To construct the Crank–Nicolson scheme, we define the following function:
$$\begin{aligned} H\bigl(Du_{h}^{n}\bigr)=\frac{1}{4} \bigl(1- \bigl\vert Du_{h}^{n} \bigr\vert ^{2} \bigr)^{2}, \end{aligned}$$
(35)
where \(H(Du_{h}^{n})\) is a double well potential function. Obviously, \(H'(Du_{h})=|Du_{h}|^{2} Du_{h}-Du_{h}\).
The fully discrete finite element scheme for problem (1) is: Find \(u_{h}^{n}\in U_{h}\ (n=1,2,\ldots, N)\) such that
$$\begin{aligned} \textstyle\begin{cases} (\partial _{t}u_{h}^{n},v_{h})+(\alpha ^{n-\frac{1}{2}} D^{2}u_{h}^{n- \frac{1}{2}},D^{2}v_{h}) + ( \frac{H(Du_{h}^{n})-H(Du_{h}^{n-1})}{Du_{h}^{n}-Du_{h}^{n-1}},Dv_{h} )=0, \\ \forall v_{h}\in U_{h}, \\ (u(0)-u_{h}^{0},v_{h})=0, \quad \forall v_{h}\in U_{h}, \end{cases}\displaystyle \end{aligned}$$
(36)
where N is a given positive integer, \(\Delta t=T/N\) denotes the time step size, \(t_{n}=n\Delta t\) and
$$\begin{aligned} &\partial _{t} u_{h}^{n}= \bigl(u_{h}^{n}-u_{h}^{n-1}\bigr)/\Delta t, \\ &\alpha ^{n-\frac{1}{2}}=\alpha \bigl(x,t^{n-\frac{1}{2}}\bigr), \\ &u_{h}^{n-\frac{1}{2}}=\bigl(u_{h}^{n}+u_{h}^{n-1} \bigr)/2, \\ &t^{n-\frac{1}{2}}=\bigl(t^{n}+t^{n-1}\bigr)/2. \end{aligned}$$
Firstly, we analyze the boundedness of the fully discrete scheme (36). It is a key step for deducing the error estimate.
Theorem 4.1
Let\(u_{h}^{0}\in H^{2}_{0}(I)\cap W^{1,4}(I)\), then there exists a unique solution\(u_{h}^{n}\)for problem (36) such that
$$\begin{aligned} \bigl\Vert u_{h}^{n} \bigr\Vert _{2}\leq C \bigl\Vert u_{h}^{0} \bigr\Vert _{2},\quad 0\leq t\leq T, \end{aligned}$$
(37)
whereCis a positive constant depending on\(\alpha (x,t)\)andT, independent ofhand Δt.
Proof
A direct calculation gives
$$\begin{aligned} &\frac{H(Du_{h}^{n})-H(Du_{h}^{n-1})}{Du_{h}^{n}-Du_{h}^{n-1}} \\ &\quad =\frac{1}{4}\bigl(Du_{h}^{n}+Du_{h}^{n-1} \bigr) \bigl( \bigl\vert Du_{h}^{n} \bigr\vert ^{2}+ \bigl\vert Du_{h}^{n-1} \bigr\vert ^{2}\bigr) -\frac{1}{2}\bigl(Du_{h}^{n}+Du_{h}^{n-1} \bigr). \end{aligned}$$
(38)
Setting \(v_{h}=u_{h}^{n}+u_{h}^{n-1}\) in (36), we get
$$\begin{aligned} &\frac{1}{\Delta t}\bigl( \bigl\Vert u_{h}^{n} \bigr\Vert ^{2}- \bigl\Vert u_{h}^{n-1} \bigr\Vert ^{2}\bigr)+ \frac{s}{2} \bigl\Vert D^{2}u_{h}^{n}+D^{2}u_{h}^{n-1} \bigr\Vert ^{2} \\ & \quad{}+\frac{1}{4}\bigl(\bigl(Du_{h}^{n}+Du_{h}^{n-1} \bigr)^{2}, \bigl\vert Du_{h}^{n} \bigr\vert ^{2}+ \bigl\vert Du_{h}^{n-1} \bigr\vert ^{2}\bigr) \leq \frac{1}{2} \bigl\Vert Du_{h}^{n}+Du_{h}^{n-1} \bigr\Vert ^{2}. \end{aligned}$$
(39)
Using Cauchy’s inequality, we obtain
$$\begin{aligned} &\frac{1}{\Delta t}\bigl( \bigl\Vert u_{h}^{n} \bigr\Vert ^{2}- \bigl\Vert u_{h}^{n-1} \bigr\Vert ^{2}\bigr)+ \frac{s}{2} \bigl\Vert D^{2}u_{h}^{n}+D^{2}u_{h}^{n-1} \bigr\Vert ^{2} \\ &\quad \leq \frac{s}{4} \bigl\Vert D^{2}u_{h}^{n}+D^{2}u_{h}^{n-1} \bigr\Vert ^{2}+\frac{1}{4s} \bigl\Vert u_{h}^{n}+u_{h}^{n-1} \bigr\Vert ^{2}. \end{aligned}$$
Further, we derive
