline{\bf u},\underline{\bf v})=-a_\Gamma(\underline{\bf v},\underline{\bf u}),\quad\quad a_\Gamma(\underline{\bf u},\underline{\bf u})=0, \quad\forall\ \underline{\bf u},\underline{\bf v}\in {\bf U}. \end{matrix}$

(2.12)

下面给出Poincar $\acute{e}$ , Trace, Sobolev 不等式: 存在常数 $C_p$ , $C_t$ 和 $C_s$ 仅仅依赖于区域 $\Omega_f$ , $\tilde{C}_p$ , $\tilde{C}_t$ 和 $\tilde{C}_s$ 仅仅依赖于区域 $\Omega_p$ , 使得 $\forall$ ${\bf v}_f\in{\bf H}_f$ 和 $\psi_p\in{\bf H}_p$ ,有

$\begin{matrix}\label{pts} &&\|{\bf v}_f\|_{f}\leq C_p\|\nabla{\bf v}_f\|_{f},\qquad\qquad\qquad\quad \|\psi_p\|_{f}\leq \tilde{C}_p\|\nabla\psi_p\|_{p},\nonumber\\ &&\|{\bf v}_f\|_{\Gamma}\leq C_t\|\nabla{\bf v}_f\|_{f},\qquad\qquad\qquad\quad \|\psi_p\|_{\Gamma}\leq \tilde{C}_t\|\nabla\psi_p\|_{p},\\ &&\|{\bf v}_f\|_{H^\frac{1}{2}(\partial\Omega_f)}\leq C_s\|\nabla{\bf v}_f\|_{f},\qquad\qquad\|\psi_p\|_{H^\frac{1}{2}(\partial\Omega_p)}\leq \tilde{C}_s\|\nabla\psi_p\|_{p}.\nonumber \end{matrix}$

(2.13)

定常的耦合Stokes/Darcy 模型的适定性已被证明. 这里, 假定在非稳态的情况下也同样成立. 本文主要研究其数值解.

3 数值算法和稳定性

对任意给定的小参数 $h>0$ , 构造 $\Omega$ , $\Omega_f$ 和 $\Omega_p$ 的正则三角剖分 ${\cal T}_h$ , ${\cal T}_{fh}$ 和 ${\cal T}_{ph}$ . 为了简便, 设 $\Omega_f$ 和 $\Omega_p$ 是足够光滑的区域. 通过 MINI 元 (P1b-P1) 离散的 ${\bf H}_{fh}\subset{\bf H}_f$ , $Q_{fh}\subset Q_f$ 和通过线性拉格朗日元 (P1) 离散的 ${\bf H}_{ph}\subset {\bf H}_p$ 是有限元空间, 使得空间对 $({\bf H}_{fh},Q_{fh})$ 满足离散的 LBB 条件

$\begin{equation}\label{DLBB} \inf\limits_{q_{fh}\in Q_{fh}}\sup\limits_{{\bf v}_{fh}\in {\bf H}_{fh}}\frac{(q_{fh},\nabla\cdot {\bf v}_{fh})_{\Omega_f}}{\|q_{fh}\|_{Q_f}\,\|{\bf v}_{fh}\|_{{\bf H}_f}}\geq \beta, \end{equation}$

(3.1)

并定义 ${\bf u}_h={\bf H}_{fh}\times {\bf H}_{ph}$ .

定义线性投影算子(见文献[31]): $\forall$ $\underline{\bf v}_h\in{\bf u}_h$ , $q_{fh}\in Q_{fh}$ 和 $t\in(0,T]$ , $P_h:(\underline{\bf u}(t),p_f(t)) \in {\bf U}\times Q_f\to(P_h^{\underline{\bf u}}\underline{\bf u}(t), P_h^{p}p_f(t)) \in {\bf u}_h\times Q_{fh}$ 满足

$\begin{eqnarray*}\label{projection} &&a(P_h^{\underline{\bf u}}\underline{\bf u}(t),\underline{\bf v}_h)+b(\underline{\bf v}_h,P_h^{p}p_f(t))=a(\underline{\bf u}(t),\underline{\bf v}_h)+b(\underline{\bf v}_h,p_f(t)),\\\nonumber &&b(P_h^{\underline{\bf u}}\underline{\bf u}(t),q_{fh})=0. \end{eqnarray*}$

(3.2)

假设 $(\underline{\bf u}(t),p_f(t))$ 足够光滑, $(\underline{\bf u}(t),p_f(t))$ 的投影 $(P_h^{\underline{\bf u}}\underline{\bf u}(t)$ , $P_h^{p}p_f(t))$ 满足性质

$\begin{matrix}\label{app} &&\|P_h^{\underline{\bf u}}\underline{\bf u}(t)-\underline{\bf u}(t)\|_0\leq Ch^2\|\underline{\bf u}(t)\|_{H^2},\nonumber\\ &&\|P_h^{\underline{\bf u}}\underline{\bf u}(t)-\underline{\bf u}(t)\|_{\bf U}\leq Ch\|\underline{\bf u}(t)\|_{H^2},\\ &&\|P_h^{p}p_f(t)-p_f(t)\|_{L^2}\leq Ch\|p_f(t)\|_{H^1}.\nonumber \end{matrix}$

(3.3)

Inverse 不等式: 存在常数 $C_I$ , $\tilde{C}_I$ 依赖于有限元网格中的角度, 使得 $\forall$ ${\bf v}_{fh}\in{\bf H}_{fh}$ , $\psi_{ph}\in {\bf H}_{ph}$ ,

$\begin{matrix}\label{inverse} \|{\bf v}_{fh}\|_{{\bf H}_f}\leq C_I h^{-1}\|{\bf v}_{fh}\|_{f},\qquad \|\psi_{ph}\|_{{\bf H}_p}\leq \tilde{C}_I h^{-1}\|\psi_{ph}\|_{p}. \end{matrix}$

(3.4)

理论分析时将用到下面两个引理.

${\bf引理3.1}$ (文献[30,第4章引理 3.3]) 设 $\delta =\beta_2-\frac{\alpha_2}{2}>0$ . 系数 $\alpha_i$ 和 $\beta_i$ 满足下面关系

$\begin{eqnarray*} 2\bigg(\sum_{i=0}^{2}\alpha_i\zeta_i\bigg) \bigg(\sum_{i=0}^{2}\beta_i\zeta_i\bigg) &\geq &(\alpha_2^2+\delta )\zeta_2^2-(2\alpha_2-1)\zeta_1^2-\left((\alpha_2-1)^2+\delta\right)\zeta_0^2\\ &&-2\left(\alpha_2(\alpha_2-1)+\delta\right)(\zeta_2\zeta_1-\zeta_1\zeta_0),\quad \forall\ \zeta_0, \zeta_1, \zeta_2\in R. \end{eqnarray*}$

${\bf引理3.2}$ (文献[引理 1]) $\forall$ ${\bf v}_f\in {\bf H}_f$ , $\phi_p\in{\bf H}_p$ , 存在 $C_k>0$ , 使得 $\forall$ $\varepsilon>0$ ,有

$\begin{matrix}\label{Gammal} \vert c_\Gamma({\bf v}_f,\phi_p)\vert \leq\frac{1}{4\varepsilon}\Vert {\bf v}_f \Vert_{{\bf H}_f}^2 +C\varepsilon\Vert\phi_p\Vert_{{\bf H}_p}^2. \end{matrix}$

(3.5)

另外, $\forall$ ${\bf v}_{fh}\in {\bf H}_{fh}$ , $\phi_{ph}\in{\bf H}_{ph}$ , 存在 $\tilde{C}_k>0$ , 使得 $\forall$ $\tilde{\varepsilon}>0$ , 有

$\begin{matrix}\label{Gamma2} \vert c_\Gamma({\bf v}_{fh},\phi_{ph})\vert \leq\frac{1}{4\tilde{\varepsilon}}\Vert{\bf v}_{fh}\Vert_{{\bf H}_f}^2 +C\tilde{\varepsilon} h^{-1}\Vert\phi_{ph}\Vert_{p}^2. \end{matrix}$

(3.6)

这里引入相关的记号

$\begin{eqnarray*} &&A(\underline{\bf u}_h^{m+1},\theta)= \frac{3-2\theta}{2}\underline{\bf u}_h^{m+1}-2(1-\theta)\underline{\bf u}_h^{m}+\frac{1-2\theta}{2}\underline{\bf u}_h^{m-1},\\ &&B(\underline{\bf u}_h^{m+1},\theta)=\frac{(1-\theta)(3-2\theta)}{2}\underline{\bf u}_h^{m+1} +(4\theta-1-2\theta^2)\underline{\bf u}_h^{m} +\frac{(1-\theta)(1-2\theta)}{2}\underline{\bf u}_h^{m-1},\\ &&F({\bf F}^{m+1},\theta)=-\frac{(1-\theta)(1-2\theta)}{2}{\bf F}^{m+1} +(1-\theta)(1-2\theta){\bf F}^{m} -\frac{(1-\theta)(1-2\theta)}{2}{\bf F}^{m-1},\\ &&W(\eta_{{\phi}_p}^{m+1},\theta)=(1-\theta)(3-2\theta)\eta_{{\phi}_p}^{m} -\frac{(1-\theta)(3-2\theta)}{2}\eta_{{\phi}_p}^{m-1} -\frac{(1-\theta)(3-2\theta)}{2}\eta_{{\phi}_p}^{m+1}. \end{eqnarray*}$

本节其余部分将给出非定常 Stokes/Darcy 模型耦合和解耦的线性多步法加 time filter 算法以及其稳定性. 将时间段 $[T]$ 进行均匀剖分, 并令 $t_m=m\Delta t$ , $m=0,1,\cdots,N$ , 其中 $\Delta t=\frac{T}{N}$ 是时间步长. $(\underline{\bf u}_h^{m+1},p_{fh}^{m+1})=({\bf u}_{fh}^{m+1},p_{fh}^{m+1},\phi_{ph}^{m+1})$ 表示 $(\underline{\bf u}_h(t_{m+1}), p_{fh}(t_{m+1}))=({\bf u}_{fh}(t_{m+1}),p_{fh}(t_{m+1}),\phi_{ph}(t_{m+1}))$ 的全离散格式的数值解.

3.1 耦合算法及其稳定性

耦合的线性多步法加 time filter 算法描述如下.

$\blacktriangledown$ 线性多步法 (一阶收敛)

给定 $({\underline{\bf u}}_h^0,{p}_{fh}^0)$ 和 $({\underline{\bf u}}_h^1,{p}_{fh}^1)$ , 求 $\hat{\underline{\bf u}}_h^{m+1}=(\hat{\bf u}_{fh}^{m+1}$ , $\hat{\phi}_{ph}^{m+1})\in{\bf u}_h$ , $\hat{p}_{fh}^{m+1}\in Q_{fh}$ , $m=1,2,$ $\cdots,N-1$ , 使得 $\forall$ $\underline{\bf v}_h\in{\bf u}_h$ 和 $q_{fh}\in Q_{fh}$ ,

$\begin{equation} \begin{array}{ll} \label{first} \bigg(\frac{\hat{\underline{\bf u}}_h^{m+1}-\underline{\bf u}_h^{m}}{\Delta t},\underline{\bf v}_h\bigg) +a\left((1-\theta)\hat{\underline{\bf u}}_h^{m+1}+\theta\underline{\bf u}_h^{m},\underline{\bf v}_h\right) +b\left(\underline{\bf v}_h,(1-\theta)\hat{p}_{fh}^{m+1}+\theta p_{fh}^{m}\right)\\ =\langle (1-\theta){\bf F}^{m+1}+\theta{\bf F}F^{m},\underline{\bf v}_h\rangle,\\ b\left((1-\theta)\hat{\underline{\bf u}}_h^{m+1}+\theta\underline{\bf u}_h^{m},q_{fh}\right)=0. \end{array} \end{equation}$

(3.7)

$\blacktriangledown$ Time Filter 算法 (二阶收敛)

通过 time filter 算法更新线性多步法的解 $(\hat{\underline{\bf u}}_h^{m+1},\hat{p}_{fh}^{m+1})$ ,

$\begin{equation} \begin{array}{ll} \label{tf} \underline{\bf u}_h^{m+1}=\hat{\underline{\bf u}}_h^{m+1}-\frac{1-2\theta}{3-2\theta}(\hat{\underline{\bf u}}_h^{m+1}-2\underline{\bf u}_h^{m}+\underline{\bf u}_h^{m-1}),\\[3mm] p_{fh}^{m+1}=\hat{p}_{fh}^{m+1}-\frac{1-2\theta}{3-2\theta}(\hat{p}_{fh}^{m+1}-2p_{fh}^{m}+p_{fh}^{m-1}). \end{array} \end{equation}$

(3.8)

${\bf注3.1}$ 无论压力 $\hat{p}_{fh}^{m+1}$ 是否经 time filter 更新, 最终的结论几乎一致.

