Since there is huge computational work in computing \((OCP\mbox{-}OPT)^{hk}\), we need to develop a highly efficient algorithm to compute \((OCP\mbox{-}OPT)^{hk}\). Here different partitions of meshes are used for the state and control.
First, we present the following two lemmas. Lemma 4.1 can be found in [24], and Lemma 4.2 can be found, for example, in Chapter 3 of [25].
Lemma 4.1
[24]
Let
\({\pi}_{h}\)
be the average interpolation operator defined in [24]. Then, for
\(m =0\)
or 1, \(1\leq q \leq\infty\), and
\(v \in W^{1,q}(\Omega^{h})\),
$$ | v - {\pi}_{h}v| _{m,q, \tau} \leq\sum _{\overline{\tau}'\cap\overline{\tau}\neq \emptyset}Ch_{\tau}^{1-m}|v|_{1,q,\tau'}. $$
(4.1)
Lemma 4.2
[25]
For all
\(v \in W^{1,q} (\Omega)\), \(1\leq q <\infty\),
$$ \| v \|_{0,q, \partial\tau} \leq C\bigl(h_{\tau}^{-\frac{1}{q}} \| v \|_{0,q, \tau}+ h_{\tau}^{1-\frac{1}{q}}|v|_{1,q, \tau}\bigr). $$
(4.2)
Upper bound estimates
To obtain upper bound estimates, we need to introduce the following auxiliary system:
$$ \left \{ \textstyle\begin{array}{@{}l} (\frac{\partial}{\partial t}y(U_{h}),w)+a(\frac{\partial}{\partial t}y(U_{h}),w)+d(y(U_{h}),w)+\int_{0}^{t} c(t,\tau; y(U_{h})(\tau),w )\,d\tau\\ \quad=(f+U_{h},w) , \quad \forall w\in H_{0}^{1}(\Omega), \\ y(U_{h})(0)=y_{0}(x), \quad x\in\Omega, \\ -(q,\frac{\partial}{\partial t}p(U_{h}))-a(q,\frac{\partial}{\partial t}p(U_{h}))+d( q , p(U_{h})) + \int_{t}^{T} c(\tau,t;q,p(U_{h})(\tau))\,d\tau\\ \quad =(y(U_{h})-z_{d},q), \quad \forall q\in H_{0}^{1}(\Omega), \\ p(U_{h})(T)=0, \quad x\in\Omega. \end{array}\displaystyle \right . $$
(4.3)
Lemma 4.3
Suppose that
\((y,p,u)\)
and
\((Y_{h},P_{h},U_{h})\)
are the solutions of (2.4) and (3.7), respectively. Then we have
$$ \| u-U_{h}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}\leq C \bigl\{ \eta_{1}^{2}+ \bigl\| P_{h}-p(U_{h}) \bigr\| ^{2}_{L^{2}(0,T;L^{2}(\Omega))}\bigr\} , $$
(4.4)
where
\(\eta_{1}^{2}\)
is defined as follows:
$$ \eta_{1}^{2}=\sum _{i=1}^{N} \int_{t_{i-1}}^{t_{i}} \biggl\{ \sum _{\tau_{U}} \int_{\tau_{U}} ( P_{h}-\Pi_{h}P_{h} )^{2} \biggr\} \,dt. $$
(4.5)
Proof
First, we have
$$\begin{aligned} \alpha\|u-U_{h}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))} \leq{}& \inf_{v_{h}(t)\in K^{h}} \int_{0}^{T}(P_{h}+\alpha U_{h},v_{h}-u)\,dt \\ &{} + \int_{0}^{T}\bigl(P_{h}-p(U_{h}),u-U_{h} \bigr)\,dt+ \int_{0}^{T}\bigl(p(U_{h})-p,u-U_{h} \bigr)\,dt \\ ={}& I_{1}+I_{2}+I_{3}. \end{aligned}$$
(4.6)
Letting \(v_{h}=\Pi_{h}u \in U^{h}\) in \(I_{1}\), we have
$$ \begin{aligned} & I_{1} \leq C(\delta)\sum _{i=1}^{N} \int_{t_{i-1}}^{t_{i}} \biggl\{ \sum _{\tau_{U}} \int_{\tau_{U}}(-\Pi_{h}P_{h}+P_{h})^{2} \biggr\} \,dt +\delta \| u-U_{h}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}, \\ &I_{2}\leq C\bigl\| P_{h}-p(U_{h}) \bigr\| ^{2}_{L^{2}(0,T;L^{2}(\Omega))}+\delta\| u-U_{h}\|^{2} _{L^{2}(0,T;L^{2}(\Omega))}. \end{aligned} $$
(4.7)
From (4.3) and (2.4), for any \(t\in\ (0,T]\), we have
$$\begin{aligned} &\biggl(\frac{\partial}{\partial t}\bigl(y-y(U_{h})\bigr),w \biggr)+a\biggl(\frac{\partial}{\partial t}\bigl(y-y(U_{h})\bigr),w\biggr)+d \bigl(y-y(U_{h}),w\bigr) \\ &\quad{} + \int_{0}^{t} c\bigl(t,\tau; \bigl(y-y(U_{h}) \bigr) (\tau),w\bigr)\,d\tau =(u-U_{h},w),\quad \forall w\in V=H_{0}^{1}(\Omega), \end{aligned}$$
(4.8)
and
$$\begin{aligned} & {-}\biggl(q,\frac{\partial}{\partial t}\bigl(p-p(U_{h})\bigr) \biggr)-a\biggl(q,\frac{\partial }{\partial t}\bigl(p-p(U_{h})\bigr)\biggr)+d \bigl(q,p-p(U_{h})\bigr) \\ &\quad{} + \int_{t}^{T} c\bigl(\tau,t;q,\bigl(p-p(U_{h}) \bigr) (\tau)\bigr)\,d\tau=\bigl(y-y(U_{h}),q\bigr) ,\quad \forall q\in H_{0}^{1}(\Omega). \end{aligned}$$
(4.9)
From (4.8) and (4.9) we obtain
$$\begin{aligned} I_{3} ={}& \int_{0}^{T}\bigl(p(U_{h})-p,u-U_{h} \bigr)\,dt \\ ={}& \int_{0}^{T} \biggl\{ \biggl(\frac{\partial}{\partial t} \bigl(y-y(U_{h})\bigr),p(U_{h})-p\biggr)+a\biggl( \frac{\partial}{\partial t}\bigl(y-y(U_{h})\bigr),p(U_{h})-p\biggr) \\ &{} +d\bigl(y-y(U_{h}),p(U_{h})-p\bigr)+ \int _{0}^{t} c\bigl(t,\tau; \bigl(y-y(U_{h}) \bigr) (\tau),\bigl(p(U_{h})-p\bigr) (t)\bigr)\,d\tau \biggr\} \,dt \\ ={}& \int_{0}^{T} \biggl\{ -\biggl(y-y(U_{h}), \frac{\partial}{\partial t}\bigl(p(U_{h})-p\bigr)\biggr)-a\biggl(y-y(U_{h}), \frac{\partial}{\partial t}\bigl(p(U_{h})-p\bigr)\biggr) \\ &{} +d\bigl(y-y(U_{h}),p(U_{h})-p\bigr)+ \int _{t}^{T} c\bigl(\tau,t; \bigl(y-y(U_{h}) \bigr) (t),\bigl(p(U_{h})-p\bigr) (\tau)\bigr)\,d\tau \biggr\} \,dt \\ ={}& \int_{0}^{T} -\bigl(y-y(U_{h}),y-y(U_{h}) \bigr)\,dt \leq0. \end{aligned}$$
(4.10)
We can obtain (4.4) from (4.6)-(4.10). This completes the proof. □
Since \(y-Y_{h}=y-y(U_{h})+y(U_{h})-Y_{h}\), \(p-P_{h}=p-p(U_{h})+p(U_{h})-P_{h}\), we now first present the estimates of \(Y_{h}-y(U_{h})\) and \(P_{h}-p(U_{h})\).