$$\begin{aligned} \frac{1}{\Delta t}\bigl( \bigl\Vert u_{h}^{n} \bigr\Vert ^{2}- \bigl\Vert u_{h}^{n-1} \bigr\Vert ^{2}\bigr)\leq \frac{1}{4s} \bigl\Vert u_{h}^{n}+u_{h}^{n-1} \bigr\Vert ^{2}\leq \frac{1}{2s}\bigl( \bigl\Vert u_{h}^{n} \bigr\Vert ^{2}+ \bigl\Vert u_{h}^{n-1} \bigr\Vert ^{2}\bigr). \end{aligned}$$
(40)
Letting \(\gamma =\frac{1}{2s}\), we have
$$\begin{aligned} \bigl\Vert u_{h}^{n} \bigr\Vert ^{2}\leq \frac{1+\gamma \Delta t}{1-\gamma \Delta t} \bigl\Vert u_{h}^{n-1} \bigr\Vert ^{2} \leq \cdots \leq \biggl( \frac{1+\gamma \Delta t}{1-\gamma \Delta t} \biggr)^{n} \bigl\Vert u_{h}^{0} \bigr\Vert ^{2}. \end{aligned}$$
(41)
It is easy to show
$$\begin{aligned} \biggl(\frac{1+\gamma \Delta t}{1-\gamma \Delta t} \biggr)^{n} = \biggl(1+\frac{2\gamma \Delta t}{1-\gamma \Delta t} \biggr)^{ \frac{1-\gamma \Delta t}{2\gamma \Delta t}\cdot \frac{2\gamma n\Delta t}{1-\gamma \Delta t}}. \end{aligned}$$
If Δt is small enough, we conclude
$$\begin{aligned} \bigl\Vert u_{h}^{n} \bigr\Vert ^{2}\leq C \bigl\Vert u_{h}^{0} \bigr\Vert ^{2}. \end{aligned}$$
(42)
Choosing \(v_{h}=\partial _{t}u_{h}^{n}\) in (36), we have
$$\begin{aligned} & \bigl\Vert \partial _{t} u_{h}^{n} \bigr\Vert ^{2}+\frac{1}{2\Delta t} \bigl(\alpha \bigl(x,t^{n- \frac{1}{2}}\bigr) \bigl( \bigl\vert D^{2}u_{h}^{n} \bigr\vert ^{2}- \bigl\vert D^{2}u_{h}^{n-1} \bigr\vert ^{2} \bigr),1\bigr) \\ &\quad{}+\frac{1}{\Delta t}\bigl(H\bigl(Du_{h}^{n}\bigr)-H \bigl(Du_{h}^{n-1}\bigr),1\bigr)=0. \end{aligned}$$
(43)
Then we get
$$\begin{aligned} &\biggl(\frac{1}{2}\alpha \bigl(x,t^{n-\frac{1}{2}}\bigr) \bigl\vert D^{2}u_{h}^{n} \bigr\vert ^{2}+H\bigl(Du_{h}^{n}\bigr),1 \biggr) \\ &\quad \leq \biggl(\frac{1}{2}\alpha \bigl(x,t^{n-\frac{1}{2}}\bigr) \bigl\vert D^{2}u_{h}^{n-1} \bigr\vert ^{2}+H\bigl(Du_{h}^{n-1}\bigr),1\biggr). \end{aligned}$$
(44)
Define the function
$$\begin{aligned} G\bigl(u_{h}^{n},t^{n-\frac{1}{2}}\bigr)= \biggl(\frac{1}{2} \alpha \bigl(x,t^{n-\frac{1}{2}}\bigr) \bigl\vert D^{2}u_{h}^{n} \bigr\vert ^{2}+H \bigl(Du_{h}^{n}\bigr),1 \biggr), \end{aligned}$$
(45)
then \(G(u_{h}^{n},t^{n-\frac{1}{2}})\geq 0\). By (44) and (45), we have
$$\begin{aligned} G\bigl(u_{h}^{n},t^{n-\frac{1}{2}}\bigr)\leq G \bigl(u_{h}^{n-1},t^{n-\frac{3}{2}}\bigr) +\frac{1}{2} \bigl(\bigl( \alpha \bigl(x,t^{n-\frac{1}{2}}\bigr)-\alpha \bigl(x,t^{n-\frac{3}{2}} \bigr)\bigr) \bigl\vert D^{2}u_{h}^{n-1} \bigr\vert ^{2},1\bigr). \end{aligned}$$
With the differential mean value theorem and the boundedness of variable coefficient, we obtain
$$\begin{aligned} G\bigl(u_{h}^{n},t^{n-\frac{1}{2}} \bigr) \leq {}& G\bigl(u_{h}^{n-1},t^{n-\frac{3}{2}}\bigr) + \frac{\Delta t}{2} \biggl\vert \frac{\partial \alpha }{\partial t}(x,\xi ) \biggr\vert \bigl\Vert D^{2}u_{h}^{n-1} \bigr\Vert ^{2} \\ \leq {}& G\bigl(u_{h}^{n-1},t^{n-\frac{3}{2}}\bigr)+ \frac{M_{1}\Delta t}{2} \bigl\Vert D^{2}u_{h}^{n-1} \bigr\Vert ^{2}, \end{aligned}$$