下面给出耦合的线性多步法加 time filter 算法的稳定性分析.

${\bf定理3.1}$ (耦合算法的稳定性) 设 $\underline{\bf u}_h^{m+1}$ 是线性多步法加 time filter 算法的解. 对于 $N\geq 2$ , 有

$\begin{eqnarray*} &&\|\underline{\bf u}_h^{N}\|_0^2+\frac{4(1-\theta)C_{coe}\Delta t}{3-2\theta}\sum_{m=1}^{N-1} \left\|B(\underline{\bf u}_h^{m+1},\theta)\right\|_{\bf U}^2\\ &\leq &C (\|{\bf g}_{f}\|^2_{L^2(0,T;L^2(\Omega_f))}+\|{\bf g}_{p}\|^2_{L^2(0,T;L^2(\Omega_p))} +\|\underline{\bf u}_h^{1}\|^2_0+\|\underline{\bf u}_h^{0}\|^2_0+\|\underline{\bf u}_h^{1}\|_0\|\underline{\bf u}_h^{0}\|_0). \end{eqnarray*}$

其中 $C$ 是一个常数, 与 $h$ , $\Delta t$ 以及其他参数无关, 并且在不同的情况下可能取不同的值.

${\bf证}$ 由 (3.8) 式可得

$\hat{\underline{\bf u}}_h^{m+1}=\frac{3-2\theta}{2}\underline{\bf u}_h^{m+1}-(1-2\theta)\underline{\bf u}_h^{m}+\frac{1-2\theta}{2}\underline{\bf u}_h^{m-1},$

$\hat{p}_{fh}^{m+1}=\frac{3-2\theta}{2}p_{fh}^{m+1}-(1-2\theta)p_{fh}^{m}+\frac{1-2\theta}{2}p_{fh}^{m-1},$

将上述两式代入 (3.7) 式, $\forall\ \underline{\bf v}_h\in {\bf u}_h$ , $q_{fh}\in Q_{fh}$ ,有

$\begin{matrix} \label{TF} &&\frac{1}{\Delta t}\bigg(A(\underline{\bf u}_h^{m+1},\theta),\underline{\bf v}_h\bigg) +a\left(B(\underline{\bf u}_h^{m+1},\theta),\underline{\bf v}_h\right) +b\left(\underline{\bf v}_h,B(p_{fh}^{m+1},\theta)\right)\nonumber\\ &=&\langle(1-\theta){\bf F}^{m+1}+\theta{\bf F}^{m},\underline{\bf v}_h\rangle,\\ &&b\left(B(\underline{\bf u}_h^{m+1},\theta), q_{fh}\right)=0.\nonumber \end{matrix}$

(3.9)

假设 $\underline{\bf v}_h=2\Delta t B(\underline{\bf u}_h^{m+1},\theta)$ , $q_{fh}=2\Delta t B(p_{fh}^{m+1},\theta)$ , 下面对 (3.8) 式的每一项进行分析. 首先, 由于 $\frac{(1-\theta)(3-2\theta)}{2}-\frac{1}{2}\frac{3-2\theta}{2}>0$ , 因此借助引理3.1, 有

$\begin{matrix}\label{L1} &&2\left(A(\underline{\bf u}_h^{m+1},\theta),B(\underline{\bf u}_h^{m+1},\theta)\right)\nonumber\\ &\geq& (2\theta^2-5\theta+3)\|\underline{\bf u}_h^{m+1}\|^2_0 -2(1-\theta)\|\underline{\bf u}_h^{m}\|^2_0 -(1-2\theta)(1-\theta)\|\underline{\bf u}_h^{m-1}\|^2_0\nonumber\\ &&-(1-2\theta)(3-2\theta)(\|\underline{\bf u}_h^{m+1}\|_0\|\underline{\bf u}_h^{m}\|_0 -\|\underline{\bf u}_h^{m}\|_0\|\underline{\bf u}_h^{m-1}\|_0). \end{matrix}$

(3.10)

其次, 利用双线性形式 $a(\cdot,\cdot)$ 的强制性可得

$\begin{matrix}\label{L2} 2\Delta t a\left(B(\underline{\bf u}_h^{m+1},\theta),B(\underline{\bf u}_h^{m+1},\theta)\right) \geq2C_{coe}\Delta t \left\|B(\underline{\bf u}_h^{m+1},\theta)\right\|_{\bf U}^2. \end{matrix}$

(3.11)

由无散度特性可知, 左边第三项等于 0. 最后, 根据 Young 不等式和 Cauchy Schwarz 不等式, 有

$\begin{aligned} & 2 \Delta t\left\langle(1-\theta) \mathbf{F}^{m+1}+\theta \mathbf{F}^{m}, B\left(\underline{\mathbf{u}}_{h}^{m+1}, \theta\right)\right\rangle \\ \leq & \frac{2 \theta^{2} \Delta t}{C_{\text {coe }}}\left\|\mathbf{F}^{m}\right\|_{\mathbf{U}^{\prime}}^{2}+\frac{2(1-\theta)^{2} \Delta t}{C_{\text {coe }}}\left\|\mathbf{F}^{m+1}\right\|_{\mathbf{U}^{\prime}}^{2}+C_{c o e} \Delta t\left\|B\left(\underline{\mathbf{u}}_{h}^{m+1}, \theta\right)\right\|_{\mathbf{U}^{2}}^{2}. \end{aligned}$

(3.12)

结合 (3.10)-(3.12) 式, 并将 (3.9) 式从 $m=1$ 直到 $m=N-1$ 求和, 整理得

$\begin{eqnarray*} &&(2\theta^2-5\theta+3)\|\underline{\bf u}_h^{N}\|^2_0 +(1-2\theta)(1-\theta)\|\underline{\bf u}_h^{N-1}\|^2_0\\ &&-(1-2\theta)(3-2\theta)\|\underline{\bf u}_h^{N}\|_0\|\underline{\bf u}_h^{N-1}\|_0 +C_{coe}\Delta t\sum_{m=1}^{N-1} \left\|B(\underline{\bf u}_h^{m+1},\theta)\right\|_{\bf U}^2\\ &\leq&\frac{2\theta^2\Delta t}{C_{coe}}\sum_{m=1}^{N-1}\|{\bf F}^{m}\|_{{\bf U}'}^2 +\frac{2(1-\theta)^2\Delta t}{C_{coe}}\sum_{m=1}^{N-1}\|{\bf F}^{m+1}\|_{{\bf U}'}^2\\ & &+(2\theta^2-5\theta+3)\|\underline{\bf u}_h^{1}\|^2_0 +(1-2\theta)(1-\theta)\|\underline{\bf u}_h^{0}\|^2_0 -(1-2\theta)(3-2\theta)\|\underline{\bf u}_h^{1}\|_0\|\underline{\bf u}_h^{0}\|_0. \end{eqnarray*}$

由于

$-(1-2\theta)(3-2\theta)\|\underline{\bf u}_h^{N}\|_0\|\underline{\bf u}_h^{N-1}\|_0 \geq -\frac{(1-2\theta)(3-2\theta)^2}{4(1-\theta)}\|\underline{\bf u}_h^{N}\|_0^2 -(1-2\theta)(1-\theta)\|\underline{\bf u}_h^{N-1}\|^2_0,$

因此

$\begin{eqnarray*} &&\frac{3-2\theta}{4(1-\theta)}\|\underline{\bf u}_h^{N}\|^2_0 +C_{coe}\Delta t\sum_{m=1}^{N-1} \left\|B(\underline{\bf u}_h^{m+1},\theta)\right\|_{\bf U}^2\\ &\leq&\frac{2\theta^2\Delta t}{C_{coe}}\sum_{m=1}^{N-1}\|{\bf F}^{m}\|_{{\bf U}'}^2 +\frac{2(1-\theta)^2\Delta t}{C_{coe}}\sum_{m=1}^{N-1}\|{\bf F}^{m+1}\|_{{\bf U}'}^2\\ & &+(2\theta^2-5\theta+3)\|\underline{\bf u}_h^{1}\|^2_0 +(1-2\theta)(1-\theta)\|\underline{\bf u}_h^{0}\|^2_0 -(1-2\theta)(3-2\theta)\|\underline{\bf u}_h^{1}\|_0\|\underline{\bf u}_h^{0}\|_0, \end{eqnarray*}$

从而得到耦合算法的稳定性结论.证毕.

3.2 解耦算法及其稳定性

解耦的线性多步法加 time filter 算法表述如下.

$\blacktriangledown$ 线性多步法 (一阶收敛)

给定 $({\bf u}_{fh}^0,p_{fh}^0)$ 和 $({\bf u}_{fh}^1,p_{fh}^1)$ , 求 $(\hat{\bf u}_{fh}^{m+1},\hat{p}_{fh}^{m+1})\in({\bf H}_{fh},Q_{fh})$ , $m=1,2,\cdots,N-1$ , 使得 $\forall\ {\bf v}_{fh} \in{\bf H} _{fh}$ , $q_{fh}\in Q_{fh}$ ,有

$\begin{eqnarray*} \label{DS} &&\bigg(\frac{\hat{\bf u}_{fh}^{m+1}-{\bf u}_{fh}^m}{\Delta t},{\bf v}_{fh}\bigg)_{\Omega_f} +a_{\Omega_f}\left((1-\theta)\hat{\bf u}_{fh}^{m+1}+\theta{\bf u}_{fh}^{m},{\bf v}_{fh}\right) +b\left({\bf v}_{fh},(1-\theta)\hat{p}_{fh}^{m+1}+\theta p_{fh}^m\right)\nonumber\\ &=&\left((1-\theta){\bf g}_{f}^{m+1}+\theta{\bf g}_{f}^{m},{\bf v}_{fh}\right)_{\Omega_f} -c_{\Gamma}\left({\bf v}_{fh},(2-\theta)\phi_{ph}^m-(1-\theta)\phi_{ph}^{m-1}\right),\\ &&b\left((1-\theta)\hat{\bf u}_{fh}^{m+1}+\theta{\bf u}_{fh}^{m},q_{fh}\right)=0.\nonumber \end{eqnarray*}$

(3.13)

给定 $\phi_{ph}^0$ 和 $\phi_{ph}^1$ , 求 $\hat{\phi}_{ph}^{m+1}\in{\bf H}_{ph}$ , $m=1,2,\cdots,N-1$ , 使得 $\forall$ $\psi_{ph} \in{\bf H}_{ph}$ ,有

$\begin{matrix} \label{DD} &&g\bigg(\frac{\hat{\phi}_{ph}^{m+1}-\phi_{ph}^m}{\Delta t},\psi_{ph}\bigg)_{{\Omega}_p} +a_{\Omega_p}\left((1-\theta)\hat{\phi}_{ph}^{m+1}+\theta\phi_{ph}^{m},\psi_{ph}\right)\\ &=&g\left((1-\theta){\bf g}_{p}^{m+1}+\theta {\bf g}_{p}^{m},\psi_{ph}\right)_{\Omega_p} +c_{\Gamma}\left((2-\theta){\bf u}_{fh}^m-(1-\theta){\bf u}_{fh}^{m-1},\psi_{ph}\right). \end{matrix}$

(3.14)

$\blacktriangledown$ Time Filter 算法 (二阶收敛)

通过 time filter 算法更新线性多步法的解 $(\hat{\bf u}_{fh}^{m+1},\hat{p}_{fh}^{m+1},\hat{\phi}_{ph}^{m+1})$ ,

$\begin{matrix} \label{time filter} &&{\bf u}_{fh}^{m+1}=\hat{\bf u}_{fh}^{m+1}-\frac{1-2\theta}{3-2\theta}(\hat{\bf u}_{fh}^{m+1}-2{\bf u}_{fh}^{m}+{\bf u}_{fh}^{m-1}),\\ &&p_{fh}^{m+1}=\hat{p}_{fh}^{m+1}-\frac{1-2\theta}{3-2\theta}(\hat{p}_{fh}^{m+1}-2p_{fh}^{m}+p_{fh}^{m-1}),\\ &&\phi_{ph}^{m+1}=\hat{\phi}_{ph}^{m+1}-\frac{1-2\theta}{3-2\theta}(\hat{\phi}_{ph}^{m+1}-2\phi_{ph}^m+\phi_{ph}^{m-1}). \end{matrix}$

(3.15)

${\bf注3.2}$ 交界面项利用二阶外推方法近似 $\hat{\bf u}_{fh}^{m+1}$ 和 $\hat{\phi}_{ph}^{m+1}$ , 即 $\hat{\bf u}_{fh}^{m+1}\approx 2{\bf u}_{fh}^{m}-{\bf u}_{fh}^{m-1}$ , $\hat{\phi}_{ph}^{m+1}\approx 2\phi_{ph}^{m}-\phi_{ph}^{m-1}$ . 类似地, 无论压力 $\hat{p}_{fh}^{m+1}$ 是否经 time filter 更新, 最终的结果几乎一致.