Lemma 4.4
Suppose that
\((Y_{h},P_{h},U_{h})\)
and
\((y(U_{h}),p(U_{h}))\)
are the solutions of (3.7) and (4.3), respectively. Then we have
$$\begin{aligned} &\bigl\| Y_{h}-y(U_{h})\bigr\| ^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}+ \bigl\| Y_{h}-y(U_{h})\bigr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad{}+\biggl\| \frac{\partial}{\partial t}\bigl(Y_{h}-y(U_{h})\bigr) \biggr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +\bigl\| P_{h}-p(U_{h}) \bigr\| ^{2}_{L^{\infty}(0,T;L^{2}(\Omega))} \\ &\quad{}+\bigl\| P_{h}-p(U_{h})\bigr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))}+\biggl\| \frac{\partial }{\partial t}\bigl(P_{h}-p(U_{h})\bigr) \biggr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))}\leq C\sum_{i=2}^{13} \eta_{i}^{2}, \end{aligned}$$
(4.11)
where
\(\eta_{i} \) (\(i=2,\ldots,13\)) is defined as follows:
$$\begin{aligned} &\eta_{2}^{2}=\sum _{i=1}^{N}k_{i} \Biggl\{ \sum _{\tau}h_{\tau}^{2} \int_{\tau}\Biggl(\hat{Y}_{h}-z_{d}+ \frac{\partial}{\partial t}P_{h}-\operatorname{div}\biggl(A^{*}\nabla\frac{\partial}{\partial t} P_{h}\biggr)+\operatorname{div}\bigl(D^{*}\nabla\tilde{{P}}_{h}\bigr) \\ &\hphantom{\eta_{2}^{2}=}+\sum_{m=i}^{N}k_{m}\operatorname{div} \bigl(C^{*}(t_{m},t_{i-1})\nabla{{P}}_{h}^{m} \bigr) \Biggr)^{2} +\sum_{l} h_{l} \int_{l}\Biggl[\biggl(A^{*}\nabla\frac{\partial}{\partial t}P_{h} \biggr)\cdot n-\bigl(D^{*}\nabla\tilde{{P}}_{h}\bigr)\cdot n \\ &\hphantom{\eta_{2}^{2}=} -\sum_{m=i}^{N}k_{m} \bigl(C^{*}(t_{m},t_{i-1})\nabla {{P}}_{h}^{m} \bigr)\cdot n\Biggr]^{2} \Biggr\} , \\ &\eta_{3}^{2}=\sum_{i=1}^{N} \frac{k_{i}}{3}\bigl| P_{h}^{i-1}-P_{h}^{i}\bigr| _{1,\Omega}^{2}, \\ &\eta_{4}^{2}=\sum_{i=1}^{N} \int_{t_{i-1}}^{t_{i}} \Biggl(\sum _{m=i}^{N} \int_{t_{m-1}}^{t_{m}}\bigl\| \bigl(C^{*}(t_{m},t_{i-1})-C^{*}( \tau,t)\bigr)\nabla P_{h}^{m}\bigr\| _{0,\Omega}^{2}\,d\tau \Biggr)\,dt, \\ &\eta_{5}^{2}=\sum_{i=1}^{N}k_{i} \sum_{m=i}^{N}\frac{k_{m}}{3}\bigl| P_{h}^{m-1}-P_{h}^{m}\bigr| _{1,\Omega}^{2}, \\ &\eta_{6}^{2}= \sum_{i=1}^{N}k_{i}^{3}| \tilde{{P}}_{h} | _{1,\Omega}^{2}, \\ &\eta_{7}^{2} =\sum_{i=1}^{N}k_{i} \Biggl\{ \sum_{\tau}h_{\tau}^{2} \int_{\tau}\Biggl(\hat{f}+U_{h}- \frac{\partial}{\partial t}Y_{h}+\operatorname{div}\biggl(A\nabla\frac{\partial}{\partial t}Y_{h} \biggr) \\ &\hphantom{\eta_{7}^{2} =}{}+\operatorname{div}(D\nabla\hat{Y}_{h})+\sum _{m=1}^{i}k_{m} \operatorname{div} \bigl(C(t_{i},t_{m-1})\nabla Y_{h}^{m-1} \bigr) \Biggr)^{2} +\sum_{l}h_{l} \int_{l}\Biggl[\biggl(A\nabla\frac{\partial }{\partial t}Y_{h} \biggr)\cdot n \\ &\hphantom{\eta_{7}^{2} =}{} +(D\nabla\hat{Y}_{h})\cdot n+\sum _{m=1}^{i}k_{m}\bigl(C(t_{i},t_{m-1}) \nabla Y_{h}^{m-1}\bigr)\cdot n\Biggr]^{2} \Biggr\} , \\ &\eta_{8}^{2}= \| f- \hat{f}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}, \\ &\eta_{9}^{2}= \sum_{i=1}^{N} \frac{k_{i}}{3}\bigl| Y_{h}^{i-1}-Y_{h}^{i}\bigr| _{1,\Omega}^{2}, \\ &\eta_{10}^{2}=\sum_{i=1}^{N} \int_{t_{i-1}}^{t_{i}} \Biggl(\sum _{m=1}^{i} \int_{t_{m-1}}^{t_{m}}\bigl\| \bigl(C(t_{i},t_{m-1})-C(t, \tau )\bigr)\nabla Y_{h}^{m-1}\bigr\| _{0,\Omega}^{2}\,d\tau \Biggr)\,dt, \\ &\eta_{11}^{2}=\sum_{i=1}^{N}k_{i} \sum_{m=1}^{i}\frac {k_{m}}{3}\bigl| Y_{h}^{m-1}-Y_{h}^{m}\bigr| _{1,\Omega}^{2}, \\ &\eta_{12}^{2} =\sum_{i=1}^{N}k_{i}^{3}| \hat{{Y}}_{h} | _{1,\Omega}^{2}, \\ &\eta_{13}^{2}=\bigl\| Y_{h}(x,0)- y_{0}(x) \bigr\| ^{2}_{0,\Omega}, \end{aligned}$$
(4.12)
where
l
is a face of an element
τ, \(h_{l}\)
is the maximal diameter of
l, and we denote the normal derivative jumps over the interior face
l
by
\([\nabla\tilde{P_{h}} \cdot n]\)
and
\([\nabla \hat{Y_{h}} \cdot n]\)
as follows:
$$\begin{aligned} & [\nabla\tilde{P}_{h} \cdot n]_{l}=(\nabla \tilde{P}_{h}|_{{\tau }_{l}^{1}}-\nabla\tilde{P}_{h}|_{{\tau}_{l}^{2}}) \cdot n, \\ &[\nabla\hat{Y}_{h}\cdot n]_{l}=(\nabla\hat {Y}_{h}|_{{\tau}_{l}^{1}}-\nabla\hat{Y}_{h}|_{{\tau}_{l}^{2}}) \cdot n, \end{aligned}$$
where
n
is the unit normal vector on
\(l={\tau}_{l}^{1}\cap{\tau}_{l}^{2}\). For simplicity, we suppose that when
\(l \subset \partial\Omega\), \([\nabla\tilde{P}_{h} \cdot n]_{l}=0\)
and
\([\nabla\hat{Y}_{h}\cdot n]_{l}=0\).
Proof
First, we estimate \(P_{h}-p(U_{h})\). Suppose that the average interpolation operator \(\pi_{h}\) is defined as in [24] and define
$$\begin{aligned} \bigl\langle R(U_{h}),v\bigr\rangle ={}&{-} \biggl(\frac{\partial}{\partial t}\bigl(p(U_{h})-P_{h}\bigr),v \biggr)-a\biggl(v,\frac{\partial}{\partial t}\bigl(p(U_{h})-P_{h} \bigr)\biggr)+d\bigl(v,p(U_{h})-P_{h}\bigr) \\ &{} + \int_{t}^{T}c\bigl(\tau,t;v,p(U_{h}) ( \tau)\bigr)\,d\tau-\hat{h}(v,{P}_{h}). \end{aligned}$$
Then, letting \(q=\pi_{h}v\) in (4.3) and \(q_{h}=\pi_{h}v\) in (3.7), respectively, then subtracting (3.7) from (4.3), for all \(t\in [t_{i-1},t_{i}]\), we have
$$\begin{aligned} &{-}\biggl(\pi_{h}v,\frac{\partial}{\partial t} \bigl(p(U_{h})-P_{h}\bigr)\biggr)-a\biggl( \pi_{h}v,\frac {\partial}{\partial t}\bigl(p(U_{h})-P_{h} \bigr)\biggr)+d\bigl(\pi_{h}v, p(U_{h})- \tilde{P}_{h}\bigr) \\ &\quad{} + \int_{t}^{T} c\bigl(\tau,t;\pi_{h}v,p(U_{h}) (\tau)\bigr)\,d\tau-\hat {h}(\pi_{h}v,{P}_{h} ) = \bigl(y(U_{h})-\hat{Y}_{h},\pi_{h}v\bigr). \end{aligned}$$
(4.13)
So we have
$$\begin{aligned} &-\biggl(\pi_{h}v,\frac{\partial}{\partial t} \bigl(p(U_{h})-P_{h}\bigr)\biggr)-a\biggl( \pi_{h}v,\frac {\partial}{\partial t}\bigl(p(U_{h})-P_{h} \bigr)\biggr)+d\bigl(\pi_{h}v, p(U_{h})-{P}_{h} \bigr) \\ &\qquad{} + \int_{t}^{T} c\bigl(\tau,t;\pi_{h}v,p(U_{h}) (\tau)\bigr)\,d\tau-\hat {h}(\pi_{h}v,{P}_{h} ) \\ &\quad= \bigl(y(U_{h})-\hat{Y}_{h},\pi_{h}v\bigr)+d( \pi_{h}v, \tilde{P}_{h}-{P}_{h}). \end{aligned}$$
(4.14)
Then from (4.14) and (4.3) we obtain