where \(t^{n-\frac{3}{2}}<\xi <t^{n-\frac{1}{2}}\). Then
$$\begin{aligned} G\bigl(u_{h}^{n},t^{n-\frac{1}{2}}\bigr)-G \bigl(u_{h}^{n-1},t^{n-\frac{3}{2}}\bigr)\leq \frac{M_{1}\Delta t}{2} \bigl\Vert D^{2}u_{h}^{n-1} \bigr\Vert ^{2}. \end{aligned}$$
Taking the sum over n, we get
$$\begin{aligned} G\bigl(u_{h}^{n},t^{n-\frac{1}{2}}\bigr)-G \bigl(u_{h}^{1},t^{\frac{1}{2}}\bigr)\leq \frac{M_{1}\Delta t}{2} \sum_{j=2}^{n-1} \bigl\Vert D^{2}u_{h}^{j} \bigr\Vert ^{2}. \end{aligned}$$
(46)
It is obvious that
$$\begin{aligned} G\bigl(u_{h}^{n},t^{n-\frac{1}{2}}\bigr)\geq \frac{s}{2} \bigl\Vert D^{2}u_{h}^{n} \bigr\Vert ^{2}+\bigl(H\bigl(Du_{h}^{n}\bigr),1 \bigr) \geq \frac{s}{2} \bigl\Vert D^{2}u_{h}^{n} \bigr\Vert ^{2}. \end{aligned}$$
Therefore we know
$$\begin{aligned} G\bigl(u_{h}^{n},t^{n-\frac{1}{2}}\bigr)-G \bigl(u_{h}^{1},t^{\frac{1}{2}}\bigr)\leq \frac{M_{1}\Delta t}{s} \sum_{j=2}^{n-1} G\bigl(u_{h}^{j},t^{j-\frac{1}{2}} \bigr). \end{aligned}$$
Based on (44) and \(u_{h}^{0}\in H^{2}_{0}(I)\cap W^{1,4}(I)\), we have
$$\begin{aligned} G\bigl(u_{h}^{1},t^{\frac{1}{2}} \bigr)&= \biggl(\frac{1}{2}\alpha \bigl(x,t^{\frac{1}{2}}\bigr) \bigl\vert D^{2}u_{h}^{1} \bigr\vert ^{2}+H\bigl(Du_{h}^{1}\bigr),1 \biggr) \\ &\leq \biggl(\frac{1}{2}\alpha \bigl(x,t^{\frac{1}{2}}\bigr) \bigl\vert D^{2}u_{h}^{0} \bigr\vert ^{2}+H\bigl(Du_{h}^{0}\bigr),1 \biggr) \leq C \bigl(u_{h}^{0}\bigr), \end{aligned}$$
where \(C(u_{h}^{0})\) is a constant depending on \(u_{h}^{0}\). Then
$$\begin{aligned} G\bigl(u_{h}^{n},t^{n-\frac{1}{2}}\bigr)\leq C\bigl(u_{h}^{0}\bigr)+\frac{M_{1}\Delta t}{s} \sum _{j=2}^{n-1} G\bigl(u_{h}^{n},t^{n-\frac{1}{2}} \bigr). \end{aligned}$$
(47)
Using discrete Gronwall’s inequality, we derive
$$\begin{aligned} G\bigl(u_{h}^{n},t^{n-\frac{1}{2}}\bigr)\leq C, \qquad C=C\bigl(u_{h}^{0},s,M_{1},T\bigr). \end{aligned}$$
(48)
Based on (48), it is easy to see
$$\begin{aligned} \bigl\Vert D^{2}u_{h}^{n} \bigr\Vert \leq C \bigl\Vert D^{2}u^{0}_{h} \bigr\Vert . \end{aligned}$$
(49)
We also know
$$\begin{aligned} \bigl\Vert Du_{h}^{n} \bigr\Vert ^{2}\leq \frac{1}{2}\bigl( \bigl\Vert u_{h}^{n} \bigr\Vert ^{2}+ \bigl\Vert D^{2}u_{h}^{n} \bigr\Vert ^{2}\bigr). \end{aligned}$$
By (42) and (49), we obtain (37). The proof is completed. □
Next, we give the error estimate in \(L^{2}\) norm.
Theorem 4.2
Let\(u^{n}\)be the solution to problem (5), \(u_{h}^{n}\)be the solution to the fully discrete scheme (36), \(u(0)\in H^{4}(I)\), \(u_{t}\in L^{2}(0,T;H^{4}(I))\cap L^{2}(0,T;W^{1,4}(I)) \), \(u_{ttt}\in L^{2}(0,T;L^{2}(I))\)and\(u_{h}^{0}\in U_{h}\)satisfying
$$\begin{aligned} \bigl\Vert u(0)-u_{h}^{0} \bigr\Vert \leq Ch^{4} \bigl\Vert u(0) \bigr\Vert _{4}. \end{aligned}$$
(50)
Then we have the following error estimate:
$$\begin{aligned} \bigl\Vert u^{n}-u_{h}^{n} \bigr\Vert \leq C\bigl((\Delta t)^{2}+h^{3}\bigr), \end{aligned}$$
(51)
whereCis a positive constant depending on\(\alpha (x,t)\)andT, independent of mesh sizeh.