下面给出解耦的线性多步法加 time filter 算法的稳定性分析.

${\bf定理3.2}$ (解耦算法的稳定性) 设 ${\bf u}_{fh}^{m+1}$ , $\phi_{ph}^{m+1}$ 是线性多步法加 time filter 算法的解. 对于 $N\geq 2$ , 有

其中 $C(T)=\exp\Big(\sum\limits_{m=1}^{N-1}\max\left\{\frac{8(1-\theta)\Delta t}{(3-2\theta)\hat{C}_{coe}},\frac{8(1-\theta)\Delta t}{(3-2\theta)\check{C}_{coe}} \right\}\Big)$ .

${\bf证}$ 由 (3.15) 式可得

$\begin{matrix} &&\hat{\bf u}_{fh}^{m+1}=\frac{3-2\theta}{2}{\bf u}_{fh}^{m+1}-(1-2\theta){\bf u}_{fh}^{m}+\frac{1-2\theta}{2}{\bf u}_{fh}^{m-1},\nonumber\\ &&\hat{p}_{fh}^{m+1}=\frac{3-2\theta}{2}{p}_{fh}^{m+1}-(1-2\theta){p}_{fh}^{m}+\frac{1-2\theta}{2}{p}_{fh}^{m-1},\nonumber\\ &&\hat{\phi}_{ph}^{m+1}=\frac{3-2\theta}{2}\phi_{ph}^{m+1}-(1-2\theta)\phi_{ph}^{m}+\frac{1-2\theta}{2}\phi_{ph}^{m-1},\nonumber \end{matrix}$

将上述三式分别代入 (3.13) 和 (3.14) 式后相加, $\forall\ {\bf v}_{fh}\in{\bf H}_{fh}, \psi_{ph}\in{\bf H}_{ph}, q_{fh}\in Q_{fh}$ ,有

$\begin{eqnarray*} \label{TT} &&\frac{1}{\Delta t}\bigg(A({\bf u}_{fh}^{m+1},\theta),{\bf v}_{fh}\bigg)_{\Omega_f} +a_{\Omega_f}\left(B({\bf u}_{fh}^{m+1},\theta),{\bf v}_{fh}\right) +b\left({\bf v}_{fh},B(p_{fh}^{m+1},\theta)\right)\nonumber\\ &&+g\frac{1}{\Delta t}\bigg(A(\phi_{ph}^{m+1},\theta),\psi_{ph}\bigg)_{\Omega_p} +a_{\Omega_p}\left(B(\phi_{ph}^{m+1},\theta),\psi_{ph}\right)\nonumber\\ &=&\left((1-\theta){\bf g}_{f}^{m+1}+\theta{\bf g}_{f}^{m},{\bf v}_{fh}\right)_{\Omega_f} +g\left((1-\theta){\bf g}_{p}^{m+1}+\theta {\bf g}_{p}^{m},\psi_{ph}\right)_{\Omega_p}\\ &&-c_{\Gamma}\left({\bf v}_{fh},(2-\theta)\phi_{ph}^m-(1-\theta)\phi_{ph}^{m-1}\right) +c_{\Gamma}\left((2-\theta){\bf u}_{fh}^m-(1-\theta){\bf u}_{fh}^{m-1},\psi_{ph}\right),\nonumber\\ &&b\left(B({\bf u}_{fh}^{m+1},\theta), q_{fh}\right)=0.\nonumber \end{eqnarray*}$

(3.16)

假设 ${\bf v}_{fh}=2\Delta t B({\bf u}_{fh}^{m+1},\theta)$ , $\psi_{ph}=2\Delta t B(\phi_{ph}^{m+1},\theta)$ , $q_{fh}=2\Delta t B(p_{fh}^{m+1},\theta)$ . 类似定理 3.1 的证明, 有以下不等式成立

$\begin{aligned} & 2\left(A\left(\mathbf{u}_{f h}^{m+1}, \theta\right), B\left(\mathbf{u}_{f h}^{m+1}, \theta\right)\right)_{\Omega_f} \\ \geq & \left(2 \theta^2-5 \theta+3\right)\left\|\mathbf{u}_{f h}^{m+1}\right\|_f^2-2(1-\theta)\left\|\mathbf{u}_{f h}^m\right\|_f^2-(1-2 \theta)(1-\theta)\left\|\underline{\mathbf{u}}_h^{m-1}\right\|_f^2 \\ & -(1-2 \theta)(3-2 \theta)\left(\left\|\mathbf{u}_{f h}^{m+1}\right\|_f\left\|\mathbf{u}_{f h}^m\right\|_f-\left\|\mathbf{u}_{f h}^m\right\|_f\left\|\mathbf{u}_{f h}^{m-1}\right\|_f\right),\end{aligned}$

(3.17)

$\begin{aligned} & 2g\big(A(\phi_{ph}^{m+1},\theta),B(\phi_{ph}^{m+1},\theta)\big)_{\Omega_{p}}\nonumber\\ \geq & g\big[(2\theta^2-5\theta+3)\|\phi_{ph}^{m+1}\|^2_p -2(1-\theta)\|\phi_{ph}^{m}\|^2_p -(1-2\theta)(1-\theta)\|\phi_{ph}^{m-1}\|^2_p\nonumber\\ &-(1-2\theta)(3-2\theta)\left(\|\phi_{ph}^{m+1}\|_p\|\phi_{ph}^{m}\|_p -\|\phi_{ph}^{m}\|_p\|\phi_{ph}^{m-1}\|_p\right)\big],\end{aligned}$

(3.18)

$\begin{aligned} & 2\Delta t a_{\Omega_f} \left(B({\bf u}_{fh}^{m+1},\theta),B({\bf u}_{fh}^{m+1},\theta)\right) \geq 2\hat{C}_{coe}\Delta t \left\|B({\bf u}_{fh}^{m+1},\theta)\right\|_{H_f}^2,\end{aligned}$

(3.19)

$\begin{aligned} & 2\Delta t a_{\Omega_p} \left(B(\phi_{ph}^{m+1},\theta),B(\phi_{ph}^{m+1},\theta)\right) \geq2g\check{C}_{coe}\Delta t \left\|B(\phi_{ph}^{m+1},\theta)\right\|_{H_p}^2,\end{aligned}$

(3.20)

$\begin{aligned} & 2 \Delta t\left((1-\theta) \mathbf{g}_f^{m+1}+\theta \mathbf{g}_f^m, B\left(\mathbf{u}_{f h}^{m+1}, \theta\right)\right)_{\Omega_f} \\ \leq & \frac{2 \theta^2 \Delta t}{\hat{C}_{\text {coe }}}\left\|\mathbf{g}_f^m\right\|_{\mathbf{H}_f^{\prime}}^2+\frac{2(1-\theta)^2 \Delta t}{\hat{C}_{\text {coe }}}\left\|\mathbf{g}_f^{m+1}\right\|_{\mathbf{H}_f^{\prime}}^2+\frac{\hat{C}_{c o e} \Delta t}{2}\left\|B\left(\mathbf{u}_{f h}^{m+1}, \theta\right)\right\|_{\mathbf{H}_f}^2,\end{aligned}$

(3.21)

$\begin{aligned} & 2g\Delta t\left((1-\theta){\bf g}_{p}^{m+1}+\theta {\bf g}_{p}^{m}, B(\phi_{ph}^{m+1},\theta)\right)_{\Omega_{p}}\\ \leq & \frac{2 g \theta^2 \Delta t}{\check{C}_{\text {coe }}}\left\|{\bf g}_p^m\right\|_{\mathbf{H}_p^{\prime}}^2+\frac{2 g(1-\theta)^2 \Delta t}{\check{C}_{\text {coe }}}\left\|{\bf g}_p^{m+1}\right\|_{\mathbf{H}_p^{\prime}}^2+\frac{g \check{C}_{c o e} \Delta t}{2}\left\|B\left(\phi_{p h}^{m+1}, \theta\right)\right\|_{\mathbf{H}_p}^2.\end{aligned}$

(3.22)

这里主要分析交界面项. 借助引理 3.2, 并代入 $\varepsilon_1=\frac{1}{\hat{C}_{coe}}$ , $\varepsilon_2=\frac{g}{\check{C}_{coe}}$ , 有

$\begin{eqnarray*}\label{R12} &&-2\Delta tc_{\Gamma} \left(B({\bf u}_{fh}^{m+1},\theta), (2-\theta)\phi_{ph}^m-(1-\theta)\phi_{ph}^{m-1}\right)\nonumber\\ &&+2\Delta tc_{\Gamma} \left((2-\theta){\bf u}_{fh}^m-(1-\theta){\bf u}_{fh}^{m-1}, B(\phi_{ph}^{m+1},\theta)\right)\nonumber\\ &\leq &\frac{\hat{C}_{coe}\Delta t}{2} \left\|B({\bf u}_{fh}^{m+1},\theta)\right\|_{{\bf H}_f}^2 +\frac{2C_1h^{-1}\Delta t}{\hat{C}_{coe}} \|(2-\theta)\phi_{ph}^m-(1-\theta)\phi_{ph}^{m-1}\|_p^2\\ &&+\frac{g\check{C}_{coe}\Delta t}{2} \left\| B(\phi_{ph}^{m+1},\theta)\right\|_{{\bf H}_p}^2 +\frac{2gC_2h^{-1}\Delta t}{\check{C}_{coe}} \|(2-\theta){\bf u}_{fh}^m-(1-\theta){\bf u}_{fh}^{m-1}\|_f^2\nonumber\\ &\leq &\frac{\hat{C}_{coe}\Delta t}{2} \left\|B({\bf u}_{fh}^{m+1},\theta)\right\|_{{\bf H}_f}^2 +\frac{g\check{C}_{coe}\Delta t}{2} \left\| B(\phi_{ph}^{m+1},\theta)\right\|_{{\bf H}_p}^2\nonumber\\ &&+\frac{2g\Delta t}{\hat{C}_{coe}} \|(2-\theta)\phi_{ph}^m-(1-\theta)\phi_{ph}^{m-1}\|_p^2 +\frac{2\Delta t}{\check{C}_{coe}} \|(2-\theta){\bf u}_{fh}^m-(1-\theta){\bf u}_{fh}^{m-1}\|_f^2, \end{eqnarray*}$

(3.23)

其中, 最后一个不等式是在选择合适的常量 $C_1$ 和 $C_2$ 下得到的.

结合 (3.17)-(3.23) 式, 并将 (3.16) 式从 $m=1$ 直到 $m=N-1$ 求和, 整理可得

$\begin{eqnarray*} &&(2\theta^2-5\theta+3)\|{\bf u}_{fh}^{N}\|^2_f +(1-2\theta)(1-\theta)\|{\bf u}_{fh}^{N-1}\|^2_f\\ &&+g(2\theta^2-5\theta+3)\|\phi_{ph}^{N}\|^2_p +g(1-2\theta)(1-\theta)\|\phi_{ph}^{N-1}\|^2_p\\ &&-(1-2\theta)(3-2\theta)\|{\bf u}_{fh}^{N}\|_f\|{\bf u}_{fh} ^{N-1}\|_f -g(1-2\theta)(3-2\theta){2}\|\phi_{ph}^{N}\|_p\|\phi_{ph}^{N-1}\|_p\\ &&+\hat{C}_{coe}\Delta t\sum_{m=1}^{N-1} \left\|B({\bf u}_{fh}^{m+1},\theta)\right\|_{{\bf H}_f}^2 +g\check{C}_{coe}\Delta t\sum_{m=1}^{N-1} \left\|B(\phi_{ph}^{m+1},\theta)\right\|_{{\bf H}_p}^2\\ &\leq &\frac{2\theta^2\Delta t}{\hat{C}_{coe}}\sum_{m=1}^{N-1}\|{\bf g}_f^m\|_{{\bf H}_f'}^2 +\frac{2(1-\theta)^2\Delta t}{\hat{C}_{coe}}\sum_{m=1}^{N-1}\|{\bf g}_f^{m+1}\|_{{\bf H}_f'}^2\\ &&+\frac{2g\theta^2\Delta t}{\check{C}_{coe}}\sum_{m=1}^{N-1}\|{\bf g}_p^m\|_{{\bf H}_p'}^2 +\frac{2g(1-\theta)^2\Delta t}{\check{C}_{coe}}\sum_{m=1}^{N-1}\|{\bf g}_p^{m+1}\|_{{\bf H}_p'}^2\\ &&+\frac{2g\Delta t}{\hat{C}_{coe}} \|(2-\theta)\phi_{ph}^m-(1-\theta)\phi_{ph}^{m-1}\|_p^2 +\frac{2\Delta t}{\check{C}_{coe}} \|(2-\theta){\bf u}_{fh}^m-(1-\theta){\bf u}_{fh}^{m-1}\|_f^2\\ &&+(2\theta^2-5\theta+3)\|{\bf u}_{fh}^1\|_f^2 +(1-2\theta)(1-\theta)\|{\bf u}_{fh}^0\|_f^2\\ &&+g(2\theta^2-5\theta+3)\|\phi_{ph}^1\|_p^2 +g(1-2\theta)(1-\theta)\|\phi_{ph}^0\|_p^2\\ &&-(1-2\theta)(3-2\theta)\|{\bf u}_{fh}^{1}\|_f\|{\bf u}_{fh}^{0}\|_f -g(1-2\theta)(3-2\theta)\|\phi_{ph}^{1}\|_p\|\phi_{ph}^{0}\|_p. \end{eqnarray*}$