$$\begin{aligned} &{-}\biggl(\frac{\partial}{\partial t}\bigl(p(U_{h})-P_{h} \bigr),v\biggr)-a\biggl(v,\frac{\partial}{\partial t}\bigl(p(U_{h})-P_{h} \bigr)\biggr)+d\bigl(v,p(U_{h})-P_{h}\bigr) \\ &\qquad{} + \int_{t}^{T}c\bigl(\tau,t;v,p(U_{h}) ( \tau)\bigr)\,d\tau-\hat{h}(v,{P}_{h}) \\ &\quad=-\biggl(\frac{\partial}{\partial t}\bigl(p(U_{h})-P_{h}\bigr),v- \pi_{h}v\biggr)-a\biggl(v-\pi_{h}v,\frac{\partial}{\partial t} \bigl(p(U_{h})-P_{h}\bigr)\biggr) \\ &\qquad{}+d\bigl(v- \pi_{h}v,p(U_{h})-P_{h}\bigr) + \int_{t}^{T}c\bigl(\tau,t;v-\pi_{h}v,p(U_{h}) (\tau)\bigr)\,d\tau-\hat{h}(v-\pi _{h}v,{P}_{h}) \\ &\qquad{}-\biggl(\frac{\partial}{\partial t}\bigl(p(U_{h})-P_{h}\bigr), \pi_{h} v\biggr)-a\biggl(\pi_{h}v,\frac{\partial}{\partial t} \bigl(p(U_{h})-P_{h}\bigr)\biggr)+d\bigl( \pi_{h}v,p(U_{h})-P_{h}\bigr) \\ &\qquad{} + \int_{t}^{T} c\bigl(\tau,t;\pi_{h}v,p(U_{h}) (\tau)\bigr)\,d\tau-\hat{h}(\pi _{h}v,{P}_{h}) \\ &\quad=-\biggl(\frac{\partial}{\partial t}\bigl(p(U_{h})-P_{h}\bigr),v- \pi_{h}v\biggr)-a\biggl(v-\pi_{h}v,\frac{\partial}{\partial t} \bigl(p(U_{h})-P_{h}\bigr)\biggr) \\ &\qquad{}+d\bigl(v- \pi_{h}v,p(U_{h})-P_{h}\bigr) + \int_{t}^{T}c\bigl(\tau,t;v-\pi_{h}v,p(U_{h}) (\tau)\bigr)\,d\tau-\hat{h}(v-\pi _{h}v,{P}_{h}) \\ &\qquad{}+ \bigl(y(U_{h})-\hat{Y}_{h},\pi_{h}v\bigr)+d( \pi_{h}v,\tilde{P}_{h}-P_{h}) \\ &\quad=-\biggl(\frac{\partial}{\partial t}p(U_{h}),v-\pi_{h}v\biggr)-a \biggl(v-\pi_{h}v,\frac{\partial}{\partial t}p(U_{h})\biggr)+d \bigl(v-\pi_{h}v,p(U_{h})\bigr) \\ &\qquad{}+ \int_{t}^{T}c\bigl(\tau,t;v-\pi_{h}v,p(U_{h}) (\tau )\bigr)\,d\tau +\biggl(\frac{\partial}{\partial t}{P}_{h},v-\pi_{h}v\biggr)+a \biggl(v-\pi_{h}v,\frac{\partial}{\partial t}{P}_{h}\biggr) \\ &\qquad{}-d(v- \pi_{h}v,{P}_{h}) -\hat{h}(v-\pi_{h}v,P_{h})+ \bigl(y(U_{h})-\hat{Y}_{h},\pi_{h}v\bigr) +d(\pi_{h}v,\tilde{P}_{h}-P_{h}) \\ &\quad=\bigl(y(U_{h})-z_{d},v-\pi_{h}v\bigr)+ \bigl(y(U_{h})-\hat{Y}_{h},\pi_{h}v\bigr)+ d( \pi_{h}v,\tilde{{P}}_{h}-P_{h}) \\ &\qquad{} +\biggl(\frac{\partial}{\partial t}P_{h},v-\pi_{h}v\biggr)+a \biggl(v-\pi_{h}v,\frac{\partial}{\partial t}P_{h}\biggr)-d(v- \pi_{h}v,P_{h})-\hat{h}(v-\pi_{h}v,{P}_{h}) \\ &\quad=\bigl(y(U_{h})-\hat{Y}_{h}+\hat{Y}_{h}-z_{d},v- \pi_{h}v\bigr)+\bigl(y(U_{h})-\hat{Y}_{h}, \pi_{h}v\bigr)+ d(\pi_{h}v,\tilde{{P}}_{h}-P_{h}) \\ &\qquad{} +\biggl(\frac{\partial}{\partial t}P_{h},v-\pi_{h}v\biggr)+a \biggl(v-\pi_{h}v,\frac{\partial}{\partial t}P_{h}\biggr)-d(v- \pi_{h}v,P_{h})-\hat{h}(v-\pi_{h}v,{P}_{h}) \\ &\quad=\bigl(y(U_{h})-\hat{Y}_{h},v\bigr)+( \hat{Y}_{h}-z_{d},v-\pi_{h}v)+\biggl( \frac{\partial }{\partial t}P_{h},v-\pi_{h}v\biggr)+a\biggl(v- \pi_{h}v,\frac{\partial}{\partial t}P_{h}\biggr) \\ &\qquad{} -d(v-\pi_{h}v,\tilde{P}_{h})+d(v,\tilde{P}_{h}-{P}_{h})- \hat{h}(v-\pi _{h}v,{P}_{h}) \\ &\quad=\bigl(y(U_{h})-\hat{Y}_{h},v\bigr)+d(v, \tilde{{P}}_{h}-{P}_{h})+(\hat{Y}_{h}-z_{d},v- \pi _{h}v)+\biggl(\frac{\partial}{\partial t}P_{h},v- \pi_{h}v\biggr) \\ &\qquad{} +a\biggl(v-\pi_{h}v,\frac{\partial}{\partial t}P_{h}\biggr)-d(v- \pi_{h}v,\tilde{{P}}_{h}) -\hat{h}(v-\pi_{h}v,{P}_{h}). \end{aligned}$$
(4.15)
From the definition of a, d, ĥ, the Green formula, and (4.15) we have
$$\begin{aligned} &{-}\biggl(\frac{\partial}{\partial t}\bigl(p(U_{h})-P_{h} \bigr),v\biggr)-a\biggl(v,\frac{\partial}{\partial t}\bigl(p(U_{h})-P_{h} \bigr)\biggr)+d\bigl(v,p(U_{h})-P_{h}\bigr) \\ &\qquad{} + \int_{t}^{T}c\bigl(\tau,t;v,p(U_{h}) ( \tau)\bigr)\,d\tau-\hat{h}(v,{P}_{h}) \\ &\quad=\bigl(y(U_{h})-\hat{Y}_{h},v\bigr)+d(v, \tilde{{P}}_{h}-{P}_{h})+\sum_{\tau}\int _{\tau}\Biggl(\hat{Y}_{h}-z_{d}+ \frac{\partial}{\partial t}P_{h}-\operatorname{div}\biggl(A^{*}\nabla\frac{\partial}{\partial t}P_{h} \biggr) \\ &\qquad{} +\operatorname{div}\bigl(D^{*}\nabla\tilde{{P}}_{h}\bigr) +\sum _{m=i}^{N}k_{m}\operatorname{div}\bigl(C^{*}(t_{m},t_{i-1}) \nabla{{P}}_{h}^{m}\bigr) \Biggr) (v-\pi_{h}v) \\ &\qquad{} +\sum_{l} \int_{l}\Biggl[\biggl(A^{*}\nabla\frac{\partial P_{h}}{\partial t}\biggr)\cdot n-\bigl(D^{*}\nabla\tilde{{P}}_{h}\bigr)\cdot n -\sum _{m=i}^{N}k_{m}\bigl(C^{*}(t_{m},t_{i-1}) \nabla{{P}}_{h}^{m}\bigr)\cdot n\Biggr](v- \pi_{h}v) \\ &\quad=I_{1}+\cdots+I_{4}. \end{aligned}$$
(4.16)
Let
$$ G=-\biggl(\frac{\partial}{\partial t}\bigl(p(U_{h})-P_{h} \bigr),v\biggr)-a\biggl(v,\frac{\partial}{\partial t}\bigl(p(U_{h})-P_{h} \bigr)\biggr)+d\bigl(v,p(U_{h})-P_{h}\bigr). $$
(4.17)
Then we have
$$ G=I_{1}+\cdots+I_{4}+\hat{h}(v,{P}_{h})- \int_{t}^{T}c\bigl(\tau,t;v,p(U_{h}) ( \tau)\bigr)\,d\tau, $$
(4.18)
where, for all \(t\in[t_{i-1},t_{i}]\),
$$\begin{aligned} J={}&\hat{h}(v,{P}_{h})- \int_{t}^{T}c\bigl(\tau,t;v,p(U_{h}) ( \tau)\bigr)\,d\tau \\ ={}&\sum_{m=i}^{N} \int_{t_{m-1}}^{t_{m}} \bigl(\bigl(C(t_{m},t_{i-1})-C( \tau ,t)\bigr)\nabla v,\nabla P_{h}^{m} \bigr)\,d\tau+\sum _{m=i}^{N} \int_{t_{m-1}}^{t_{m}}c\bigl(\tau ,t;v,P_{h}^{m}-P_{h} \bigr)\,d\tau \\ &{} +\sum_{m=i}^{N} \int_{t_{m-1}}^{t_{m}}c\bigl(\tau ,t;v,\bigl(P_{h}-P(U_{h}) \bigr) (\tau)\bigr)\,d\tau + \int_{t_{i-1}}^{t}c\bigl(\tau,t;v,p(U_{h})-P_{h} \bigr)\,d\tau \\ &{} + \int_{t_{i-1}}^{t}c(\tau,t;v,P_{h}- \tilde{P}_{h})\,d\tau + \int_{t_{i-1}}^{t}c(\tau,t;v,\tilde{P}_{h})\,d\tau \\ ={}&J_{1}+\cdots+J_{6}. \end{aligned}$$
(4.19)
Then from Theorem 3.1, Lemma 4.1, and 4.2, setting \(v=p(U_{h})-P_{h}\) in (4.16)-(4.19), we have
$$ G \geq -\frac{1}{2}\frac{d}{dt}\bigl\| p(U_{h})-P_{h} \bigr\| _{0,\Omega}^{2}-\frac{1}{2}\frac{d}{dt}a\bigl( p(U_{h})-P_{h},p(U_{h})-P_{h}\bigr)+c \bigl\| p(U_{h})-P_{h}\bigr\| _{1,\Omega}^{2}. $$
(4.20)
Then from [23], for all \(t\in[t_{i-1},t_{i}]\), using
$$\tilde{P}_{h}-P_{h}=\frac{t-t_{i-1}}{k_{i}} \bigl(P_{h}^{i-1}-P_{h}^{i}\bigr), $$
we have the following estimates:
$$\begin{aligned} &I_{1}+I_{2} \leq\bigl\| y(U_{h})- \hat{Y}_{h}\bigr\| _{0,\Omega}^{2}+C| \tilde{{P}}_{h}-{P}_{h}| _{1,\Omega}^{2}+\varepsilon\bigl\| p(U_{h})-P_{h} \bigr\| _{1,\Omega}^{2}, \\ &I_{3}+I_{4} \leq C \Biggl\{ \sum _{\tau}h_{\tau}^{2} \int_{\tau}\Biggl(\hat{Y}_{h}-z_{d}+ \frac{\partial}{\partial t}P_{h}-\operatorname{div}\biggl(A^{*}\nabla\frac{\partial}{\partial t}P_{h} \biggr) +\operatorname{div}\bigl(D^{*}\nabla\tilde{{P}}_{h}\bigr) \\ &\hphantom{I_{3}+I_{4} \leq}{} +\sum_{m=i}^{N}k_{m} \operatorname{div}\bigl(C^{*}(t_{m},t_{i-1}) \nabla{{P}}_{h}^{m}\bigr) \Biggr)^{2} +\sum _{l} h_{l} \int_{l}\Biggl[\biggl(A^{*}\nabla\frac{\partial}{\partial t}P_{h} \biggr)\cdot n \\ &\hphantom{I_{3}+I_{4} \leq}{} -\bigl(D^{*}\nabla\tilde{{P}}_{h}\bigr)\cdot n-\sum _{m=i}^{N}k_{m} \bigl(C^{*}(t_{m},t_{i-1})\nabla{{P}}_{h}^{m} \bigr)\cdot n\Biggr]^{2} \Biggr\} +\varepsilon\bigl\| p(U_{h})-P_{h} \bigr\| _{1,\Omega}^{2}, \\ &J_{1} \leq C\sum_{m=i}^{N} \int_{t_{m-1}}^{t_{m}}\bigl\| \bigl(C^{*}(t_{m},t_{i-1})-C^{*}( \tau,t)\bigr)\nabla P_{h}^{m}\bigr\| _{0,\Omega}^{2}\,d\tau+\varepsilon \bigl\| p(U_{h})-P_{h} \bigr\| _{1,\Omega}^{2}, \\ &J_{2} \leq C\sum_{m=i}^{N} \frac{k_{m}}{3}\bigl| P_{h}^{m-1}-P_{h}^{m}\bigr| _{1,\Omega}^{2}+\varepsilon \bigl\| p(U_{h})-P_{h} \bigr\| _{1,\Omega}^{2}, \\ &J_{3} \leq C \int_{t_{i-1}}^{T}\bigl\| \bigl(p(U_{h})-P_{h} \bigr) (\tau) \bigr\| _{1,\Omega}^{2}\,d\tau+\varepsilon \bigl\| p(U_{h})-P_{h} \bigr\| _{1,\Omega}^{2}, \\ & J_{4} \leq C \int_{t_{i-1}}^{t}\bigl\| p(U_{h})-P_{h} \bigr\| _{1,\Omega}^{2}\,d\tau+\varepsilon \bigl\| p(U_{h})-P_{h} \bigr\| _{1,\Omega}^{2}, \\ &J_{5}\leq C \int_{t_{i-1}}^{t_{i}}| P_{h}- \tilde{P}_{h}| _{1,\Omega}^{2}\,d\tau+ \varepsilon \bigl\| p(U_{h})-P_{h} \bigr\| _{1,\Omega}^{2} \\ &\hphantom{J_{5}}\leq C\frac{k_{i}}{3}\bigl| P_{h}^{i-1}-P_{h}^{i}\bigr| _{1,\Omega}^{2}+\varepsilon \bigl\| p(U_{h})-P_{h} \bigr\| _{1,\Omega}^{2}, \\ &J_{6} \leq Ck_{i}^{2} | \tilde{{P}}_{h}| _{1,\Omega}^{2}+\varepsilon \bigl\| p(U_{h})-P_{h} \bigr\| _{1,\Omega}^{2}. \end{aligned}$$
(4.21)
Setting ε small enough, from (4.20)-(4.21) we obtain:
$$\begin{aligned} &{-}\frac{1}{2}\frac{d}{dt}\bigl\| p(U_{h})-P_{h} \bigr\| _{0,\Omega }^{2}-\frac{1}{2}\frac{d}{dt}a\bigl( p(U_{h})-P_{h},p(U_{h})-P_{h}\bigr)+c \bigl\| p(U_{h})-P_{h}\bigr\| _{1,\Omega}^{2} \\ &\quad\leq C \Biggl\{ \bigl\| y(U_{h})-\hat{Y}_{h} \bigr\| _{0,\Omega}^{2}+ | \tilde{P}_{h}-{P}_{h}| _{1,\Omega}^{2}+\sum_{\tau}h_{\tau}^{2} \int_{\tau}\Biggl(\hat{Y}_{h}-z_{d}+ \frac{\partial}{\partial t}P_{h} \\ &\qquad{} -\operatorname{div}\biggl(A^{*}\nabla\frac{\partial}{\partial t}P_{h}\biggr) +\operatorname{div}\bigl(D^{*} \nabla\tilde{{P}}_{h}\bigr) +\sum_{m=i}^{N}k_{m}\operatorname{div} \bigl(C^{*}(t_{m},t_{i-1})\nabla{{P}}_{h}^{m} \bigr) \Biggr)^{2} \\ &\qquad{} +\sum_{l} h_{l} \int_{l}\Biggl[\biggl(A^{*}\nabla\frac{\partial}{\partial t}P_{h} \biggr)\cdot n-\bigl(D^{*}\nabla\tilde{{P}}_{h}\bigr)\cdot n-\sum _{m=i}^{N}k_{m} \bigl(C^{*}(t_{m},t_{i-1})\nabla{{P}}_{h}^{m} \bigr)\cdot n\Biggr]^{2} \\ &\qquad{} +\sum_{m=i}^{N} \int_{t_{m-1}}^{t_{m}}\bigl\| \bigl(C^{*}(t_{m},t_{i-1})-C^{*}( \tau,t)\bigr)\nabla P_{h}^{m}\bigr\| _{0,\Omega}^{2}\,d\tau \\ &\qquad{} +\sum _{m=i}^{N}\frac{k_{m}}{3}\bigl| P_{h}^{m-1}-P_{h}^{m}\bigr| _{1,\Omega}^{2}+ \int_{t_{i-1}}^{T}\bigl\| p(U_{h})-P_{h} \bigr\| _{1,\Omega}^{2}\,d\tau+k_{i}^{2}| \tilde{{P}}_{h} | _{1,\Omega}^{2} \Biggr\} . \end{aligned}$$
(4.22)
Then integrating (4.22) from 0 to T in time, we have
$$\begin{aligned} &\bigl\| p(U_{h})-P_{h} \bigr\| _{L^{\infty}(0,T;L^{2}(\Omega))}^{2}+\bigl\| p(U_{h})-P_{h} \bigr\| _{L^{2}(0,T;H^{1}(\Omega))}^{2} \\ &\quad\leq C \Biggl\{ \sum_{i=1}^{N}k_{i} \Biggl(\sum_{\tau}h_{\tau}^{2} \int_{\tau}\Biggl(\hat{Y}_{h}-z_{d}+ \frac{\partial}{\partial t}P_{h}-\operatorname{div}\biggl(A^{*}\nabla\frac{\partial}{\partial t}P_{h} \biggr) +\operatorname{div}\bigl(D^{*}\nabla\tilde{{P}}_{h}\bigr) \\ &\qquad{} +\sum_{m=i}^{N}k_{m}\operatorname{div} \bigl(C^{*}(t_{m},t_{i-1})\nabla{{P}}_{h}^{m} \bigr) \Biggr)^{2} +\sum_{l} h_{l} \int_{l}\Biggl[\biggl(A^{*}\nabla\frac{\partial}{\partial t}P_{h} \biggr)\cdot n-\bigl(D^{*}\nabla\tilde{{P}}_{h}\bigr)\cdot n \\ &\qquad{} -\sum_{m=i}^{N}k_{m} \bigl(C^{*}(t_{m},t_{i-1})\nabla{{P}}_{h}^{m} \bigr)\cdot n\Biggr]^{2} \Biggr)+ \sum_{i=1}^{N} \frac{k_{i}}{3}\bigl| P_{h}^{i-1}-P_{h}^{i}\bigr| _{1,\Omega}^{2}+\sum_{i=1}^{N}k_{i}^{3}| \tilde{{P}}_{h} | _{1,\Omega}^{2} \\ &\qquad{} +\sum_{i=1}^{N} \int_{t_{i-1}}^{t_{i}}\Biggl(\sum _{m=i}^{N} \int _{t_{m-1}}^{t_{m}}\bigl\| \bigl(C^{*}(t_{m},t_{i-1})-C^{*}( \tau,t)\bigr)\nabla P_{h}^{m}\bigr\| _{0,\Omega}^{2}\,d\tau \Biggr)\,dt \\ &\qquad{} + \sum_{i=1}^{N}k_{i}\sum _{m=i}^{N}\frac{k_{m}}{3}\bigl| P_{h}^{m-1}-P_{h}^{m}\bigr| _{1,\Omega}^{2}+\sum_{i=1}^{N} \bigl\| y(U_{h})-\hat{Y}_{h}\bigr\| _{L^{2}(t_{i-1},{t_{i}};L^{2}(\Omega))}^{2} \Biggr\} \\ &\quad=C\Biggl(\eta_{2}^{2}+\cdots+\eta_{6}^{2}+ \sum_{i=1}^{N}\bigl\| y(U_{h})- \hat{Y}_{h}\bigr\| _{L^{2}(t_{i-1},{t_{i}};L^{2}(\Omega))}^{2}\Biggr). \end{aligned}$$
(4.23)
In the same way, setting \(v=\frac{\partial}{\partial t}(p(U_{h})-P_{h})\) in (4.16)-(4.19), we obtain:
$$ \biggl\| \frac{\partial}{\partial t}\bigl(p(U_{h})-P_{h} \bigr)\biggr\| _{L^{2}(0,T;H^{1}(\Omega))}^{2} \leq C\Biggl(\eta_{2}^{2}+ \cdots+\eta_{6}^{2}+\sum_{i=1}^{N} \bigl\| y(U_{h})-\hat{Y}_{h}\bigr\| _{L^{2}(t_{i-1},{t_{i}};L^{2}(\Omega))}^{2} \Biggr), $$
(4.24)
where
$$\begin{aligned} &\sum_{i=1}^{N} \bigl\| y(U_{h})-\hat{Y}_{h}\bigr\| _{L^{2}(t_{i-1},{t_{i}};L^{2}(\Omega))}^{2}\\ &\quad\leq C\Biggl(\sum_{i=1}^{N}\bigl\| y(U_{h})-{Y}_{h}\bigr\| _{L^{2}(t_{i-1},{t_{i}};L^{2}(\Omega))}^{2}+\sum _{i=1}^{N}\| {Y}_{h}- \hat{Y}_{h}\|_{L^{2}(t_{i-1},{t_{i}};L^{2}(\Omega))}^{2}\Biggr)\\ &\quad\leq C\Biggl(\bigl\| y(U_{h})-{Y}_{h}\bigr\| _{L^{2}(0,T;L^{2}(\Omega))}^{2}+ \sum_{i=1}^{N}\frac{k_{i}}{3}\bigl| Y_{h}^{i-1}-Y_{h}^{i}\bigr| _{1,\Omega}^{2}\Biggr). \end{aligned}$$
Next, we estimate \(y(U_{h})-Y_{h}\). Let