Proof
Denote \(u_{t}^{n}=u_{t}(x,t^{n})\) and \(u^{n}=u(x,t^{n})\). Setting \(t=t^{n-1}\) and \(t=t^{n}\) in (5), respectively, we obtain
$$\begin{aligned} & \biggl(\frac{u_{t}^{n}+u_{t}^{n-1}}{2},v_{h} \biggr)+ \biggl( \frac{\alpha (x,t^{n})D^{2}u^{n}+\alpha (x,t^{n-1})D^{2}u^{n-1}}{2},D^{2}v_{h} \biggr) \\ &\quad{}+ \biggl( \frac{ \vert Du^{n} \vert ^{2}Du^{n}+ \vert Du^{n-1} \vert ^{2}Du^{n-1}-Du^{n}-Du^{n-1}}{2},Dv_{h} \biggr)=0. \end{aligned}$$
(52)
Denote
$$\begin{aligned} &\varPhi \bigl(D^{2}u^{n},D^{2}u^{n-1},D^{2}u_{h}^{n-\frac{1}{2}} \bigr) \\ &\quad =\frac{\alpha (x,t^{n})D^{2}u^{n}+\alpha (x,t^{n-1})D^{2}u^{n-1}}{2} - \alpha \bigl(x,t^{n-\frac{1}{2}}\bigr)D^{2}u_{h}^{n-\frac{1}{2}} \end{aligned}$$
(53)
and
$$\begin{aligned} &F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \\ &\quad =\frac{ \vert Du^{n} \vert ^{2}Du^{n}+ \vert Du^{n-1} \vert ^{2}Du^{n-1}-Du^{n}-Du^{n-1}}{2} - \frac{H(Du_{h}^{n})-H(Du_{h}^{n-1})}{Du_{h}^{n}-Du_{h}^{n-1}}. \end{aligned}$$
(54)
It follows from (52)–(54) and (36) that
$$\begin{aligned} & \biggl(\frac{u_{t}^{n}+u_{t}^{n-1}}{2}-\partial _{t} u_{h}^{n},v_{h} \biggr) + \bigl( \varPhi \bigl(D^{2}u^{n},D^{2}u^{n-1},D^{2}u_{h}^{n-\frac{1}{2}} \bigr),D^{2}v_{h} \bigr) \\ &\quad{}+ \bigl(F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr),Dv_{h} \bigr)=0. \end{aligned}$$
(55)
Let \(\rho ^{n}=u^{n}-R_{h} u^{n}\) and \(\theta ^{n}=R_{h} u^{n}-u_{h}^{n}\), then \(u^{n}-u_{h}^{n}=\rho ^{n}+\theta ^{n}\). It is clear to get
$$\begin{aligned} \frac{u_{t}^{n}+u_{t}^{n-1}}{2}-\partial _{t} u_{h}^{n}={}& \frac{u_{t}^{n}+u_{t}^{n-1}}{2}-\partial _{t} u^{n}+\partial _{t} u^{n}- \partial _{t} u_{h}^{n} \\ ={}&\frac{u_{t}^{n}+u_{t}^{n-1}}{2}-\partial _{t} u^{n}+\partial _{t}\bigl(u^{n}-R_{h}u^{n}+R_{h}u^{n}-u_{h}^{n} \bigr) =\partial _{t}\theta ^{n}-r^{n}, \end{aligned}$$
(56)
where
$$\begin{aligned} r^{n}=\partial _{t}R_{h}u^{n}- \partial _{t} u^{n}+\partial _{t} u^{n}- \frac{u_{t}(t_{j})+u_{t}(t_{j-1})}{2}. \end{aligned}$$
An easy calculation gives
$$\begin{aligned} &\varPhi \bigl(D^{2}u^{n},D^{2}u^{n-1},D^{2}u_{h}^{n-\frac{1}{2}} \bigr) \\ &\quad =\frac{1}{2}\bigl(\bigl(\alpha \bigl(x,t^{n}\bigr)-\alpha \bigl(x,t^{n-\frac{1}{2}}\bigr)\bigr)D^{2}u^{n} +\bigl( \alpha \bigl(x,t^{n-1}\bigr)-\alpha \bigl(x,t^{n-\frac{1}{2}}\bigr) \bigr)D^{2}u^{n-1} \\ &\qquad{}+\alpha \bigl(x,t^{n-\frac{1}{2}}\bigr) \bigl(D^{2}u^{n}+D^{2}u^{n-1}-D^{2}u_{h}^{n}-D^{2}u_{h}^{n-1} \bigr)\bigr) \\ &\quad=\frac{1}{2}\bigl(\bigl(\alpha \bigl(x,t^{n}\bigr)-\alpha \bigl(x,t^{n-\frac{1}{2}}\bigr)\bigr)D^{2}u^{n} +\bigl( \alpha \bigl(x,t^{n-1}\bigr)-\alpha \bigl(x,t^{n-\frac{1}{2}}\bigr) \bigr)D^{2}u^{n-1} \\ &\qquad{}+\alpha \bigl(x,t^{n-\frac{1}{2}}\bigr) \bigl(D^{2}\theta ^{n}+D^{2}\theta ^{n-1}+D^{2} \rho ^{n}+D^{2}\rho ^{n-1}\bigr)\bigr). \end{aligned}$$
Using Taylor’s theorem, we have