(3.24)

由于

$\begin{eqnarray*} &&-(1-2\theta)(3-2\theta)\|{\bf u}_{fh}^{N}\|_f\|{\bf u}_{fh}^{N-1}\|_f\\ &\geq &-\frac{(1-2\theta)(3-2\theta)^2}{4(1-\theta)}\|{\bf u}_{fh}^{N}\|_f^2 -(1-2\theta)(1-\theta)\|{\bf u}_{fh}^{N-1}\|^2_f, \end{eqnarray*}$

以及

$\begin{eqnarray*} &&-g(1-2\theta)(3-2\theta)\|\phi_{ph}^{N}\|_p\|\phi_{ph}^{N-1}\|_p\\ &\geq &-g\frac{(1-2\theta)(3-2\theta)^2}{4(1-\theta)}\|\phi_{ph}^{N}\|_p^2 -g(1-2\theta)(1-\theta)\|\phi_{ph}^{N-1}\|^2_p, \end{eqnarray*}$

因此

$\begin{eqnarray*} &&\frac{3-2\theta}{4(1-\theta)}\|{\bf u}_{fh}^{N}\|^2_f +\hat{C}_{coe}\Delta t\sum_{m=1}^{N-1} \left\|B({\bf u}_{fh}^{m+1},\theta)\right\|_{{\bf H}_f}^2\\ &&+\frac{g(3-2\theta)}{4(1-\theta)}\|\phi_{ph}^{N}\|^2_p +g\check{C}_{coe}\Delta t\sum_{m=1}^{N-1} \left\|B(\phi_{ph}^{m+1},\theta)\right\|_{{\bf H}_p}^2\\ &\leq &\frac{2\theta^2\Delta t}{\hat{C}_{coe}}\sum_{m=1}^{N-1}\|{\bf g}_f^m\|_{{\bf H}_f'}^2 +\frac{2(1-\theta)^2\Delta t}{\hat{C}_{coe}}\sum_{m=1}^{N-1}\|{\bf g}_f^{m+1}\|_{{\bf H}_f'}^2\\ &&+\frac{2g\theta^2\Delta t}{\check{C}_{coe}}\sum_{m=1}^{N-1}\|{\bf g}_p^m\|_{{\bf H}_p'}^2 +\frac{2g(1-\theta)^2\Delta t}{\check{C}_{coe}}\sum_{m=1}^{N-1}\|{\bf g}_p^{m+1}\|_{{\bf H}_p'}^2\\ &&+\frac{8(1-\theta)\Delta t}{(3-2\theta)\hat{C}_{coe}}\sum_{m=1}^{N-1} \frac{g(3-2\theta)}{4(1-\theta)}\|\phi_{ph}^m\|_p^2 +\frac{8(1-\theta)\Delta t}{(3-2\theta)\check{C}_{coe}}\sum_{m=1}^{N-1} \frac{(3-2\theta)}{4(1-\theta)}\Vert {\bf u}_{fh}^m \Vert_f^2\\ &&+(2\theta^2-5\theta+3)\|{\bf u}_{fh}^1\|_f^2 +(1-2\theta)(1-\theta)\|{\bf u}_{fh}^0\|_f^2\nonumber\\ &&+g(2\theta^2-5\theta+3)\|\phi_{ph}^1\|_p^2 +g(1-2\theta)(1-\theta)\|\phi_{ph}^0\|_p^2\nonumber\\ &&-(1-2\theta)(3-2\theta)\|{\bf u}_{fh}^{1}\|_f\|{\bf u}_{fh}^{0}\|_f -g(1-2\theta)(3-2\theta)\|\phi_{ph}^{1}\|_p\|\phi_{ph}^{0}\|_p. \end{eqnarray*}$

结合离散的 Gronwall 引理可得解耦算法的稳定性结论. 证毕.

4 误差估计

本节将证明耦合和解耦算法的误差估计. 记算法的精确解为 $(\underline{\bf u}^{m+1},p_f^{m+1})=$ $(\underline{\bf u}(t_{m+1}),$ $p_f(t_{m+1}))$ , 其中 $\underline{\bf u}^{m+1}=({\bf u}_f^{m+1},\phi_f^{m+1})$ , $\underline{\bf u}(t_{m+1})=({\bf u}_f(t_{m+1}),\phi_f(t_{m+1}))$ , 同时定义误差函数

$\begin{eqnarray*} &&e_{\underline{\bf u}}^{m}=\underline{\bf u}_h^{m}-\underline{\bf u}^{m}=\underline{\bf u}_h^{m}-P_h^{\underline{\bf u}}\underline{\bf u}^{m}+P_h^{\underline{\bf u}}\underline{\bf u}^{m}-\underline{\bf u}^{m}=\eta_{\underline{\bf u}}^m+\xi_{\underline{\bf u}}^m,\\ &&e_{p}^{m}=p_{fh}^{m}-p_{f}^{m}=p_{fh}^{m}-P_h^{p}p_{f}^{m}+P_h^{p}p_{f}^{m}-p_{f}^{m}=\eta_{p}^m+\xi_{p}^m,\\ &&e_{{\bf u}_f}^{m}={\bf u}_{fh}^{m}-{\bf u}_f^{m}={\bf u}_{fh}^{m}-P_h^{{\bf u}_f}{\bf u}_f^{m}+P_h^{{\bf u}_f}{\bf u}_f^{m}-{\bf u}_f^{m}=\eta_{{\bf u}_f}^m+\xi_{{\bf u}_f}^m,\\ &&e_{\phi_p}^{m}=\phi_{ph}^{m}-\phi_{p}^{m}=\phi_{ph}^{m}-P_h^{\phi_p}\phi_{p}^{m}+P_h^{\phi_p}\phi_{p}^{m}-\phi_{p}^{m}=\eta_{\phi_{p}}^m+\xi_{\phi_{p}}^m. \end{eqnarray*}$

显然地, 下述关系式成立

$\begin{matrix} \label{theta} &&\|\xi_{\underline{\bf u}}^m\|_0\leq Ch^2\|\underline{\bf u}(t)\|_{H^2},\quad \|\xi_{\underline{\bf u}}^m\|_{\bf U}\leq Ch\|\underline{\bf u}(t)\|_{H^2},\quad \|\xi_{p}^m\|_{L^2}\leq Ch\|p_f(t)\|_{H^1},\nonumber\\ &&\|\xi_{{\bf u}_f}^m\|_{f}\leq Ch^2\|{\bf u}_f(t)\|_{H^2},\quad \|\xi_{{\bf u}_f}^m\|_{{\bf H}_f}\leq Ch\|{\bf u}_f(t)\|_{H^2},\\ &&\|\xi_{\phi_p}^m\|_{p}\leq Ch^2\|\phi_p(t)\|_{H^2},\quad \|\xi_{\phi_p}^m\|_{{\bf H}_p}\leq Ch\|\phi_p(t)\|_{H^2}.\nonumber \end{matrix}$

(4.1)

需注意 $\eta_{\underline{\bf u}}^0=0$ , $\eta_{p}^0=0$ , $\eta_{{\bf u}_f}^0=0$ , $\eta_{\phi_p}^0=0$ .

假设所求解满足下述正则性条件

$\begin{matrix} \label{regularity} \begin{array}{ll} {\bf u}_f\in L^{\infty}(0,T;{\bf H}^2), {\bf u}_{f,t}\in {\bf L}^{2}(0,T;{\bf H}^1)\cap L^{\infty}(0,T;{\bf L}^2),\\ {\bf u}_{f,tt}\in {\bf L}^{2}(0,T;{\bf L}^2), {\bf u}_{f,ttt}\in L^{2}(0,T;{\bf H}'),\\ \phi_p\in L^{\infty}(0,T;{\bf H}^2), \phi_{p,t}\in {\bf L}^{2}(0,T;{\bf H}^1)\cap L^{\infty}(0,T;{\bf L}^2),\\ \phi_{p,tt}\in L^{2}(0,T;{\bf L}^2), \phi_{p,ttt}\in {\bf L}^{2}(0,T;{\bf H}'). \end{array} \end{matrix}$

(4.2)

外力项 ${\bf g}_f$ 和 ${\bf g}_p$ 满足

$\begin{matrix} \label{regularity1} \begin{array}{ll} {\bf g}_{f,t}\in {\bf L}^{2}(0,T;{\bf L}^2),\ {\bf g}_{f,tt}\in {\bf L}^{2}(0,T;{\bf H}_f'),\\ {\bf g}_{p,t}\in {\bf L}^{2}(0,T;{\bf L}^2),\ {\bf g}_{p,tt}\in L^{2}(0,T;{\bf H}_p'). \end{array} \end{matrix}$

(4.3)

4.1 耦合算法的误差估计

${\bf定理4.1}$ (耦合算法的二阶收敛性) 在 (4.2)和(4.3) 式的假设下, 对于 $N\geq 2$ 有以下估计

$\begin{eqnarray*} \frac{3-2\theta}{4(1-\theta)}\|e_{\underline{\bf u}}^{N}\|_0^2+C_{coe}\Delta t\sum_{m=1}^{N-1}\left\|B(e_{\underline{\bf u}}^{m+1},\theta)\right\|_{\bf U}^2\leq C (\Delta t^4+h^4). \end{eqnarray*}$

${\bf证}$ 在 $t_{m+1}$ , $t_{m}$ 和 $t_{m-1}$ 时刻分别对 (2.11) 式乘 $\frac{(1-\theta)(3-2\theta)}{2}$ , $(4\theta-1-2\theta^2)$ , $\frac{(1-\theta)(1-2\theta)}{2}$ 得到三个对应时刻的等式. 从 (3.9) 式中减去三者之和, 结合误差函数的定义和投影算子的性质 (3.2) 式可得, $\forall\ \underline{\bf v}_h\in{\bf u}_h$ , $\ q_{fh}\in Q_{fh}$ ,有

$\begin{aligned} & \frac{1}{\Delta t}\left(A\left(\eta_{\underline{\mathbf{u}}}^{m+1}, \theta\right), \underline{\mathbf{v}}_{h}\right)+a\left(B\left(\eta_{\underline{\mathbf{u}}}^{m+1}, \theta\right), \underline{\mathbf{v}}_{h}\right)+b\left(\underline{\mathbf{v}}_{h}, B\left(\eta_{p}^{m+1}, \theta\right)\right) \\ = & -\left(\frac{1}{\Delta t} A\left(\underline{\mathbf{u}}^{m+1}, \theta\right)-B\left(\underline{\mathbf{u}}_{t}^{m+1}, \theta\right), \underline{\mathbf{v}}_{h}\right)-\left(\frac{1}{\Delta t} A\left(\xi_{\underline{\mathbf{u}}}^{m+1}, \theta\right), \underline{\mathbf{v}}_{h}\right)+\left\langle F\left(\mathbf{F}^{m+1}, \theta\right), \underline{\mathbf{v}}_{h}\right\rangle, \\ & b\left(B\left(\eta_{\underline{\mathbf{u}}}^{m+1}, \theta\right), q_{f h}\right)=0. \end{aligned}$

(4.4)

假设 $\underline{\bf v}_h=2\Delta t B(\eta_{\underline{\bf u}}^{m+1},\theta)$ , $q_{fh}=2\Delta t B(\eta_{p}^{m+1},\theta)$ . 类似定理 3.1 的证明, 对 (4.4) 式的每一项进行分析.