$$\begin{aligned} \bigl\langle Q(U_{h}),v\bigr\rangle ={}&\biggl(\frac{\partial}{\partial t} \bigl(y(U_{h})-Y_{h}\bigr),v\biggr)+a\biggl( \frac{\partial}{\partial t}\bigl(y(U_{h})-Y_{h}\bigr),v\biggr)+d \bigl(y(U_{h})-Y_{h},v\bigr)\\ &{}+ \int_{0}^{t}c\bigl(t,\tau;y(U_{h}) ( \tau),v\bigr)\,d\tau-\tilde{{g}}({Y}_{h},v). \end{aligned}$$
We obviously have
$$\begin{aligned} &\biggl(\frac{\partial}{\partial t} \bigl(y(U_{h})-Y_{h}\bigr),v\biggr)+a\biggl( \frac{\partial}{\partial t}\bigl(y(U_{h})-Y_{h}\bigr),v\biggr)+d \bigl(y(U_{h})-Y_{h},v\bigr)-\tilde{{g}}({Y}_{h},v) \\ &\qquad{}+ \int_{0}^{t}c\bigl(t,\tau;y(U_{h}) ( \tau),v\bigr)\,d\tau \\ &\quad=(f-\hat{f},v)+d(\hat{Y}_{h}-{Y}_{h},v)+ \Biggl\{ \sum _{\tau}\int_{\tau}\Biggl(\hat {f}+U_{h}- \frac{\partial}{\partial t}Y_{h}+\operatorname{div}\biggl(A\nabla\frac{\partial}{\partial t}Y_{h} \biggr) \\ &\qquad{}+\operatorname{div}(D\nabla\hat{Y}_{h})+\sum_{m=1}^{i}k_{m}\operatorname{div} \bigl(C(t_{i},t_{m-1})\nabla Y_{h}^{m-1} \bigr) \Biggr) \\ &\qquad{}-\sum_{l} \int_{l}\Biggl[\biggl(A\nabla\frac{\partial}{\partial t}Y_{h} \biggr)\cdot n +(D\nabla\hat{Y}_{h})\cdot n \\ &\qquad{}+\sum_{m=1}^{i}k_{m} \bigl(C(t_{i},t_{m-1})\nabla Y_{h}^{m-1} \bigr)\cdot n\Biggr] \Biggr\} (v-\pi_{h}v) \\ &\quad =L_{1}+L_{2}+L_{3}. \end{aligned}$$
(4.25)
Therefore,
$$\begin{aligned} &\biggl(\frac{\partial}{\partial t} \bigl(y(U_{h})-Y_{h}\bigr),v\biggr)+a\biggl( \frac{\partial}{\partial t}\bigl(y(U_{h})-Y_{h}\bigr),v\biggr)+d \bigl(y(U_{h})-Y_{h},v\bigr) \\ &\quad =L_{1}+L_{2}+L_{3} +\tilde{{g}}({Y}_{h},v)- \int_{0}^{t}c\bigl(t,\tau;y(U_{h}) ( \tau),v\bigr)\,d\tau. \end{aligned}$$
(4.26)
For \(t\in[t_{i-1},t_{i}]\), let
$$\begin{aligned} &\tilde{{g}}({Y}_{h},v)- \int_{0}^{t}c\bigl(t,\tau;y(U_{h}) ( \tau),v\bigr)\,d\tau \\ &\quad =\sum_{m=1}^{i} \int_{t_{m-1}}^{t_{m}} \bigl(\bigl(C(t_{i},t_{m-1})-C(t, \tau)\bigr)\nabla Y_{h}^{m-1},\nabla v \bigr)\,d\tau+\sum _{m=1}^{i} \int_{t_{m-1}}^{t_{m}}c\bigl(t,\tau ;Y_{h}^{m-1}-Y_{h},v \bigr)\,d\tau \\ &\qquad{} + \sum_{m=1}^{i} \int_{t_{m-1}}^{t_{m}}c\bigl(t,\tau;Y_{h}-y(U_{h}),v \bigr)\,d\tau + \int_{t}^{t_{i}}c\bigl(t,\tau;y(U_{h})-Y_{h},v \bigr)\,d\tau \\ &\qquad{} + \int_{t}^{t_{i}}c(t,\tau;Y_{h}- \hat{Y}_{h},v)\,d\tau+ \int_{t}^{t_{i}}c(t,\tau ;\hat{Y}_{h},v)\,d\tau. \end{aligned}$$
(4.27)
Then we set \(v=y(U_{h})-Y_{h}\) and \(v=\frac{\partial}{\partial t}(y(U_{h})-Y_{h})\) in (4.26) and (4.27), respectively. Similarly to (4.23), integrating in time from 0 to T, we obtain:
$$\begin{aligned} &\bigl\| y(U_{h})-Y_{h} \bigr\| _{L^{\infty}(0,T;L^{2}(\Omega))}^{2}+\bigl\| y(U_{h})-Y_{h} \bigr\| _{L^{2}(0,T;H^{1}(\Omega))}^{2} \\ &\quad \leq C \Biggl\{ \sum_{i=1}^{N}k_{i} \Biggl(\sum_{\tau}h_{\tau}^{2} \int_{\tau}\Biggl(\hat{f}+U_{h}- \frac{\partial}{\partial t}Y_{h}+\operatorname{div}\biggl(A\nabla\frac{\partial}{\partial t}Y_{h} \biggr)+\operatorname{div}(D\nabla\hat{Y}_{h}) \\ &\qquad{} +\sum_{m=1}^{i}k_{m}\operatorname{div} \bigl(C(t_{i},t_{m-1})\nabla Y_{h}^{m-1} \bigr) \Biggr)^{2} +\sum_{l}h_{l} \int_{l}\Biggl[\biggl(A\nabla\frac{\partial }{\partial t}Y_{h} \biggr)\cdot n+(D\nabla\hat{Y}_{h})\cdot n \\ &\qquad{} +\sum_{m=1}^{i}k_{m} \bigl(C(t_{i},t_{m-1})\nabla Y_{h}^{m-1} \bigr)\cdot n\Biggr]^{2} \Biggr) + \| f- \hat{f}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))} \\ &\qquad{} +\sum_{i=1}^{N} \int_{t_{i-1}}^{t_{i}} \Biggl(\sum _{m=1}^{i} \int_{t_{m-1}}^{t_{m}}\bigl\| \bigl(C(t_{i},t_{m-1})-C(t, \tau)\bigr)\nabla Y_{h}^{m-1}\bigr\| _{0,\Omega}^{2}\,d\tau \Biggr)\,dt \\ &\qquad{} + \sum_{i=1}^{N}\frac{k_{i}}{3}\bigl| Y_{h}^{i-1}-Y_{h}^{i}\bigr| _{1,\Omega}^{2} +\sum_{i=1}^{N}k_{i} \sum_{m=1}^{i}\frac{k_{m}}{3}\bigl| Y_{h}^{m-1}-Y_{h}^{m}\bigr| _{1,\Omega}^{2} \\ &\qquad{} +\sum_{i=1}^{N}k_{i}^{3}| \hat{{Y}}_{h} | _{1,\Omega}^{2}+\bigl\| Y_{h}(x,0)-y_{0}(x) \bigr\| ^{2}_{0,\Omega} \Biggr\} \\ &\quad =C\bigl(\eta_{7}^{2}+\cdots+\eta_{13}^{2} \bigr) \end{aligned}$$
(4.28)
and
$$ \biggl\| \frac{\partial}{\partial t} \bigl(y(U_{h})-Y_{h}\bigr)\biggr\| _{L^{2}(0,T;H^{1}(\Omega))}^{2} \leq C\bigl(\eta_{7}^{2}+\cdots+\eta_{13}^{2} \bigr). $$
(4.29)
Combing (4.23)-(4.24) and (4.28)-(4.29), we can prove Lemma 4.4. This completes the proof. □
Lemma 4.5
Suppose that
\((y,p,u)\)
and
\((y(U_{h}),p(U_{h}))\)
are the solutions of (2.4) and (4.3), respectively. Then we have:
$$\begin{aligned}& \begin{aligned}[b] &\bigl\| y-y(U_{h}) \bigr\| ^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}+\bigl\| y-y(U_{h})\bigr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))}\\ &\quad{} + \biggl\| \frac{\partial}{\partial t}\bigl(y-y(U_{h})\bigr)\biggr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \leq C \| u-U_{h}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}, \end{aligned} \end{aligned}$$
(4.30)
$$\begin{aligned}& \begin{aligned}[b] &\bigl\| p-p(U_{h}) \bigr\| ^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}+ \bigl\| p-p(U_{h})\bigr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))}\\ &\quad{} + \biggl\| \frac{\partial}{\partial t}\bigl(p-p(U_{h})\bigr)\biggr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \leq C \bigl\| y-y(U_{h})\bigr\| ^{2}_{L^{2}(0,T;L^{2}(\Omega))}. \end{aligned} \end{aligned}$$
(4.31)
Proof
Letting \(w=y-y(U_{h})\) and \(w=\frac{\partial}{\partial t}(y-y(U_{h}))\) in (4.8), we have
$$ \begin{aligned} &\bigl\| y-y(U_{h}) \bigr\| ^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}+\bigl\| y-y(U_{h})\bigr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \leq C \| u-U_{h}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}, \\ & \biggl\| \frac{\partial}{\partial t}\bigl(y-y(U_{h})\bigr)\biggr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \leq C \| u-U_{h}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}. \end{aligned} $$
(4.32)
We can similarly obtain the estimate for \(p-p(U_{h})\). This completes the proof. □
We now present an upper bound of the a posteriori error estimates.