$$\begin{aligned} \alpha \bigl(x,t^{n}\bigr)=\alpha \bigl(x,t^{n-\frac{1}{2}}\bigr)+ \frac{\Delta t}{2} \frac{\partial \alpha }{\partial t}\bigl(x,t^{n-\frac{1}{2}}\bigr) + \frac{(\Delta t)^{2}}{6}\frac{\partial ^{2}\alpha }{\partial ^{2} t}\biggl(x,t^{n- \frac{1}{2}}+\xi _{1} \frac{\Delta t}{2}\biggr),\quad 0< \xi _{1}< 1 \end{aligned}$$
and
$$\begin{aligned} \alpha \bigl(x,t^{n-1}\bigr)=\alpha \bigl(x,t^{n-\frac{1}{2}}\bigr)- \frac{\Delta t}{2} \frac{\partial \alpha }{\partial t}\bigl(x,t^{n-\frac{1}{2}}\bigr) + \frac{(\Delta t)^{2}}{6}\frac{\partial ^{2}\alpha }{\partial ^{2} t}\biggl(x,t^{n- \frac{1}{2}}+\xi _{2} \frac{\Delta t}{2}\biggr), \quad -1< \xi _{2}< 0. \end{aligned}$$
With (4), we get
$$\begin{aligned} &\varPhi \bigl(D^{2}u^{n},D^{2}u^{n-1},D^{2}u_{h}^{n-\frac{1}{2}} \bigr) \\ &\quad =\frac{\Delta t}{2}\frac{\partial \alpha }{\partial t}\bigl(x,t^{n- \frac{1}{2}}\bigr) \bigl(D^{2}u^{n}-D^{2}u^{n-1}\bigr)+O \bigl((\Delta t)^{2}\bigr) \\ &\qquad{}+\frac{1}{2}\alpha \bigl(x,t^{n-\frac{1}{2}}\bigr) \bigl(D^{2} \theta ^{n}+D^{2}\theta ^{n-1}+D^{2} \rho ^{n}+D^{2}\rho ^{n-1}\bigr). \end{aligned}$$
(57)
From (7), we have
$$\begin{aligned} &\bigl(\partial _{t}\theta ^{n},v_{h}\bigr)+\frac{1}{2}\bigl(\alpha ^{n-\frac{1}{2}}\bigl( D^{2} \theta ^{n}+D^{2} \theta ^{n-1}\bigr),D^{2}v_{h}\bigr) \\ &\qquad{}+\frac{\Delta t}{2} \biggl(\frac{\partial \alpha }{\partial t}\bigl(x,t^{n- \frac{1}{2}}\bigr) \bigl(D^{2}u^{n}-D^{2}u^{n-1} \bigr),D^{2}v_{h} \biggr) +\bigl(O\bigl((\Delta t)^{2}\bigr),D^{2}v_{h}\bigr) \\ &\quad =\bigl(r^{n},v_{h}\bigr)-\bigl(F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr),Dv_{h}\bigr). \end{aligned}$$
(58)
Setting \(v_{h}=\theta ^{n}+\theta ^{n-1}\) in (58), we get
$$\begin{aligned} &\frac{1}{\Delta t}\bigl( \bigl\Vert \theta ^{n} \bigr\Vert ^{2}- \bigl\Vert \theta ^{n-1} \bigr\Vert ^{2}\bigr)+ \frac{s}{2} \bigl\Vert D^{2} \theta ^{n}+D^{2}\theta ^{n-1} \bigr\Vert ^{2} \\ &\quad \leq \bigl\Vert r^{n} \bigr\Vert ^{2}+ \frac{1}{4} \bigl\Vert \theta ^{n}+\theta ^{n-1} \bigr\Vert ^{2}+ \bigl\Vert F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert ^{2} \\ &\qquad{}+\frac{1}{4} \bigl\Vert D\theta ^{n}+D\theta ^{n-1} \bigr\Vert ^{2}+\frac{M_{1}\Delta t}{2} \bigl\Vert D^{2}u^{n}-D^{2}u^{n-1} \bigr\Vert \bigl\Vert D^{2}\theta ^{n}+D^{2}\theta ^{n-1} \bigr\Vert \\ &\quad \leq \bigl\Vert r^{n} \bigr\Vert ^{2}+ \bigl\Vert F \bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert ^{2} +\frac{1}{4} \biggl(1+\frac{1}{2s} \biggr) \bigl\Vert \theta ^{n}+\theta ^{n-1} \bigr\Vert ^{2} \\ &\qquad{}+\frac{s}{4} \bigl\Vert D^{2}\theta ^{n}+D^{2} \theta ^{n-1} \bigr\Vert ^{2}+ \frac{(M_{1}\Delta t)^{2}}{2s} \bigl\Vert D^{2}u^{n}-D^{2}u^{n-1} \bigr\Vert ^{2}. \end{aligned}$$
Based on the Newton–Leibniz formula and Hölder’s inequality, we have
$$\begin{aligned} \bigl\vert D^{2}u^{n}-D^{2}u^{n-1} \bigr\vert = \biggl\vert \int _{t_{n-1}}^{t_{n}}D^{2}u_{t}(t)\,dt \biggr\vert \leq \Delta t^{\frac{1}{2}} \biggl( \int _{t_{n-1}}^{t_{n}} \bigl\vert D^{2}u_{t}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}. \end{aligned}$$
Thus