首先, 借助引理 3.1, 有

$\begin{eqnarray*} \label{errorl1} &&2\left(A(\eta_{\underline{\bf u}}^{m+1},\theta),B(\eta_{\underline{\bf u}}^{m+1},\theta)\right)\\ &\geq &(2\theta^2-5\theta+3)\|\eta_{\underline{\bf u}}^{m+1}\|^2_0 -2(1-\theta)\|\eta_{\underline{\bf u}}^{m}\|^2_0 -(1-2\theta)(1-\theta)\|\eta_{\underline{\bf u}}^{m-1}\|^2_0\nonumber\\ &&-(1-2\theta)(3-2\theta)(\|\eta_{\underline{\bf u}}^{m+1}\|_0\|\eta_{\underline{\bf u}}^{m}\|_0 -\|\eta_{\underline{\bf u}}^{m}\|_0\|\eta_{\underline{\bf u}}^{m-1}\|_0). \end{eqnarray*}$

(4.5)

利用双线性形式 $a(\cdot,\cdot)$ 的强制性可得

$2\Delta t a\left(B(\eta_{\underline{\bf u}}^{m+1},\theta),B(\eta_{\underline{\bf u}}^{m+1},\theta)\right) \geq 2C_{coe}\Delta t \left\|B(\eta_{\underline{\bf u}}^{m+1},\theta)\right\|_{\bf U}^2.\nonumber$

由无散度特性可知, 左边第三项的结果为 0.

其次, 利用积分型余项的泰勒展开式,有

$\begin{eqnarray*} \label{taylor} &&\underline{\bf u}^{m}=\underline{\bf u}^{m+1}-\Delta t\underline{\bf u}^{m+1}_t+\frac{\Delta t^2}{2}\underline{\bf u}^{m+1}_{tt}+\frac{1}{2}\int_{t^{m+1}}^{t^{m}}(t^{m}-t)^2\underline{\bf u}_{ttt}{\rm d}t,\\ &&\underline{\bf u}^{m-1}=\underline{\bf u}^{m+1}-2\Delta t\underline{\bf u}^{m+1}_t+2\Delta t^2\underline{\bf u}^{m+1}_{tt}+\frac{1}{2}\int_{t^{m+1}}^{t^{m-1}}(t^{m-1}-t)^2\underline{\bf u}_{ttt}{\rm d}t,\\ &&\underline{\bf u}^{m}_t=\underline{\bf u}^{m+1}_t-\Delta t\underline{\bf u}^{m+1}_{tt}+\int_{t^{m+1}}^{t^{m}}(t^{m}-t)\underline{\bf u}_{ttt}{\rm d}t,\\ &&\underline{\bf u}^{m-1}_t=\underline{\bf u}^{m+1}_t-2\Delta t\underline{\bf u}^{m+1}_{tt}+\int_{t^{m+1}}^{t^{m-1}}(t^{m-1}-t)\underline{\bf u}_{ttt}{\rm d}t, \end{eqnarray*}$

代入并整理得

$\begin{eqnarray*} &&\frac{1}{\Delta t}A(\underline{\bf u}^{m+1},\theta)-B(\underline{\bf u}_{t}^{m+1},\theta)\\ &=&\frac{1}{\Delta t} \bigg[(1-\theta)\int_{t^{m}}^{t^{m+1}}(t^{m}-t)^2\underline{\bf u}_{ttt}{\rm d}t -\frac{1-2\theta}{4}\int_{t^{m-1}}^{t^{m+1}}(t^{m-1}-t)^2\underline{\bf u}_{ttt}{\rm d}t\bigg]\\ &&-(4\theta-1-2\theta^2)\int_{t^{m}}^{t^{m+1}}(t^{m}-t)\underline{\bf u}_{ttt}{\rm d}t -\frac{(1-\theta)(1-2\theta)}{2}\int_{t^{m-1}}^{t^{m+1}}(t^{m-1}-t)\underline{\bf u}_{ttt}{\rm d}t. \end{eqnarray*}$

再结合 Cauchy-Schwarz 不等式,有

$\begin{eqnarray*} &&\bigg(\int_{t^{m}}^{t^{m+1}}(t^{m}-t)^2\underline{\bf u}_{ttt}{\rm d}t\bigg)^2\leq \frac{\Delta t^5}{5}\int_{t^{m}}^{t^{m+1}}\underline{\bf u}_{ttt}^2{\rm d}t,\\ &&\bigg(\int_{t^{m-1}}^{t^{m+1}}(t^{m-1}-t)^2\underline{\bf u}_{ttt}{\rm d}t\bigg)^2 \leq \frac{32\Delta t^5}{5}\int_{t^{m-1}}^{t^{m+1}}\underline{\bf u}_{ttt}^2{\rm d}t,\\ &&\bigg(\int_{t^{m}}^{t^{m+1}}(t^{m}-t)\underline{\bf u}_{ttt}{\rm d}t\bigg)^2\leq \frac{\Delta t^3}{3}\int_{t^{m}}^{t^{m+1}}\underline{\bf u}_{ttt}^2{\rm d}t,\\ &&\bigg(\int_{t^{m-1}}^{t^{m+1}}(t^{m-1}-t)\underline{\bf u}_{ttt}{\rm d}t\bigg)^2 \leq \frac{8\Delta t^3}{3}\int_{t^{m-1}}^{t^{m+1}}\underline{\bf u}_{ttt}^2{\rm d}t, \end{eqnarray*}$

可以得到

$\begin{eqnarray*} &&2\Delta t\left( \frac{1}{\Delta t}A(\underline{\bf u}^{m+1},\theta)-B(\underline{\bf u}_{t}^{m+1},\theta), B(\eta_{\underline{\bf u}}^{m+1},\theta)\right)\nonumber\\ &\leq& \frac{3\Delta t}{C_{coe}} \left\|\frac{1}{\Delta t}A(\underline{\bf u}^{m+1},\theta)-B(\underline{\bf u}_{t}^{m+1},\theta)\right\|_{\bf U}^2 +\frac{C_{coe}\Delta t}{3} \left\|B(\eta_{\underline{\bf u}}^{m+1},\theta)\right\|_{\bf U}^2\\ &\leq & \frac{ (60\theta^4-200\theta^3+257\theta^2-130\theta+24)\Delta t^4}{5C_{coe}} \int_{t^{m-1}}^{t^{m+1}}\|\underline{\bf u}_{ttt}\|^2_{\bf U'}{\rm d}t +\frac{C_{coe}\Delta t}{3} \left\|B(\eta_{\underline{\bf u}}^{m+1},\theta)\right\|_{\bf U}^2.\nonumber \end{eqnarray*}$

(4.6)

类似地, 借助积分型余项的泰勒展开式,有

$\begin{eqnarray*} \underline{\bf u}^{m}=\underline{\bf u}^{m+1}+\int_{t^{m+1}}^{t^{m}}\underline{\bf u}_{t}{\rm d}t, \underline{\bf u}^{m-1}=\underline{\bf u}^{m+1}+\int_{t^{m+1}}^{t^{m-1}}\underline{\bf u}_{t}{\rm d}t. \end{eqnarray*}$

代入并整理得

$\begin{eqnarray*} \frac{1}{\Delta t}A(\xi_{\underline{\bf u}}^{m+1},\theta) &=&\frac{1}{\Delta t}\left[(P_h^{\underline{\bf u}}-I) A(\underline{\bf u}^{m+1},\theta)\right]\\ &=&\frac{1}{\Delta t}\bigg[2(1-\theta)\int_{t^{m}}^{t^{m+1}}(P_h^{\underline{\bf u}}-I)\underline{\bf u}_{t}{\rm d}t -\frac{1-2\theta}{2}\int_{t^{m-1}}^{t^{m+1}}(P_h^{\underline{\bf u}}-I)\underline{\bf u}_{t}{\rm d}t\bigg]. \end{eqnarray*}$

从而

$\begin{eqnarray*}\label{E2} &&2\Delta t\left(\frac{1}{\Delta t}A(\xi_{\underline{\bf u}}^{m+1},\theta),B(\eta_{\underline{\bf u}}^{m+1},\theta)\right)\nonumber\\ &\leq& \frac{3\Delta t}{C_{coe}} \|\frac{1}{\Delta t}A(\xi_{\underline{\bf u}}^{m+1},\theta)\|^2_{\bf U'} +\frac{C_{coe}\Delta t}{3}\left\|B(\eta_{\underline{\bf u}}^{m+1},\theta)\right\|_{\bf U}^2\nonumber\\ &\leq& \frac{3(20\theta^2-36\theta+17)}{4C_{coe}} \int_{t^{m-1}}^{t^{m+1}}\|(P_h^{\underline{\bf u}}-I)\underline{\bf u}_t\|^2_{\bf U'}{\rm d}t +\frac{C_{coe}\Delta t}{3} \left\|B(\eta_{\underline{\bf u}}^{m+1},\theta)\right\|_{\bf U}^2. \end{eqnarray*}$

(4.7)

类似地, 根据积分型余项的泰勒展开式,有

$\begin{eqnarray*} & &{\bf F}^{m+1}={\bf F}^{m}+\Delta t{\bf F}_{t}^{m}+\int_{t^{m}}^{t^{m+1}}(t^{m+1}-t){\bf F}_{tt}{\rm d}t,\\ &&{\bf F}^{m-1}={\bf F}^{m}-\Delta t{\bf F}_{t}^{m}+\int_{t^{m}}^{t^{m-1}}(t^{m-1}-t){\bf F}_{tt}{\rm d}t, \end{eqnarray*}$

代入并整理得

$\begin{eqnarray*} F({\bf F}^{m+1},\theta) &=&-\frac{(1-\theta)(1-2\theta)}{2}\int_{t^m}^{t^{m+1}}(t^{m+1}-t){\bf F}_{tt}{\rm d}t\\ &&+\frac{(1-\theta)(1-2\theta)}{2}\int_{t^{m-1}}^{t^{m}}(t^{m-1}-t){\bf F}_{tt}{\rm d}t. \end{eqnarray*}$

从而

$\begin{matrix}\label{E3} &&2\Delta t\left(F({\bf F}^{m+1},\theta),B(\eta_{\underline{\bf u}}^{m+1},\theta)\right)\nonumber\\ &\leq &\frac{(1-\theta)^2(1-2\theta)^2\Delta t^4}{4C_{coe}}\int_{t^{m-1}}^{t^{m+1}} \|{\bf F}_{tt}\|_{\bf U'}^2{\rm d}t +\frac{C_{coe}\Delta t}{3}\left\|B(\eta_{\underline{\bf u}}^{m+1},\theta)\right\|_{\bf U}^2. \end{matrix}$

(4.8)

最后, 结合 (4.5)-(4.8) 式, 将 (4.4) 式从 $m=1$ 直到 $m=N-1$ 求和, 类似定理3.1的证明有

$\begin{eqnarray*} &&\frac{3-2\theta}{4(1-\theta)}\|\eta_{\underline{\bf u}}^{N}\|_0^2+C_{coe}\Delta t\sum_{m=1}^{N-1}\left\|B(\eta_{\underline{\bf u}}^{m+1},\theta)\right\|_{\bf U}^2\\ &\leq&\sum_{m=1}^{N-1} \bigg(\frac{ (60\theta^4-200\theta^3+257\theta^2-130\theta+24)\Delta t^4}{5C_{coe}} \int_{t^{m-1}}^{t^{m+1}}\|\underline{\bf u}_{ttt}\|^2_{\bf U'}{\rm d}t\\ &&+\frac{3(20\theta^2-36\theta+17)}{4C_{coe}} \int_{t^{m-1}}^{t^{m+1}}\|(P_h^{\underline{\bf u}}-I)\underline{\bf u}_t\|^2_{\bf U'}{\rm d}t +\frac{(1-\theta)^2(1-2\theta)^2\Delta t^4}{4C_{coe}} \int_{t^{m-1}}^{t^{m+1}} \|{\bf F}_{tt}\|_{\bf U'}^2{\rm d}t\bigg). \end{eqnarray*}$

由三角不等式可得耦合算法误差估计的结论. 证毕.