Theorem 4.6
Suppose that
\((y,p,u)\)
and
\((Y_{h},P_{h},U_{h})\)
are the solutions of (2.4) and (3.7), respectively. Then we have:
$$\begin{aligned} &\| u-U_{h} \|^{2}_{L^{2}(0,T;L^{2}(\Omega))}+\| y-Y_{h}\|^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}+ \| y-Y_{h}\|^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad{}+ \biggl\| \frac{\partial}{\partial t}(y-Y_{h})\biggr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +\| p-P_{h}\|^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}+\| p-P_{h} \|^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad{}+ \biggl\| \frac{\partial}{\partial t}(p-P_{h})\biggr\| _{L^{2}(0,T;H^{1}(\Omega))} \leq C\sum _{i=1}^{13}\eta_{i}^{2}, \end{aligned}$$
(4.33)
where
\(\eta_{i}\) (\(i=1,\ldots,13\)) is defined in (4.5) and (4.12)
Proof
From Lemmas 4.3 and 4.4 we have
$$ \| u-U_{h}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}\leq C \eta_{1}^{2}+ C\bigl\| P_{h}-p(U_{h})\bigr\| ^{2}_{L^{2}(0,T;L^{2}(\Omega))}\leq C\sum _{i=1}^{13}\eta_{i}^{2}. $$
(4.34)
By the triangle inequality we have
$$\begin{aligned} &\| y-Y_{h} \|^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}+\| y-Y_{h}\|^{2}_{L^{2}(0,T;H^{1}(\Omega))} + \biggl\| \frac{\partial}{\partial t}(y-Y_{h})\biggr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad \leq\bigl\| y-y(U_{h})\bigr\| ^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}+\bigl\| y-y(U_{h})\bigr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\qquad{} + \biggl\| \frac{\partial}{\partial t}\bigl(y-y(U_{h})\bigr)\biggr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +\bigl\| y(U_{h})-Y_{h}\bigr\| ^{2}_{L^{\infty}(0,T;L^{2}(\Omega))} \\ &\qquad{} +\bigl\| y(U_{h})-Y_{h}\bigr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))}+ \biggl\| \frac{\partial}{\partial t}\bigl(y(U_{h})-Y_{h}\bigr) \biggr\| ^{2}_{L^{2}(0,T;H^{-1}(\Omega))}. \end{aligned}$$
(4.35)
We have the same results for \(p-P_{h}\). Then from (4.35), Lemmas 4.3-4.5, combined with (4.34), we obtain (4.33). This completes the proof. □
Lower bound estimates
For convenience, we first define the average integration of v in element τ, \(\bar{v}|_{\tau}=\frac{\int_{\tau}v}{\int_{\tau}1}\), and define \(R_{\tau}\), \(R_{l}\), \(T_{\tau}\), \(T_{l}\) as follows:
$$\begin{aligned} &R_{\tau}= \hat{f}+U_{h}-\frac{\partial}{\partial t}Y_{h}+\operatorname{div}\biggl(A\nabla \frac{\partial}{\partial t}Y_{h}\biggr)+\operatorname{div}(D\nabla\hat{Y}_{h}) +\sum _{m=1}^{i}k_{m}\operatorname{div} \bigl(C(t_{i},t_{m-1})\nabla Y_{h}^{m-1} \bigr), \\ &R_{l}=\Biggl[\biggl(A\nabla\frac{\partial}{\partial t}Y_{h}\biggr) \cdot n+(D\nabla\hat{Y}_{h})\cdot n+\sum_{m=1}^{i}k_{m} \bigl(C(t_{i},t_{m-1})\nabla Y_{h}^{m-1} \bigr)\cdot n\Biggr], \\ &T_{\tau}= \hat{Y}_{h}-z_{d}+\frac{\partial}{\partial t}P_{h}-\operatorname{div} \biggl(A^{*}\nabla\frac{\partial}{\partial t}P_{h}\biggr)+\operatorname{div}\bigl(D^{*}\nabla\tilde {{P}}_{h}\bigr)+\sum_{m=i}^{N}k_{m}\operatorname{div} \bigl(C^{*}(t_{m},t_{i-1})\nabla{{P}}_{h}^{m} \bigr), \\ &T_{l}= \Biggl[\biggl(A^{*}\nabla\frac{\partial}{\partial t}P_{h} \biggr)\cdot n-\bigl(D^{*}\nabla\tilde{{P}}_{h}\bigr) \cdot n-\sum _{m=i}^{N}k_{m} \bigl(C^{*}(t_{m},t_{i-1})\nabla{{P}}_{h}^{m} \bigr)\cdot n \Biggr]. \end{aligned}$$
In order to present the main conclusion in this subsection, we first use the standard bubble function technique in [11, 26] and give the following lemma.
Lemma 4.7
Suppose that
\((y,p,u)\)
and
\((Y_{h},P_{h},U_{h})\)
are the solutions of (2.4) and (3.7), respectively. Then we have:
$$\begin{aligned} \eta_{7}^{2}+ \eta_{9}^{2}+\eta_{11}^{2} \leq{}& C \Biggl\{ \| f-\hat{f}\|^{2}_{L^{2}(0,T;H^{-1}(\Omega))} +\| u-U_{h} \|^{2}_{L^{2}(0,T;L^{2}(\Omega))} \\ &{} +\| y-Y_{h}\|^{2}_{L^{2}(0,T;H^{1}(\Omega))}+\biggl\| \frac{\partial}{\partial t}(y-Y_{h})\biggr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))}+\sum _{i=1}^{N}k_{i}^{3}| \hat{{Y}}_{h} | _{1,\Omega}^{2} \\ &{} + \sum _{i=1}^{N} \int_{t_{i-1}}^{t_{i}} \Biggl(\sum _{m=1}^{i} \int _{t_{m-1}}^{t_{m}}\bigl\| \bigl(C(t_{i},t_{m-1})-C(t, \tau)\bigr)\nabla Y_{h}^{m-1}\bigr\| _{0,\Omega}^{2}\,d\tau \Biggr)\,dt \\ &{} +\sum_{i=1}^{N}k_{i}\sum _{\tau\in T^{h}} \int_{\tau}h_{\tau}^{2} (R_{\tau}- \bar{R}_{\tau})^{2}+\sum_{i=1}^{N}k_{i} \sum_{l\in \tilde{\varepsilon}^{h}} \int_{l}h_{l} (R_{l}- \bar{R}_{l})^{2} \Biggr\} . \end{aligned}$$
(4.36)
Proof
By using techniques in [23] we prove the lemma in three steps.
(1) First, we estimate \(\frac{k_{i}}{3}| Y_{h}^{i-1}-Y_{h}^{i}| _{1,\Omega}^{2}\).
Let
$$ \eta_{2i}^{2}=\sum _{\tau}h_{\tau}^{2} \int_{\tau}R_{\tau}^{2} +\sum _{l}h_{l} \int_{l}R_{l}^{2}. $$
(4.37)
By using the equalities \(\frac{k_{i}}{3}| Y_{h}^{i-1}-Y_{h}^{i}| _{1,\Omega}^{2}=\int_{t_{i-1}}^{t_{i}}| Y_{h}-\hat{Y}_{h}| _{1,\Omega}^{2}\,dt\) and
$$\begin{aligned} & \biggl(\frac{\partial}{\partial t}(y-Y_{h}),v \biggr)+a\biggl(\frac{\partial}{\partial t}(y-Y_{h}),v\biggr)+d(y-Y_{h},v)+ \int_{0}^{t}c\bigl(t,\tau;y(\tau),v\bigr)\,d\tau- \tilde {g}(Y_{h},v) \\ &\quad=\biggl(\frac{\partial}{\partial t}(y-Y_{h}),v\biggr)+a\biggl( \frac{\partial}{\partial t}(y-Y_{h}),v\biggr)+d(y-Y_{h},v)+ \int_{0}^{t}c\bigl(t,\tau;y(\tau),v\bigr)\,d\tau \\ &\qquad{}-\sum_{m=1}^{i}k_{m}c \bigl(t_{i},t_{m-1}; {Y_{h}^{m-1}}, v \bigr) \\ &\quad=(f+u-\hat{f}-U_{h},v)+( \hat{f}+U_{h},v-\pi_{h}v)-\biggl(\frac{\partial}{\partial t}Y_{h},v- \pi_{h}v\biggr)-a\biggl(\frac{\partial}{\partial t}Y_{h},v- \pi_{h}v\biggr) \\ &\qquad{}-d(\hat{Y}_{h},v-\pi_{h}v)-\sum _{m=1}^{i}k_{m}c\bigl(t_{i},t_{m-1}; {Y_{h}^{m-1}}, v-\pi_{h}v \bigr) -d(Y_{h}-\hat{Y}_{h},v) \end{aligned}$$
(4.38)
we have