$$\begin{aligned} &\frac{1}{\Delta t}\bigl( \bigl\Vert \theta ^{n} \bigr\Vert ^{2}- \bigl\Vert \theta ^{n-1} \bigr\Vert ^{2}\bigr)+ \frac{s}{2} \bigl\Vert D^{2}\theta ^{n}+D^{2}\theta ^{n-1} \bigr\Vert ^{2} \\ &\quad \leq \bigl\Vert r^{n} \bigr\Vert ^{2}+ \bigl\Vert F \bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert ^{2} +\frac{1}{4} \biggl(1+\frac{1}{2s} \biggr) \bigl\Vert \theta ^{n}+\theta ^{n-1} \bigr\Vert ^{2} \\ &\qquad{}+\frac{s}{4} \bigl\Vert D^{2}\theta ^{n}+D^{2} \theta ^{n-1} \bigr\Vert ^{2}+ \frac{M_{1}^{2}(\Delta t)^{3}}{2s} \int _{t_{n-1}}^{t_{n}} \bigl\Vert D^{2}u_{t}(t) \bigr\Vert ^{2}\,dt. \end{aligned}$$
(59)
A direct calculation gives
$$\begin{aligned} & \bigl\Vert F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert \\ &\quad = \biggl\Vert \frac{1}{2}\bigl(\bigl(Du^{n} \bigr)^{3}+\bigl(Du^{n-1}\bigr)^{3}\bigr)- \frac{1}{4}\bigl(Du^{n}+Du^{n-1}\bigr) \bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr) \\ &\qquad{}+\frac{1}{4}\bigl(Du^{n}+Du^{n-1}\bigr) \bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr) \\ &\qquad{}-\frac{1}{4}\bigl(Du_{h}^{n}+Du_{h}^{n-1} \bigr) \bigl( \bigl\vert Du_{h}^{n} \bigr\vert ^{2}+ \bigl\vert Du_{h}^{n-1} \bigr\vert ^{2}\bigr) \\ &\qquad{}-\frac{1}{2}\bigl(Du^{n}+Du^{n-1}\bigr)+ \frac{1}{2}\bigl(Du_{h}^{n}+Du_{h}^{n-1} \bigr) \biggr\Vert . \end{aligned}$$
From (37) and Sobolev’s embedding theorem, \(H^{2}_{0}(I)\hookrightarrow H^{1,\infty }(I)\), we know
$$\begin{aligned} \bigl\vert Du^{n} \bigr\vert _{\infty }\leq C \bigl\Vert u^{n} \bigr\Vert _{2}\leq C,\qquad \bigl\vert Du_{h}^{n} \bigr\vert _{\infty }\leq C \bigl\Vert u_{h}^{n} \bigr\Vert _{2} \leq C. \end{aligned}$$
(60)
Using Hölder’s inequality, we have
$$\begin{aligned} \bigl\vert Du^{n}-Du^{n-1} \bigr\vert = \biggl\vert \int _{t_{n-1}}^{t_{n}}Du_{t}(t)\,dt \biggr\vert \leq C(\Delta t)^{\frac{1}{2}} \biggl( \int _{t_{n-1}}^{t_{n}} \bigl\vert Du_{t}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}. \end{aligned}$$
(61)
From (60) and (61), we have
$$\begin{aligned} & \biggl\Vert \frac{1}{2}\bigl( \bigl(Du^{n}\bigr)^{3}+\bigl(Du^{n-1} \bigr)^{3}\bigr)-\frac{1}{4}\bigl(Du^{n}+Du^{n-1} \bigr) \bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr) \biggr\Vert \\ &\quad =\frac{1}{4} \bigl\Vert \bigl\vert Du^{n} \bigr\vert ^{2}Du^{n}- \bigl\vert Du^{n} \bigr\vert ^{2}Du^{n-1}-Du^{n} \bigl\vert Du^{n-1} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}Du^{n-1} \bigr\Vert \\ &\quad =\frac{1}{4} \bigl\Vert \bigl(Du^{n}+Du^{n-1} \bigr) \bigl(Du^{n}-Du^{n-1}\bigr)^{2} \bigr\Vert \\ &\quad \leq \frac{1}{4}\bigl( \bigl\vert Du^{n} \bigr\vert _{\infty }+ \bigl\vert Du^{n-1} \bigr\vert _{\infty }\bigr) \bigl\Vert \bigl(Du^{n}-Du^{n-1}\bigr)^{2} \bigr\Vert \\ &\quad \leq C\Delta t \int _{t_{n-1}}^{t_{n}} \bigl\Vert Du_{t}(t) \bigr\Vert ^{2}\,dt. \end{aligned}$$
(62)
Due to (60), we get