4.2 解耦算法的误差估计

${\bf定理4.2}$ (解耦算法的二阶收敛性) 在 (4.2) 和 (4.3) 式的假设下, 对于 $N\geq 2$ 有以下估计

$\begin{eqnarray*} &&\frac{3-2\theta}{4(1-\theta)}\|e_{{\bf u}_{f}}^{N}\|_f^2 +\hat{C}_{coe}\Delta t\sum_{m=1}^{N-1} \left\|B(e_{{\bf u}_f}^{m+1},\theta)\right\|_{{\bf H}_f}^2\\ &&+\frac{g(3-2\theta)}{4(1-\theta)}\|e_{\phi_{p}}^{N}\|_p^2 +g\check{C}_{coe}\Delta t\sum_{m=1}^{N-1} \left\|B(e_{{\phi}_p}^{m+1},\theta)\right\|_{{\bf H}_p}^2 \leq C (\Delta t^4+h^4). \end{eqnarray*}$

${\bf证}$ 在 $t_{m+1}$ , $t_{m}$ 和 $t_{m-1}$ 时刻分别对 (2.11) 式乘 $\frac{(1-\theta)(3-2\theta)}{2}$ , $(4\theta-1-2\theta^2)$ , $\frac{(1-\theta)(1-2\theta)}{2}$ 得到三个对应时刻的等式. 从 (3.16) 式中减去三者之和, 再结合误差函数的定义和投影算子的性质 (3.2) 式可得 $\forall$ ${\bf v}_{fh}\in {\bf H}_{fh}$ , $\psi_{ph}\in {\bf H}_{ph}$ , $q_{fh}\in Q_{fh}$ ,有

$\begin{eqnarray*} \label{Error d} &&\left(\frac{1}{\Delta t}A(\eta_{{\bf u}_f}^{m+1},\theta),{\bf v}_{fh}\right)_{\Omega_{f}} +a_{\Omega_f}\left(B(\eta_{{\bf u}_f}^{m+1},\theta),{\bf v}_{fh}\right) +b\left({\bf v}_{fh}, B(\eta_{p_f}^{m+1},\theta)\right)\nonumber\\ &&+g\left(\frac{1}{\Delta t}A(\eta_{{\phi}_p}^{m+1},\theta),\psi_{ph}\right)_{\Omega_{p}} +a_{\Omega_p}\left(B(\eta_{{\phi}_p}^{m+1},\theta),\psi_{ph}\right)\nonumber\\ &=&-\left(\frac{1}{\Delta t}A({\bf u}_f^{m+1},\theta)-B({\bf u}_t^{m+1},\theta),{\bf v}_{fh}\right)_{\Omega_{f}} -\left(\frac{1}{\Delta t}A(\xi_{{\bf u}_f}^{m+1},\theta),{\bf v}_{fh}\right)_{\Omega_{f}}\\ &&-g\left(\frac{1}{\Delta t}A(\phi_p^{m+1},\theta) -\frac{1}{\Delta t}B(\phi_t^{m+1},\theta),\psi_{ph}\right)_{\Omega_{p}} -g\left(\frac{1}{\Delta t}A(\xi_{{\phi}_p}^{m+1},\theta),\psi_{ph}\right)_{\Omega_{p}}\nonumber\\ &&+\left(F({\bf g}_f^{m+1},\theta),{\bf v}_{fh}\right)_{\Omega_{f}} +g\left(F({\bf g}_p^{m+1},\theta),\psi_{ph}\right)_{\Omega_{p}}\nonumber\\ &&-c_{\Gamma}\left({\bf v}_{fh},(2-\theta)\phi_{ph}^m-(1-\theta)\phi_{ph}^{m-1} -B(\phi_{p}^{m+1},\theta)\right)\nonumber\\ &&+c_{\Gamma}\left((2-\theta){\bf u}_{fh}^m-(1-\theta){\bf u}_{fh}^{m-1} -B({\bf u}_{f}^{m+1},\theta),\psi_{ph}\right),\nonumber\\ &&b\left(B(\eta_{{\bf u}_{f}}^{m+1},\theta), q_{fh}\right)=0.\nonumber \end{eqnarray*}$

(4.9)

假设 ${\bf v}_{fh}=2\Delta tB(\eta_{{\bf u}_f}^{m+1},\theta)$ , $\psi_{ph}=2\Delta tB(\eta_{{\phi}_p}^{m+1},\theta)$ , $q_{fh}=2\Delta tB(\eta_{{p}_f}^{m+1},\theta)$ . 回顾定理 4.1 的证明, 有以下不等式成立

$\begin{eqnarray*} && 2\left(A(\eta_{{u}_f}^{m+1},\theta), B(\eta_{{u}_f}^{m+1},\theta)\right)_{\Omega_{f}}\nonumber\\ &\geq & (2\theta^2-5\theta+3)\|\eta_{{u}_f}^{m+1}\|^2_f-2(1-\theta)\|\eta_{{u}_f}^{m}\|^2_f -(1-2\theta)(1-\theta)\|\eta_{{u}_f}^{m-1}\|^2_f\nonumber\\ &&-(1-2\theta)(3-2\theta)(\|\eta_{{u}_f}^{m+1}\|_f\|\eta_{{u}_f}^{m}\|_f -\|\eta_{{u}_f}^{m}\|_f\|\eta_{{u}_f}^{m-1}\|_f), \end{eqnarray*}$

(4.10)

$\begin{eqnarray*} &&2g\left(A(\eta_{{\phi}_p}^{m+1},\theta),B(\eta_{{\phi}_p}^{m+1},\theta)\right)_{\Omega_{p}}\nonumber\\ &\geq & g\left[(2\theta^2-5\theta+3)\|\eta_{{\phi}_p}^{m+1}\|^2_p-2(1-\theta)\|\eta_{{\phi}_p}^{m}\|^2_p -(1-2\theta)(1-\theta)\|\eta_{{\phi}_p}^{m-1}\|^2_p\right.\nonumber\\ &&\left.-(1-2\theta)(3-2\theta)(\|\eta_{{\phi}_p}^{m+1}\|_p\|\eta_{{\phi}_p}^{m}\|_p -\|\eta_{{\phi}_p}^{m}\|_p\|\eta_{{\psi}_p}^{m-1}\|_p)\right], \end{eqnarray*}$

(4.11)

$\begin{eqnarray*}\label{error221}&& 2\Delta t a_{\Omega_f}\left(B(\eta_{{u}_f}^{m+1},\theta),B(\eta_{{u}_f}^{m+1},\theta)\right) \geq 2\hat{C}_{coe}\Delta t \left\|B(\eta_{{u}_f}^{m+1},\theta)\right\|_{H_f}^2,\end{eqnarray*}$

(4.12)

$\begin{matrix} \label{error222} &&2\Delta t a_{\Omega_p}\left(B(\eta_{{\phi}_p}^{m+1},\theta),B(\eta_{{\phi}_p}^{m+1},\theta)\right) \geq 2g\check{C}_{coe}\Delta t \left\|B(\eta_{{\phi}_p}^{m+1},\theta)\right\|_{H_p}^2, \end{matrix}$

(4.13)

$\begin{eqnarray*}\label{error r211} &&2\Delta t\bigg(\frac{1}{\Delta t}A({\bf u}_f^{m+1},\theta)-B({\bf u}_t^{m+1},\theta), B(\eta_{{u}_f}^{m+1},\theta)\bigg)_{\Omega_{f}}\\ &\leq &\frac{\hat{C}_{coe}\Delta t}{6}\left\|B(\eta_{{u}_f}^{m+1},\theta)\right\|_{H_f}^2 +\frac{2(60\theta^4-200\theta^3+257\theta^2-130\theta+24)\Delta t^4}{5\hat{C}_{coe}} \int_{t^{m-1}}^{t^{m+1}}\|{\bf u}_{ttt}\|_{H_f'}^2{\rm d}t,\nonumber \end{eqnarray*}$

(4.14)

$\begin{aligned} & 2 g \Delta t\left(\frac{1}{\Delta t} A\left(\phi_{p}^{m+1}, \theta\right)-B\left(\phi_{t}^{m+1}, \theta\right), B\left(\eta_{\phi_{p}}^{m+1}, \theta\right)\right)_{\Omega_{p}} \\ \leq & \frac{g \check{C}_{c o e} \Delta t}{6}\left\|B\left(\eta_{\phi_{p}}^{m+1}, \theta\right)\right\|_{\mathbf{H}_{p}}^{2}+\frac{2 g\left(60 \theta^{4}-200 \theta^{3}+257 \theta^{2}-130 \theta+24\right) \Delta t^{4}}{5 \check{C}_{c o e}} \int_{t^{m-1}}^{t^{m+1}}\left\|\phi_{t t t}\right\|_{\mathbf{H}_{p}^{\prime}}^{2} \mathrm{~d} t \end{aligned}$

(4.15)

$\begin{eqnarray*}\label{error r221} &&2\Delta t \left(\frac{1}{\Delta t}A(\xi_{{u}_f}^{m+1},\theta),B(\eta_{{u}_f}^{m+1},\theta)\right)_{\Omega_{f}}\nonumber\\ &\leq &\frac{3(20\theta^2-36\theta+17)}{2\hat{C}_{coe}} \int_{t^{m-1}}^{t^{m+1}}\|(P_h^{{\bf u}_f}-I){\bf u}_t\|^2_{H_f'}{\rm d}t +\frac{\hat{C}_{coe}\Delta t}{6}\left\|B(\eta_{{u}_f}^{m+1},\theta)\right\|_{H_f}^2, \end{eqnarray*}$

(4.16)

$\begin{eqnarray*}\label{error r222} &&2g\Delta t\left(\frac{1}{\Delta t}A(\xi_{{\phi}_p}^{m+1},\theta), B(\eta_{{\phi}_p}^{m+1},\theta)\right)_{\Omega_{p}}\nonumber\\ &\leq& \frac{3g(20\theta^2-36\theta+17)}{2\check{C}_{coe}} \int_{t^{m-1}}^{t^{m+1}}\|(P_h^{{\phi}_p}-I)\phi_t\|^2_{H_p'}{\rm d}t +\frac{g\check{C}_{coe}\Delta t}{6}\left\|B(\eta_{{\phi}_p}^{m+1},\theta)\right\|_{H_p}^2, \end{eqnarray*}$

(4.17)

$\begin{eqnarray*} \label{error r231} &&2\Delta t\left(F({\bf g}_f^{m+1},\theta),B(\eta_{{u}_f}^{m+1},\theta)\right)_{\Omega_{f}}\nonumber\\ &\leq &\frac{(1-\theta)^2(1-2\theta)^2\Delta t^4}{2\hat{C}_{coe}}\int_{t^{m-1}}^{t^{m+1}} \|{\bf g}_{tt}\|_{H_f'}^2{\rm d}t+\frac{\hat{C}_{coe}\Delta t}{6}\left\|B(\eta_{{u}_f}^{m+1},\theta)\right\|_{H_f}^2, \end{eqnarray*}$

(4.18)

$\begin{eqnarray*}\label{error r232} &&2g\Delta t\left(F({\bf g}_p^{m+1},\theta),B(\eta_{{\phi}_p}^{m+1},\theta)\right)_{\Omega_{p}}\nonumber\\ &\leq &\frac{g(1-\theta)^2(1-2\theta)^2\Delta t^4}{2\check{C}_{coe}}\int_{t^{m-1}}^{t^{m+1}} \|{\bf g}_{tt}\|_{H_p'}^2{\rm d}t +\frac{g\check{C}_{coe}\Delta t}{6}\left\|B(\eta_{{\phi}_p}^{m+1},\theta)\right\|_{H_p}^2. \end{eqnarray*}$

(4.19)

下面主要分析交界面项. 交界面项可以被分解为

$\begin{eqnarray*} & &-2\Delta tc_{\Gamma} \left(B(\eta_{{\bf u}_f}^{m+1},\theta), (2-\theta)\phi_{ph}^m-(1-\theta)\phi_{ph}^{m-1} -B(\phi_{p}^{m+1},\theta)\right)\nonumber\\ &&+2\Delta tc_{\Gamma} \left((2-\theta){\bf u}_{fh}^m-(1-\theta){\bf u}_{fh}^{m-1} -B({\bf u}_{f}^{m+1},\theta),B(\eta_{{\phi}_p}^{m+1},\theta)\right)\nonumber\\ &=&-2\Delta tc_{\Gamma} \left(B(\eta_{{\bf u}_f}^{m+1},\theta),B(\eta_{\phi_{p}}^{m+1},\theta)\right) +2\Delta tc_{\Gamma} \left(B(\eta_{{\bf u}_{f}}^{m+1},\theta),B(\eta_{{\phi}_p}^{m+1},\theta)\right)\nonumber\\ &&-2\Delta tc_{\Gamma} \left(B(\eta_{{\bf u}_f}^{m+1},\theta),W(\eta_{{\phi}_p}^{m+1},\theta)\right) +2\Delta tc_{\Gamma} \left(W(\eta_{{\bf u}_{f}}^{m+1},\theta),B(\eta_{{\phi}_p}^{m+1},\theta)\right)\nonumber\\ &&-2\Delta tc_{\Gamma} \left(B(\eta_{{\bf u}_f}^{m+1},\theta),B(\xi_{\phi_{p}}^{m+1},\theta)\right) +2\Delta tc_{\Gamma} \left(B(\xi_{{\bf u}_{f}}^{m+1},\theta),B(\eta_{\phi_{p}}^{m+1},\theta)\right)\nonumber\\ &&-2\Delta tc_{\Gamma} \left(B(\eta_{{\bf u}_f}^{m+1},\theta),W(\xi_{\phi_{p}}^{m+1},\theta)\right) +2\Delta tc_{\Gamma} \left(W(\xi_{{\bf u}_{f}}^{m+1},\theta),B(\eta_{\phi_{p}}^{m+1},\theta)\right)\nonumber\\ &&-2\Delta tc_{\Gamma} \left(B(\eta_{{\bf u}_f}^{m+1},\theta),W(\phi_{p}^{m+1},\theta)\right) +2\Delta tc_{\Gamma} \left(W({\bf u}_{f}^{m+1},\theta),B(\eta_{\phi_{p}}^{m+1},\theta)\right). \end{eqnarray*}$