$$\begin{aligned} &d(Y_{h}- \hat{Y}_{h},v) \\ &\quad =(f-\hat{f},v)+(u-U_{h},v) + \Biggl\{ \sum _{\tau}\int_{\tau}\Biggl(\hat{f}+U_{h}- \frac{\partial}{\partial t}Y_{h}+\operatorname{div}\biggl(A\nabla\frac{\partial}{\partial t}Y_{h} \biggr)+\operatorname{div}(D\nabla\hat{Y}_{h}) \\ &\qquad{} +\sum_{m=1}^{i}k_{m}\operatorname{div} \bigl(C(t_{i},t_{m-1})\bigr)\nabla Y_{h}^{m-1} \Biggr) -\sum_{l} \int_{l}\Biggl[\biggl(A\nabla\frac{\partial}{\partial t}Y_{h} \biggr)\cdot n+(D\nabla\hat{Y}_{h})\cdot n \\ &\qquad{} +\sum_{m=1}^{i}k_{m} \bigl(C(t_{i},t_{m-1})\bigr)\nabla Y_{h}^{m-1} )\cdot n\Biggr] \Biggr\} (v-\Pi_{h}v) \\ &\qquad{}-\biggl(\frac{\partial}{\partial t}(y-Y_{h}),v \biggr)-a\biggl(\frac{\partial}{\partial t}(y-Y_{h}),v\biggr) \\ &\qquad{} -d(y-Y_{h},v)+\sum _{m=1}^{i}k_{m}c \bigl(t_{i},t_{m-1}; {Y_{h}^{m-1}}, v \bigr) - \int_{0}^{t}c\bigl(t,\tau;y(\tau),v\bigr)\,d\tau. \end{aligned}$$
(4.39)
Then
$$\begin{aligned} &\int_{t_{i-1}}^{t_{i}}d(Y_{h}- \hat{Y}_{h},v)\,dt \\ &\quad\leq C \Biggl\{ \| f-\hat{f}\|^{2}_{L^{2}(t_{i-1},t_{i};H^{-1}(\Omega))} +\| u-U_{h}\|^{2}_{L^{2}(t_{i-1},t_{i};L^{2}(\Omega))} \\ &\qquad{} +k_{i} \Biggl\{ \sum_{\tau\in T^{h} } \int_{\tau}h_{\tau}^{2} \Biggl( \hat{f}+U_{h}-\frac{\partial}{\partial t}Y_{h}+\operatorname{div}\biggl(A\nabla \frac{\partial}{\partial t}Y_{h}\biggr)+\operatorname{div}(D\nabla\hat{Y}_{h})+\operatorname{div}(A \nabla\hat{Y}_{h}) \\ &\qquad{}+\sum_{m=1}^{i}k_{m}\operatorname{div} \bigl(C(t_{i},t_{m-1})\nabla Y_{h}^{m-1} \bigr) \Biggr)^{2} +\sum_{l} \int_{l}h_{l}\Biggl[\biggl(A\nabla \frac{\partial}{\partial t}Y_{h}\biggr)\cdot n+(D\nabla\hat{Y}_{h}) \cdot n \\ &\qquad{}+\sum_{m=1}^{i}k_{m} \bigl(C(t_{i},t_{m-1})\nabla Y_{h}^{m-1} \bigr)\cdot n\Biggr]^{2} \Biggr\} + \| y-Y_{h} \|^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))} \\ &\qquad{} +\biggl\| \frac{\partial}{\partial t}( y-Y_{h})\biggr\| ^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))} \Biggr\} +\varepsilon| v| ^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))} \\ &\qquad{} + \int_{t_{i-1}}^{t_{i}} \Biggl\{ \sum _{m=1}^{i}k_{m}c\bigl(t_{i},t_{m-1}; {Y_{h}^{m-1}}, v \bigr) - \int_{0}^{t}c\bigl(t,\tau;y(\tau),v\bigr)\,d\tau \Biggr\} \,dt. \end{aligned}$$
(4.40)
Letting
$$\begin{aligned} J={}& \int_{t_{i-1}}^{t_{i}} \Biggl\{ \sum _{m=1}^{i}k_{m}c\bigl(t_{i},t_{m-1}; {Y_{h}^{m-1}}, v \bigr) - \int_{0}^{t}c\bigl(t,\tau;y(\tau),v\bigr)\,d\tau \Biggr\} \,dt \\ ={}& \int_{t_{i-1}}^{t_{i}} \Biggl\{ \sum _{m=1}^{i} \int _{t_{m-1}}^{t_{m}}\bigl(\bigl(C(t_{i},t_{m-1})-C(t, \tau)\bigr) \nabla{Y_{h}^{m-1}},\nabla v \bigr)\,d\tau \Biggr\} \,dt \\ &{} + \int_{t_{i-1}}^{t_{i}} \Biggl\{ \sum _{m=1}^{i} \int _{t_{m-1}}^{t_{m}}c\bigl(t,\tau; {Y_{h}^{m-1}}-Y_{h}, v \bigr)\,d\tau \Biggr\} \,dt \\ &{} + \int_{t_{i-1}}^{t_{i}} \Biggl\{ \sum _{m=1}^{i} \int _{t_{m-1}}^{t_{m}}c\bigl(t,\tau; (Y_{h}-y) ( \tau), v \bigr)\,d\tau \Biggr\} \,dt \\ &{}+ \int_{t_{i-1}}^{t_{i}} \int_{t}^{t_{i}}c\bigl(t,\tau; (y-Y_{h}) ( \tau), v \bigr)\,d\tau \,dt + \int_{t_{i-1}}^{t_{i}} \int_{t}^{t_{i}}c \bigl(t,\tau; (Y_{h}- \hat{Y}_{h}) (\tau), v \bigr)\,d\tau \,dt \\ &{} + \int_{t_{i-1}}^{t_{i}} \int_{t}^{t_{i}}c(t,\tau; \hat{Y}_{h}, v )\,d\tau \,dt \\ ={}&J_{1}+\cdots+J_{6}, \end{aligned}$$
(4.41)
we have
$$\begin{aligned} J_{1}\leq{}& C \int_{t_{i-1}}^{t_{i}} \Biggl\{ \sum _{m=1}^{i} \int_{t_{m-1}}^{t_{m}} \bigl\| \bigl(C(t_{i},t_{m-1})-C(t, \tau)\bigr) \nabla{Y_{h}^{m-1}}\bigr\| ^{2}_{0,\Omega}\,d\tau \Biggr\} \,dt \\ &{} +\varepsilon| v| ^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))}, \\ J_{2} \leq{}& Ck_{i}\sum_{m=1}^{i} \frac{k_{m}}{3}\bigl| Y_{h}^{m-1}-Y_{h}^{m}\bigr| _{1,\Omega}^{2}+\varepsilon| v| ^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))}, \\ J_{3} ={}& \int_{t_{i-1}}^{t_{i}} \biggl\{ \int_{0}^{t_{i}}c\bigl(t,\tau; (Y_{h}-y) ( \tau), v \bigr)\,d\tau \biggr\} \,dt \\ ={}& \int_{0}^{t_{i}} \biggl( \int_{t_{i-1}}^{t_{i}}c\bigl(\tau,t; (Y_{h}-y) (t), v(\tau) \bigr)\,d\tau \biggr)\,dt\\ & \leq Ck_{i}\| Y_{h}-y \|^{2}_{L^{2}(0,t_{i};H^{1}(\Omega))}+ \varepsilon| v| ^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))}, \\ J_{4} \leq{}& Ck_{i} \| y-Y_{h} \|^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))} +\varepsilon| v| ^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))}, \\ J_{5}\leq{}& C \frac{k_{i}^{2}}{3}\bigl| Y_{h}^{i-1}-Y_{h}^{i}\bigr| _{1,\Omega}^{2}+\varepsilon| v| ^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))}, \\ J_{6}\leq{}& Ck_{i}^{3}| \hat{Y}_{h}| _{1,\Omega}^{2}+\varepsilon| v| ^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))}. \end{aligned}$$
(4.42)
Setting \(v=Y_{h}-\hat{Y}_{h}\) in (4.39), from (4.38)-(4.42) we obtain:
$$\begin{aligned} \frac{k_{i}}{3}\bigl| Y_{h}^{i-1}-Y_{h}^{i}\bigr| _{1,\Omega}^{2} ={}& \int_{t_{i-1}}^{t_{i}}| Y_{h}- \hat{Y}_{h}| _{1,\Omega}^{2}\,dt \\ \leq{}& C \int_{t_{i-1}}^{t_{i}}d(Y_{h}- \hat{Y}_{h},Y_{h}-\hat{Y}_{h})\,dt \\ \leq{}& C \Biggl\{ \| f-\hat{f}\|^{2}_{L^{2}(t_{i-1},t_{i};H^{-1}(\Omega))} +\| u-U_{h}\|^{2}_{L^{2}(t_{i-1},t_{i};L^{2}(\Omega))}+k_{i} \eta^{2}_{2i} \\ &{} +\| y-Y_{h}\|^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))} +\biggl\| \frac{\partial}{\partial t}( y-Y_{h})\biggr\| ^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))} \\ &{} + \int_{t_{i-1}}^{t_{i}} \Biggl\{ \sum _{m=1}^{i} \int_{t_{m-1}}^{t_{m}} \bigl\| \bigl(C(t_{i},t_{m-1})-C(t, \tau)\bigr) \nabla{Y_{h}^{m-1}}\bigr\| ^{2}_{0,\Omega}\,d\tau \Biggr\} \,dt \\ &{} +k_{i}\sum_{m=1}^{i} \frac{k_{m}}{3}\bigl| Y_{h}^{m-1}-Y_{h}^{m}\bigr| _{1,\Omega}^{2} \\ &{}+k_{i}^{3}| \hat{Y}_{h}| _{1,\Omega }^{2}+k_{i}\|y-Y_{h}\|_{L^{2}(0,t_{i};H^{1}(\Omega))} \Biggr\} . \end{aligned}$$
(4.43)
(2) Using the bubble function methods in [11, 26] and the techniques in [23], we have:
$$\begin{aligned} k_{i} \eta_{2i}^{2} \leq{}& C \Biggl\{ \| f-\hat{f}\|^{2}_{L^{2}(t_{i-1},t_{i};H^{-1}(\Omega))} +\| u-U_{h}\|^{2}_{L^{2}(t_{i-1},t_{i};L^{2}(\Omega))} \\ &{} +\| y-Y_{h}\|^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))} +\biggl\| \frac{\partial}{\partial t}( y-Y_{h})\biggr\| ^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))} \\ &{} + \int_{t_{i-1}}^{t_{i}} \Biggl(\sum _{m=1}^{i} \int_{t_{m-1}}^{t_{m}} \bigl\| \bigl(C(t_{i},t_{m-1})-C(t, \tau)\bigr) \nabla{Y_{h}^{m-1}}\bigr\| ^{2}_{0,\Omega}\,d\tau \Biggr)\,dt \\ &{} +k_{i}\sum_{m=1}^{i} \frac{k_{m}}{3}\bigl| Y_{h}^{m-1}-Y_{h}^{m}\bigr| _{1,\Omega}^{2}+k_{i}^{3}| \hat{Y}_{h}| _{1,\Omega }^{2}+k_{i}\| y-Y_{h}\|^{2}_{L^{2}(0,t_{i};H^{1}(\Omega))} \Biggr\} \\ &{} +C_{\delta}k_{i} \biggl(\sum_{\tau\in T^{h}} \int_{\tau}h_{\tau}^{2} ({R}_{\tau}- \bar{{R}}_{\tau})^{2}+\sum_{l\in \tilde{\varepsilon}^{h}} \int_{l}h_{l} ( R_{l}- \bar{R}_{l})^{2} \biggr) . \end{aligned}$$
(4.44)
(3) From (4.43) and (4.44) we obtain:
$$\begin{aligned} & \frac{k_{i}}{3}\bigl| Y_{h}^{i-1}-Y_{h}^{i}\bigr| _{1,\Omega}^{2} \\ &\quad\leq C \Biggl\{ \| f-\hat{f}\|^{2}_{L^{2}(t_{i-1},t_{i};H^{-1}(\Omega))} +\| u-U_{h}\|^{2}_{L^{2}(t_{i-1},t_{i};L^{2}(\Omega))} \\ &\qquad{} +\| y-Y_{h}\|^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))} +\biggl\| \frac{\partial}{\partial t}( y-Y_{h})\biggr\| ^{2}_{L^{2}(t_{i-1},t_{i};H^{1}(\Omega))} \\ &\qquad{} + \int_{t_{i-1}}^{t_{i}} \Biggl(\sum _{m=1}^{i} \int_{t_{m-1}}^{t_{m}} \bigl\| \bigl(C(t_{i},t_{m-1})-C(t, \tau)\bigr) \nabla{Y_{h}^{m-1}}\bigr\| ^{2}_{0,\Omega}\,d\tau \Biggr)\,dt \\ &\qquad{} +k_{i}\sum _{m=1}^{i}\frac{k_{m}}{3}\bigl| Y_{h}^{m-1}-Y_{h}^{m}\bigr| _{1,\Omega}^{2}+k_{i}^{3}| \hat{Y}_{h}| _{1,\Omega }^{2}+k_{i}\| y-Y_{h}\|^{2}_{L^{2}(0,t_{i};H^{1}(\Omega))} \\ &\qquad{} + k_{i} \biggl(\sum_{\tau\in T^{h}} \int_{\tau}h_{\tau}^{2} ({R}_{\tau}- \bar{R}_{\tau})^{2}+\sum_{l\in \tilde{\varepsilon}^{h}} \int_{l}h_{l} ( R_{l}- \bar{R}_{l})^{2} \biggr) \Biggr\} . \end{aligned}$$
(4.45)
Further, from the Gronwall inequality, combined with (4.44), we have
$$\begin{aligned} &\eta_{7}^{2}+ \eta_{9}^{2}+\eta_{11}^{2} \\ &\quad \leq\sum _{i=1}^{N}k_{i} \eta_{2i}^{2} +(1+T)\sum_{i=1}^{N} \frac{k_{i}}{3}\bigl| Y_{h}^{i-1}-Y_{h}^{i}\bigr| _{1,\Omega}^{2} \\ &\quad\leq C \Biggl\{ \| f-\hat{f}\|^{2}_{L^{2}(0,T;H^{-1}(\Omega))} +\| u-U_{h}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))} +\| y-Y_{h} \|^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\qquad{} +\biggl\| \frac{\partial}{\partial t}(y-Y_{h})\biggr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))}+\sum _{i=1}^{N}k_{i}^{3}| \hat{{Y}}_{h} | _{1,\Omega}^{2} \\ &\qquad{}+ \sum _{i=1}^{N} \int_{t_{i-1}}^{t_{i}} \Biggl(\sum _{m=1}^{i} \int _{t_{m-1}}^{t_{m}}\bigl\| \bigl(C(t_{i},t_{m-1})-C(t, \tau)\bigr)\nabla Y_{h}^{m-1}\bigr\| _{0,\Omega}^{2}\,d\tau \Biggr)\,dt \\ &\qquad{} +\sum_{i=1}^{N}k_{i} \biggl( \sum_{\tau\in T^{h}} \int_{\tau}h_{\tau}^{2} ( R_{\tau}- \bar{R}_{\tau})^{2}+ \sum_{l\in \tilde{\varepsilon}^{h}} \int_{l}h_{l} ( R_{l}- \bar{R}_{l})^{2} \biggr) \Biggr\} . \end{aligned}$$
(4.46)
This completes the proof. □
Similarly, we can also obtain the following lemma.
Lemma 4.8
Suppose that
\((y,p,u)\)
and
\((Y_{h},P_{h},U_{h})\)
are the solutions of (2.4) and (3.7), respectively. Then we have:
$$\begin{aligned} \eta_{2}^{2}+ \eta_{3}^{2}+\eta_{5}^{2} \leq{}& C \Biggl\{ \| y-Y_{h}\|^{2}_{L^{2}(0,T;H^{1}(\Omega))}+\| p-P_{h} \|^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &{} +\biggl\| \frac{\partial}{\partial t}(p-P_{h}) \biggr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))}+\sum_{i=1}^{N}k_{i}^{3}| \tilde{{P}}_{h} | _{1,\Omega}^{2} \\ &{} + \sum_{i=1}^{N} \int_{t_{i-1}}^{t_{i}} \Biggl(\sum _{m=i}^{N} \int _{t_{m-1}}^{t_{m}}\bigl\| \bigl(C^{*}(t_{m},t_{i-1})-C^{*}(t, \tau)\bigr)\nabla P_{h}^{m}\bigr\| _{0,\Omega}^{2}\,d\tau \Biggr)\,dt \\ &{} +\sum_{i=1}^{N}k_{i}\sum _{\tau\in T^{h}} \int_{\tau}h_{\tau}^{2} (T_{\tau}- \bar{T}_{\tau})^{2} +\sum_{i=1}^{N}k_{i} \sum_{l\in \tilde{\varepsilon}^{h}} \int_{l}h_{l} (T_{l}- \bar{T}_{l})^{2} \Biggr\} . \end{aligned}$$
(4.47)
Lemma 4.9
Suppose that
\((y,p,u)\)
and
\((Y_{h},P_{h},U_{h})\)
are the solutions of (2.4) and (3.7), respectively. Then we have:
$$ \eta_{1}^{2} \leq C \bigl\{ \|u-U_{h}\|^{2}_{L^{2}(0, T;L^{2}(\Omega))}+\| p -P_{h}\| _{L^{2}(0,T;L^{2}(\Omega))}^{2} \bigr\} . $$
(4.48)
Therefore, from Lemmas 4.3-4.8 we can easily obtain the following theorem.
Theorem 4.10
Suppose that
\((y,p,u)\)
and
\((Y_{h},P_{h},U_{h})\)
are the solutions of (2.4) and (3.7), respectively. Then we have the following estimates:
$$\begin{aligned} &\eta_{1}^{2}+ \eta_{2}^{2}+\eta_{3}^{2}+ \eta_{5}^{2}+\eta_{7}^{2}+ \eta_{9}^{2}+\eta_{11}^{2} \\ &\quad\leq C \Biggl\{ \| u-U_{h}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}+\| y-Y_{h}\|^{2}_{L^{2}(0,T;H^{1}(\Omega))}+ \| y-Y_{h} \|^{2}_{L^{\infty}(0,T;L^{2}(\Omega))} \\ &\qquad{} + \biggl\| \frac{\partial}{\partial t}(y-Y_{h})\biggr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))} + \| p-P_{h}\|^{2}_{L^{2}(0,T;H^{1}(\Omega))}+\| p-P_{h} \|^{2}_{L^{\infty}(0,T;L^{2}(\Omega))} \\ &\qquad{} +\biggl\| \frac{\partial}{\partial t}(p-P_{h})\biggr\| ^{2}_{L^{2}(0,T;H^{1}(\Omega))}+\| f-\hat{f}\|^{2}_{L^{2}(0,T;H^{-1}(\Omega))} \\ &\qquad{} +\sum_{i=1}^{N} \int_{t_{i-1}}^{t_{i}} \Biggl(\sum _{m=1}^{i} \int _{t_{m-1}}^{t_{m}}\bigl\| \bigl(C(t_{i},t_{m-1})-C(t, \tau)\bigr)\nabla Y_{h}^{m-1}\bigr\| _{0,\Omega}^{2}\,d\tau \Biggr)\,dt \\ &\qquad{} +\sum_{i=1}^{N} \int_{t_{i-1}}^{t_{i}} \Biggl(\sum _{m=i}^{N} \int _{t_{m-1}}^{t_{m}}\bigl\| \bigl(C^{*}(t_{m},t_{i-1})-C^{*}(t, \tau)\bigr)\nabla P_{h}^{m}\bigr\| _{0,\Omega}^{2}\,d\tau \Biggr)\,dt \\ &\qquad{} +\sum_{i=1}^{N}k_{i}^{3}| \hat{{Y}}_{h} | _{1,\Omega}^{2}+\sum _{i=1}^{N}k_{i}^{3}| \tilde{{P}}_{h} | _{1,\Omega}^{2} +\sum _{i=1}^{N}k_{i}\sum _{\tau\in T^{h}} \int_{\tau}h_{\tau}^{2} (R_{\tau}- \bar{R}_{\tau})^{2} \\ &\qquad{} +\sum_{i=1}^{N}k_{i}\sum _{\tau\in T^{h}} \int_{\tau}h_{\tau}^{2} (T_{\tau}- \bar{T}_{\tau})^{2} \\ &\qquad{}+\sum_{i=1}^{N}k_{i} \sum_{l\in \tilde{\varepsilon}^{h}} \int_{l}h_{l} (R_{l} - \bar{R}_{l})^{2} +\sum _{i=1}^{N}k_{i}\sum _{l\in \tilde{\varepsilon}^{h}} \int_{l}h_{l} (T_{l} - \bar{T}_{l} )^{2} \Biggr\} . \end{aligned}$$
(4.49)