$$\begin{aligned} & \bigl\Vert \bigl(Du^{n}+Du^{n-1} \bigr) \bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr) \\ &\qquad{}-\bigl(Du_{h}^{n}+Du_{h}^{n-1}\bigr) \bigl( \bigl\vert Du_{h}^{n} \bigr\vert ^{2}+ \bigl\vert Du_{h}^{n-1} \bigr\vert ^{2}\bigr) \bigr\Vert \\ &\quad= \bigl\Vert \bigl(Du^{n}+Du^{n-1}\bigr) \bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr) \\ &\qquad{}-\bigl(Du_{h}^{n}+Du_{h}^{n-1}\bigr) \bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr) \\ &\qquad{}+\bigl(Du_{h}^{n}+Du_{h}^{n-1}\bigr) \bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr) \\ &\qquad{}-\bigl(Du_{h}^{n}+Du_{h}^{n-1}\bigr) \bigl( \bigl\vert Du_{h}^{n} \bigr\vert ^{2}+ \bigl\vert Du_{h}^{n-1} \bigr\vert ^{2}\bigr) \bigr\Vert \\ &\quad \leq \bigl( \bigl\vert Du^{n} \bigr\vert _{\infty }^{2}+ \bigl\vert Du^{n-1} \bigr\vert _{\infty }^{2}\bigr) \bigl\Vert \bigl(Du^{n}+Du^{n-1}\bigr)-\bigl(Du_{h}^{n}+Du_{h}^{n-1} \bigr) \bigr\Vert \\ &\qquad{}+\bigl( \bigl\vert Du_{h}^{n} \bigr\vert _{\infty }+ \bigl\vert Du_{h}^{n-1} \bigr\vert _{\infty }\bigr) \bigl\Vert \bigl(Du^{n}+Du_{h}^{n} \bigr) \bigl(Du^{n}-Du_{h}^{n}\bigr) \\ &\qquad{} +\bigl(Du^{n-1}+Du_{h}^{n-1}\bigr) \bigl(Du^{n-1}-Du_{h}^{n-1}\bigr) \bigr\Vert \\ &\quad \leq \bigl( \bigl\vert Du^{n} \bigr\vert _{\infty }^{2}+ \bigl\vert Du^{n-1} \bigr\vert _{\infty }^{2}\bigr) \bigl( \bigl\Vert D\theta ^{n}+D \theta ^{n-1} \bigr\Vert + \bigl\Vert D\rho ^{n}+D\rho ^{n-1} \bigr\Vert \bigr) \\ &\qquad{}+\bigl( \bigl\vert Du_{h}^{n} \bigr\vert _{\infty }+ \bigl\vert Du_{h}^{n-1} \bigr\vert _{\infty }\bigr) \bigl( \bigl\vert Du^{n} \bigr\vert _{ \infty }+ \bigl\vert Du_{h}^{n} \bigr\vert _{\infty }+ \bigl\vert Du^{n-1} \bigr\vert _{\infty }+ \bigl\vert Du_{h}^{n-1} \bigr\vert _{ \infty }\bigr) \\ &\qquad{}\times \bigl( \bigl\Vert D\theta ^{n}+D\theta ^{n-1} \bigr\Vert + \bigl\Vert D\rho ^{n}+D\rho ^{n-1} \bigr\Vert \bigr) \\ &\quad \leq C\bigl( \bigl\Vert D\theta ^{n}+D\theta ^{n-1} \bigr\Vert + \bigl\Vert D\rho ^{n}+D\rho ^{n-1} \bigr\Vert \bigr). \end{aligned}$$
(63)
By the triangle inequality, we obtain
$$\begin{aligned} & \bigl\Vert \bigl(Du^{n}+Du^{n-1} \bigr)-\bigl(Du_{h}^{n}+Du_{h}^{n-1} \bigr) \bigr\Vert \\ &\quad = \bigl\Vert D\theta ^{n}+D\rho ^{n}+D\theta ^{n-1}+D\rho ^{n-1} \bigr\Vert \\ &\quad \leq \bigl\Vert D\theta ^{n}+D\theta ^{n-1} \bigr\Vert + \bigl\Vert D\rho ^{n}+D\rho ^{n-1} \bigr\Vert . \end{aligned}$$
(64)
In view of (62)–(64) and (9), we have
$$\begin{aligned} & \bigl\Vert F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert \\ &\quad \leq C \biggl( \bigl\Vert D\theta ^{n}+D\theta ^{n-1} \bigr\Vert + \bigl\Vert D\rho ^{n}+D\rho ^{n-1} \bigr\Vert +\Delta t \int _{t_{n-1}}^{t_{n}} \bigl\Vert Du_{t}(t) \bigr\Vert ^{2}\,dt \biggr) \\ &\quad \leq C \biggl( \bigl\Vert D\theta ^{n}+D\theta ^{n-1} \bigr\Vert +h^{3}+\Delta t \int _{t_{n-1}}^{t_{n}} \bigl\Vert Du_{t}(t) \bigr\Vert ^{2}\,dt \biggr). \end{aligned}$$
Based on the ε-inequality and Hölder’s inequality, we obtain