(4.20)

由线性性质可知, 上式中前四项之和等于 $0$ . 下面四项借助 (2.12) 式及引理 3.2, 并代入 $\varepsilon_3=\varepsilon_5=\frac{3}{\hat{C}_{coe}}$ , $\varepsilon_4=\varepsilon_6=\frac{3g}{\check{C}_{coe}}$ , 有

$\begin{eqnarray*}\label{E42} &&-2\Delta tc_{\Gamma} \left(B(\eta_{{\bf u}_f}^{m+1},\theta),B(\xi_{\phi_{p}}^{m+1},\theta)\right) +2\Delta tc_{\Gamma} \left(B(\xi_{{\bf u}_{f}}^{m+1},\theta),B(\eta_{\phi_{p}}^{m+1},\theta)\right)\nonumber\\ &&-2\Delta tc_{\Gamma} \left(B(\eta_{{\bf u}_f}^{m+1},\theta),W(\xi_{\phi_{p}}^{m+1},\theta)\right) +2\Delta tc_{\Gamma} \left(W(\xi_{{\bf u}_{f}}^{m+1},\theta),B(\eta_{\phi_{p}}^{m+1},\theta)\right)\nonumber\\ &\leq &\frac{\hat{C}_{coe}\Delta t}{6} \left\|B(\eta_{{\bf u}_f}^{m+1},\theta)\right\|_{{\bf H}_f}^2 +\frac{6C_3h^{-1}\Delta t}{\hat{C}_{coe}} \left\|B(\xi_{\phi_{p}}^{m+1},\theta)\right\|_p^2\nonumber\\ &&+\frac{g\check{C}_{coe}\Delta t}{6} \left\|B(\eta_{\phi_{p}}^{m+1},\theta)\right\|_{{\bf H}_p}^2 +\frac{6gC_4h^{-1}\Delta t}{\check{C}_{coe}} \left\|B(\xi_{{\bf u}_{f}}^{m+1},\theta)\right\|_f^2\nonumber\\ &&+\frac{\hat{C}_{coe}\Delta t}{6} \left\|B(\eta_{{\bf u}_f}^{m+1},\theta)\right\|_{{\bf H}_f}^2 +\frac{6C_5h^{-1}\Delta t}{\hat{C}_{coe}} \left\|W(\xi_{\phi_{p}}^{m+1},\theta)\right\|_p^2\nonumber\\ & &+\frac{g\check{C}_{coe}\Delta t}{6} \left\|B(\eta_{\phi_{p}}^{m+1},\theta)\right\|_{{\bf H}_p}^2 +\frac{6gC_6h^{-1}\Delta t}{\check{C}_{coe}} \left\|W(\xi_{{\bf u}_{f}}^{m+1},\theta)\right\|_f^2\nonumber\\ &\leq &\frac{\hat{C}_{coe}\Delta t}{3} \left\|B(\eta_{{\bf u}_f}^{m+1},\theta)\right\|_{{\bf H}_f}^2 +\frac{g\check{C}_{coe}\Delta t}{3} \left\|B(\eta_{\phi_{p}}^{m+1},\theta)\right\|_{{\bf H}_p}^2\nonumber\\ &&+\frac{6g\Delta t}{\hat{C}_{coe}} \left\|B(\xi_{\phi_{p}}^{m+1},\theta)\right\|_p^2 +\frac{6\Delta t}{\check{C}_{coe}} \left\|B(\xi_{{\bf u}_{f}}^{m+1},\theta)\right\|_f^2\nonumber\\ &&+\frac{6g\Delta t}{\hat{C}_{coe}} \left\|W(\xi_{\phi_{p}}^{m+1},\theta)\right\|_p^2 +\frac{6\Delta t}{\check{C}_{coe}} \left\|W(\xi_{{\bf u}_{f}}^{m+1},\theta)\right\|_f^2, \end{eqnarray*}$

(4.21)

其中, 最后一个不等式是在选择合适的常量 $C_3,C_4,C_5$ 和 $C_6$ 下得到的.

利用带有积分型余项的泰勒展开式,有

$\begin{eqnarray*} & &\phi_p^{m+1}=\phi_p^{m}+\Delta t\phi_{t}^{m}+\int_{t^{m}}^{t^{m+1}}(t^{m+1}-t)\phi_{tt}{\rm d}t,\qquad\\ &&\phi_p^{m-1}=\phi_p^{m}-\Delta t\phi_{t}^{m}+\int_{t^{m}}^{t^{m-1}}(t^{m-1}-t)\phi_{tt}{\rm d}t, \end{eqnarray*}$

代入并整理得

$\begin{eqnarray*} W(\phi_{p}^{m+1},\theta) &=&-\frac{(1-\theta)(3-2\theta)}{2}\int_{t^m}^{t^{m+1}}(t^{m+1}-t)\phi_{tt}{\rm d}t\\ &&+\frac{(1-\theta)(3-2\theta)}{2}\int_{t^{m-1}}^{t^{m}}(t^{m-1}-t)\phi_{tt}{\rm d}t, \end{eqnarray*}$

类似地, 根据带有积分型余项的泰勒展开式,有

$\begin{eqnarray*} & &{\bf u}_f^{m+1}={\bf u}_f^{m}+\Delta t{\bf u}_{t}^{m}+\int_{t^{m}}^{t^{m+1}}(t^{m+1}-t){\bf u}_{tt}{\rm d}t,\qquad\\ &&{\bf u}_f^{m-1}={\bf u}_f^{m}-\Delta t{\bf u}_{t}^{m}+\int_{t^{m}}^{t^{m-1}}(t^{m-1}-t){\bf u}_{tt}{\rm d}t, \end{eqnarray*}$

可以得到

$\begin{eqnarray*} W({\bf u}_{f}^{m+1},\theta) &=&-\frac{(1-\theta)(3-2\theta)}{2}\int_{t^m}^{t^{m+1}}(t^{m+1}-t){\bf u}_{tt}{\rm d}t\\ &&+\frac{(1-\theta)(3-2\theta)}{2}\int_{t^{m-1}}^{t^{m}}(t^{m-1}-t){\bf u}_{tt}{\rm d}t. \end{eqnarray*}$

利用引理 3.2 并代入 $\varepsilon_7=\frac{3}{\hat{C}_{coe}},\varepsilon_8=\frac{3g}{\check{C}_{coe}}$ 可得

$\begin{eqnarray*} &&-2\Delta tc_{\Gamma} \left(B(\eta_{{\bf u}_f}^{m+1},\theta),W(\phi_{p}^{m+1},\theta)\right) +2\Delta tc_{\Gamma} \left(W({\bf u}_{f}^{m+1},\theta),B(\eta_{\phi_{p}}^{m+1},\theta)\right)\nonumber\\ &\leq &\frac{\hat{C}_{coe}\Delta t}{6} \left\|B(\eta_{{\bf u}_f}^{m+1},\theta)\right\|_{{\bf H}_f}^2 +\frac{6C_7h^{-1}\Delta t}{\hat{C}_{coe}} \left\|W(\phi_{p}^{m+1},\theta)\right\|_p^2\nonumber\\ &&+\frac{g\check{C}_{coe}\Delta t}{6} \left\|B(\eta_{\phi_{p}}^{m+1},\theta)\right\|_{{\bf H}_p}^2 +\frac{6gC_8h^{-1}\Delta t}{\check{C}_{coe}} \left\|W({\bf u}_{f}^{m+1},\theta)\right\|_f^2\nonumber\\ &\leq &\frac{\hat{C}_{coe}\Delta t}{6} \left\|B(\eta_{{\bf u}_f}^{m+1},\theta)\right\|_{{\bf H}_f}^2 +\frac{6g\Delta t}{\hat{C}_{coe}} \left\|W(\phi_{p}^{m+1},\theta)\right\|_p^2\nonumber\\ &&+\frac{g\check{C}_{coe}\Delta t}{6} \left\|B(\eta_{\phi_{p}}^{m+1},\theta)\right\|_{{\bf H}_p}^2 +\frac{6\Delta t}{\check{C}_{coe}} \left\|W({\bf u}_{f}^{m+1},\theta)\right\|_f^2\nonumber\\ &\leq &\frac{\hat{C}_{coe}\Delta t}{6} \left\|B(\eta_{{\bf u}_f}^{m+1},\theta)\right\|_{{\bf H}_f}^2 +\frac{\check{C}_{coe}\Delta t}{6} \left\|B(\eta_{\phi_{p}}^{m+1},\theta)\right\|_{{\bf H}_p}^2\nonumber\\ &&+\frac{(1-\theta)^2(3-2\theta)^2\Delta t^{4}}{2\check{C}_{coe}}\int_{t^{m-1}}^{t^{m+1}}\|{\bf u}_{tt}\|_f^2{\rm d}t +\frac{g(1-\theta)^2(3-2\theta)^2\Delta t^{4}}{2\hat{C}_{coe}}\int_{t^{m-1}}^{t^{m+1}}\|\phi_{tt}\|_p^2{\rm d}t, \end{eqnarray*}$

(4.22)

其中, 第二个不等式是在选择合适的常量 $C_7$ 和 $C_8$ 下得到的.

综上, 交界面项可化简为

$\begin{eqnarray*}\label{E4} &&-2\Delta tc_{\Gamma} \left(B(\eta_{{\bf u}_f}^{m+1},\theta), (2-\theta)\phi_{ph}^m-(1-\theta)\phi_{ph}^{m-1} -B(\phi_{p}^{m+1},\theta)\right)\nonumber\\ &&+2\Delta tc_{\Gamma} \left((2-\theta){\bf u}_{fh}^m-(1-\theta){\bf u}_{fh}^{m-1} -B({\bf u}_{f}^{m+1},\theta),B(\eta_{{\phi}_p}^{m+1},\theta)\right)\nonumber\\ &\leq &\frac{\hat{C}_{coe}\Delta t}{2} \left\|B(\eta_{{\bf u}_f}^{m+1},\theta)\right\|_{{\bf H}_f}^2 +\frac{g\check{C}_{coe}\Delta t}{2} \left\|B(\eta_{\phi_{p}}^{m+1},\theta)\right\|_{{\bf H}_p}^2\nonumber\\ &&+\frac{6g\Delta t}{\hat{C}_{coe}} \left\|B(\xi_{\phi_{p}}^{m+1},\theta)\right\|_p^2 +\frac{6\Delta t}{\check{C}_{coe}} \left\|B(\xi_{{\bf u}_{f}}^{m+1},\theta)\right\|_f^2\nonumber\\ &&+\frac{6g\Delta t}{\hat{C}_{coe}} \left\|W(\xi_{\phi_{p}}^{m+1},\theta)\right\|_p^2 +\frac{6\Delta t}{\check{C}_{coe}} \left\|W(\xi_{{\bf u}_{f}}^{m+1},\theta)\right\|_f^2\nonumber\\ &&+\frac{(1-\theta)^2(3-2\theta)^2\Delta t^{4}}{2\check{C}_{coe}}\int_{t^{m-1}}^{t^{m+1}}\|{\bf u}_{tt}\|_f^2{\rm d}t +\frac{g(1-\theta)^2(3-2\theta)^2\Delta t^{4}}{2\hat{C}_{coe}}\int_{t^{m-1}}^{t^{m+1}}\|\phi_{tt}\|_p^2{\rm d}t. \end{eqnarray*}$

(4.23)

结合 (4.10)-(4.23) 式, 将 (4.9) 式从 $m=1$ 直到 $m=N-1$ 求和, 类似定理 3.2 的证明, 有

$\begin{eqnarray*} &&\frac{3-2\theta}{4(1-\theta)}\|\eta_{\bf u}^{N}\|_f^2 +\hat{C}_{coe}\Delta t\sum_{m=1}^{N-1} \left\|B(\eta_{{\bf u}_f}^{m+1},\theta)\right\|_{{\bf H}_f}^2\\ &&+\frac{g(3-2\theta)}{4(1-\theta)}\|\eta_{\phi}^{N}\|_p^2 +g\check{C}_{coe}\Delta t\sum_{m=1}^{N-1} \left\|B(\eta_{\phi_{p}}^{m+1},\theta)\right\|_{{\bf H}_p}^2\\ &\leq &\sum_{m=1}^{N-1}\bigg( \frac{2(60\theta^4-200\theta^3+257\theta^2-130\theta+24) \Delta t^4}{5\hat{C}_{coe}} \int_{t^{m-1}}^{t^{m+1}}\|{\bf u}_{ttt}\|_{{\bf H}_f'}^2{\rm d}t\\ &&+\frac{2(60\theta^4-200\theta^3+257\theta^2-130\theta+24) g\Delta t^4}{5\check{C}_{coe}} \int_{t^{m-1}}^{t^{m+1}}\|\phi_{ttt}\|_{{\bf H}_p'}^2{\rm d}t\\ &&+\frac{3(20\theta^2-36\theta+17)}{2\hat{C}_{coe}} \int_{t^{m-1}}^{t^{m+1}}\|(P_h^{{\bf u}_f}-I){\bf u}_t\|^2_{{\bf H}_f'}{\rm d}t\\ &&+\frac{3g(20\theta^2-36\theta+17)}{2\check{C}_{coe}} \int_{t^{m-1}}^{t^{m+1}}\|(P_h^{{\phi}_p}-I)\phi_t\|^2_{{\bf H}_p'}{\rm d}t\\ &&+\frac{(1-\theta)^2(1-2\theta)^2\Delta t^4}{2\hat{C}_{coe}}\int_{t^{m-1}}^{t^{m+1}} \|{\bf g}_{tt}\|_{{\bf H}_f'}^2{\rm d}t +\frac{g(1-\theta)^2(1-2\theta)^2\Delta t^4}{2\check{C}_{coe}}\int_{t^{m-1}}^{t^{m+1}} \|{\bf g}_{tt}\|_{{\bf H}_p'}^2{\rm d}t\\ &&+\frac{(1-\theta)^2(3-2\theta)^2\Delta t^{4}}{2\check{C}_{coe}}\int_{t^{m-1}}^{t^{m+1}}\|{\bf u}_{tt}\|_f^2{\rm d}t +\frac{g(1-\theta)^2(3-2\theta)^2\Delta t^{4}}{2\hat{C}_{coe}}\int_{t^{m-1}}^{t^{m+1}}\|\phi_{tt}\|_p^2{\rm d}t\\ &&+\frac{6g\Delta t}{\hat{C}_{coe}} \left\|B(\xi_{\phi_{p}}^{m+1},\theta)\right\|_p^2 +\frac{6\Delta t}{\check{C}_{coe}} \left\|B(\xi_{{\bf u}_{f}}^{m+1},\theta)\right\|_f^2\\ &&+\frac{6g\Delta t}{\hat{C}_{coe}} \left\|W(\xi_{\phi_{p}}^{m+1},\theta)\right\|_p^2 +\frac{6\Delta t}{\check{C}_{coe}} \left\|W(\xi_{{\bf u}_{f}}^{m+1},\theta)\right\|_f^2\bigg). \end{eqnarray*}$

由三角不等式可得解耦算法误差估计的结论.证毕.

5 数值实验

这一节将进行3个数值实验. 第1个实验对耦合和解耦算法的有效性进行模拟. 第2个实验说明算法的收敛阶由一阶提高到二阶, 并通过对比体现出解耦算法的高效性. 第3个实验将耦合和解耦的向后欧拉方法和取不同 $\theta$ 值的线性多步法, 以及在此基础上加 time filter 算法进行比较, 说明了线性多步法和线性多步法加 time filter 算法的误差相对较小. 本文的数值模拟是利用 FreeFEM++ 软件实现的. 并不失一般性, 以下实验均假设相应的物理参数 $\nu=1$ , $\rho=1$ , $g=1$ , $S=1$ $\alpha=1$ 以及 ${\bf K}=k{\bf I}$ , 其中 $k=1$ . 并且所有的初始条件, 边界条件和源项遵循精确解. 为了简便, 在图例, 图注及表头中, CBEM 和 DBEM 分别表示耦合和解耦的向后欧拉方法, CLMM 和 DLMM 分别表示耦合和解耦的线性多步法, CBEMTF 和 DBEMTF 分别表示耦合和解耦的向后欧拉方法加 time filter 算法, CLMMTF 和 DLMMTF 分别表示耦合和解耦的线性多步法加 time filter 算法.

${\bf 实验 1}$ 这里利用文献[37] 中的算例，设全局区域 $\Omega$ 由不可压缩流区域 $\Omega_f=(0,\pi)\times(0,1)$ , 多孔介质区域 $\Omega_p=(0,\pi)\times(-1,0)$ , 以及交界面 $\Gamma=(0,\pi)\times{0}$ 组成. Taylor-Hood 元 (P2-P1) 和分段二次多项式 (P2) 被用于自由流体流动方程和多孔介质流动方程. 精确解由下式给出

$\begin{eqnarray*} &&{\bf u}_f=\bigg[\frac{1}{\pi}\sin(2\pi y)\cos(x)e^t, (-2+\frac{1}{\pi^2}\sin^2(\pi y))\sin(x)e^t\bigg],\\ &&p_f=0,\\ &&\phi_p=(e^y-e^{-y})\sin(x)e^t. \end{eqnarray*}$

为验证算法的有效性, 本文设置网格尺寸为 $h=1/100$ , 时间步长为 0.01 进行数值模拟. 图 2-图 4 分别展示了 $\theta=1/6$ , $1/4$ , $1/3$ 耦合和解耦的线性多步法以及线性多步法加 time filter 算法的速度等值线和速度流线, 通过观察可以发现本文提出的算法能够有效地模拟流体的流动.

图 2

图 2 $\theta=1/6$ 时, 速度等值线和速度流线

图 3

图 3 $\theta=1/4$ 时, 速度等值线和速度流线

图 4

图 4 $\theta=1/3$ 时, 速度等值线和速度流线

${\bf 实验 2}$ 这里借助文献[31]中的算例, 考虑不可压缩流区域 $\Omega_f=(0,1)\times(1,2)$ , 多孔介质区域 $\Omega_p=(0,1)\times(0,1)$ , 以及交界面 $\Gamma=(0,1)\times{1}$ , 并对自由流体流动方程和多孔介质流动方程分别使用 MINI 元 (P1b-P1) 和线性拉格朗日元 (P1). 下面给出精确解

$\begin{eqnarray*} &&{\bf u}_f=((x^2(y-1)^2+y)\cos(t),-\frac{2}{3}x(y-1)^3\cos(t)+(2-\pi\sin(\pi x))\cos(t)),\\ &&p_f=(2-\pi\sin(\pi x))\sin(\frac{1}{2}\pi y)\cos(t),\\ &&\phi_p=(2-\pi\sin(\pi x))(1-y-\cos(\pi y))\cos(t). \end{eqnarray*}$

为体现耦合和解耦算法的收敛阶, 本实验考虑 $\theta=1/4$ , 并在表1-表8 中分别列出变化时间步长 $\Delta t$ 和网格尺寸 $h$ 对应的误差, 收敛阶及CPU 时间.

表1 $T=1$ 时, 固定网格尺寸 $h=\frac{1}{8}$ , 随时间步长 $\Delta t$ 变化 CLMM 计算所得结果

表2 $T=1$ 时, 固定网格尺寸 $h=\frac{1}{8}$ , 随着时间步长 $\Delta t$ 变化 CLMMTF 计算所得结果

表3 $T=1$ 时, 固定网格尺寸 $h=\frac{1}{8}$ , 随着时间步长 $\Delta t$ 变化 DLMM 计算所得结果

表4 $T=1$ 时, 固定网格尺寸 $h=\frac{1}{8}$ , 随着时间步长 $\Delta t$ 变化 DLMMTF 计算所得结果

表5 $T=1$ 时, 固定时间步长 $\Delta t=0.01$ , 随着网格尺寸 $h$ 变化 CLMM 计算所得结果

表6 $T=1$ 时, 固定时间步长 $\Delta t=0.01$ , 随着网格尺寸 $h$ 变化 CLMMTF 计算所得结果

表7 $T=1$ 时, 固定时间步长 $\Delta t=0.01$ , 随着网格尺寸 $h$ 变化 DLMM 计算所得结果

表8 $T=1$ 时, 固定时间步长 $\Delta t=0.01$ , 随着网格尺寸 $h$ 变化 DLMMTF 计算所得结果

首先, 利用与文献[31] 中相同的方法, 通过固定网格尺寸 $h$ , 变化时间步长 $\Delta t$ 计算耦合和解耦算法的收敛阶. 这时近似误差主要由时间步长 $\Delta t$ 控制. 定义

$\rho_{v}=\frac{e_{v}(\Delta t)}{e_{v}(\frac{1}{2}\Delta t)},$

其中 $e_{v}(\Delta t)=\|v_h^{\Delta t}-v_h^{\frac{1}{2}\Delta t}\|_{L^2}$ , $v={\bf u}_f$ , $\phi_p$ , $p_f$ , $\rho_{v}\approx \frac{4^{\gamma}-2^{\gamma}}{2^{\gamma}-1}$ . 固定网格尺寸 $h=\frac{1}{8}$ , 时间步长 $\Delta t$ 由 $\frac{1}{20}$ 变化到 $\frac{1}{320}$ 得到表 1 -表 4. 将表1 和表2, 表 3 和表4分别进行比较, 发现耦合和解耦的线性多步法是一阶收敛的, 而本文提出的线性多步法加 time filter 算法在几乎不增加计算量的情况下可以达到二阶收敛. 这体现了 time filter 算法的二阶收敛性和高效性. 同时通过比较表中 CPU 时间可以发现解耦算法的计算效率相对较高.

其次, 在固定时间步长 $\Delta t$ , 变化网格尺寸 $h$ 的情况下, 计算耦合和解耦算法的收敛阶. 这时近似误差主要由时间步长 $h$ 控制. 定义

$\rho_{h,v}=\frac{\log(\frac{e_{v}(h_1)}{e_{v}(h_2)})}{\log(\frac{h_1}{h_2})}$

其中 $e_{v}(h)=\|v_{f}-v_{fh}\|_{L^2}$ 是关于网格尺寸 $h$ 的误差. 固定时间步长 $\Delta t=0.01$ , 网格尺寸 $h$ 由 $\frac{1}{4}$ 变化到 $\frac{1}{64}$ 得到表 5 -表 8. 将表 5 和表 7, 表6 和表 8 中的 CPU 时间分别进行比较, 可以看出解耦算法所用的 CPU 时间较少, 计算效率相较于耦合算法较高.

${\bf 实验 3}$ 这里, 利用文献 [38-39] 中的算例进行数值实验. 同时选择与实验 2 中相同的区域 $\Omega_f$ 和 $\Omega_p$ , 以及有限元. 该精确解是

$\begin{eqnarray*} &&{\bf u}_f=((y^2-2y+1)\cos(t),(x^2-x)\cos(t)),\\ &&p_f=[2\nu(x+y-1)+\frac{1}{3k}]\cos(t),\\ &&\phi_p=[\frac{1}{k}(x(1-x)(y-1)+\frac{1}{3}y^3-y^2+y)+2\nu x]\cos(t). \end{eqnarray*}$

图 5 分别给出了向后欧拉方法和 $\theta$ 取不同值的线性多步法在时间段为 $[T]$ ( $T=5$ ) 且时间步长为 $0.05$ 上 ${\bf u}_f$ 和 $\phi_p$ 在每个时刻的绝对误差. 从图 5 中可以看出: 与向后欧拉方法相比较, 线性多步法关于 ${\bf u}_f$ 和 $\phi_p$ 的绝对误差几乎在每个时刻都相对较小. 同时, 当 $\theta$ 取不同值时线性多步法的绝对误差会随着 $\theta$ 的增大而减小. 类似地, 图 6 分别给出了向后欧拉方法加 time filter 算法和 $\theta$ 取不同值的线性多步法加 time filter 算法在时间段为 $[T]$ ( $T=5$ ) 且时间步长为 $0.05$ 上 ${\bf u}_f$ 和 $\phi_p$ 在每个时刻的绝对误差. 由图 6 知: 与向后欧拉方法加 time filter 算法相比, 无论是耦合还是解耦线性多步法加 time filter 算法关于 ${\bf u}_f$ 和 $\phi_p$ 的绝对误差几乎在每个时刻都相对较小, 同时随着 $\theta$ 的增大, 线性多步法加 time filter 的绝对误差减小.

图 5

图 5 向后欧拉方法和线性多步法的 ${\bf u}_f$ , $\phi_p$ 的绝对误差

图 6

图 6 向后欧拉方法加 time filter 算法和线性多步法加 time filter 算法的 ${\bf u}_f$ , $\phi_p$ 的绝对误差

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