$$\begin{aligned} & \bigl\Vert F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert ^{2} \\ &\quad \leq C \biggl( \bigl\Vert \theta ^{n}+\theta ^{n-1} \bigr\Vert ^{2}+h^{6} +(\Delta t)^{3} \int _{t_{n-1}}^{t_{n}} \bigl\Vert Du_{t}(t) \bigr\Vert ^{4}\,dt \biggr) +\frac{s}{8} \bigl\Vert D^{2} \theta ^{n}+D^{2}\theta ^{n-1} \bigr\Vert ^{2}. \end{aligned}$$
(65)
Let \(r^{n}=r_{1}^{n}+r_{2}^{n}\), where
$$\begin{aligned} &r_{1}^{j}=\partial _{t}R_{h}u(t_{j})-\partial _{t}u(t_{j})= \frac{1}{\Delta t} \int _{t_{j-1}}^{t_{j}}(R_{h}-I)u_{t} \,dt, \\ &r_{2}^{j}=\partial _{t}u(t_{j})- \frac{u_{t}(t_{j})+u_{t}(t_{j-1})}{2}. \end{aligned}$$
It is clear to see that
$$\begin{aligned} \bigl\Vert r_{1}^{j} \bigr\Vert \leq \frac{1}{\Delta t}Ch^{4} \int _{t_{j-1}}^{t_{j}} \Vert u_{t} \Vert _{4}\,dt \leq C(\Delta t)^{-\frac{1}{2}}h^{4} \biggl( \int _{t_{j-1}}^{t_{j}} \Vert u_{t} \Vert _{4}^{2}\,dt \biggr)^{\frac{1}{2}}. \end{aligned}$$
Using Taylor’s formula, we derive
$$\begin{aligned} \bigl\Vert r_{2}^{j} \bigr\Vert \leq C\Delta t \int _{t_{j-1}}^{t_{j}} \Vert u_{ttt} \Vert \,dt \leq C( \Delta t)^{\frac{3}{2}} \biggl( \int _{t_{j-1}}^{t_{j}} \Vert u_{ttt} \Vert ^{2}\,dt \biggr)^{\frac{1}{2}}. \end{aligned}$$
We easily get
$$\begin{aligned} \sum_{j=1}^{n} \bigl\Vert r^{j} \bigr\Vert ^{2} \leq C(\Delta t)^{-1} \bigl((\Delta t)^{4}+h^{8}\bigr) \int _{0}^{t_{n}}\bigl( \Vert u_{t} \Vert _{4}^{2}+ \Vert u_{ttt} \Vert ^{2} \bigr)\,dt. \end{aligned}$$
(66)
Adding (59), (65), and (66), we have
$$\begin{aligned} &\bigl( \bigl\Vert \theta ^{n} \bigr\Vert ^{2}- \bigl\Vert \theta ^{n-1} \bigr\Vert ^{2} \bigr)+\frac{s\Delta t}{8} \bigl\Vert D^{2} \theta ^{n}+D^{2} \theta ^{n-1} \bigr\Vert ^{2} \\ &\quad \leq C \biggl(\Delta t\bigl( \bigl\Vert \theta ^{n}+\theta ^{n-1} \bigr\Vert ^{2}+h^{6}\bigr) \\ &\qquad{}+\bigl(({\Delta t})^{4}+h^{8}\bigr) \int _{t_{n-1}}^{t_{n}}\bigl( \Vert u_{t} \Vert _{4}^{2}+ \Vert Du_{t} \Vert ^{4}+ \bigl\Vert D^{2}u_{t} \bigr\Vert ^{2}+ \Vert u_{ttt} \Vert ^{2}\bigr)\,dt \biggr). \end{aligned}$$
We know
$$\begin{aligned} \bigl\Vert \theta ^{n}+\theta ^{n-1} \bigr\Vert ^{2}\leq 2\bigl( \bigl\Vert \theta ^{n} \bigr\Vert ^{2}+ \bigl\Vert \theta ^{n-1} \bigr\Vert ^{2} \bigr). \end{aligned}$$
Then
$$\begin{aligned} &\bigl( \bigl\Vert \theta ^{n} \bigr\Vert ^{2}- \bigl\Vert \theta ^{n-1} \bigr\Vert ^{2}\bigr)+\frac{s\Delta t}{8} \bigl\Vert D^{2} \theta ^{n}+D^{2}\theta ^{n-1} \bigr\Vert ^{2} \\ &\quad \leq C \biggl(\Delta t\bigl( \bigl\Vert \theta ^{n} \bigr\Vert ^{2}+ \bigl\Vert \theta ^{n-1} \bigr\Vert ^{2}+h^{6}\bigr) \\ &\qquad{} +\bigl(({\Delta t})^{4}+h^{8}\bigr) \int _{t_{n-1}}^{t_{n}}\bigl( \Vert u_{t} \Vert _{4}^{2}+ \Vert Du_{t} \Vert ^{4}+ \bigl\Vert D^{2}u_{t} \bigr\Vert ^{2}+ \Vert u_{ttt} \Vert ^{2}\bigr)\,dt \biggr). \end{aligned}$$
(67)
Taking the sum over n, by \(n\Delta t=t_{n}\leq T\), we have
$$\begin{aligned} & \bigl\Vert \theta ^{n} \bigr\Vert ^{2}- \bigl\Vert \theta ^{0} \bigr\Vert ^{2}+ \frac{s\Delta t}{8}\sum_{i=1}^{n} \bigl\Vert D^{2}\theta ^{i}+D^{2}\theta ^{i-1} \bigr\Vert ^{2} \\ &\quad \leq C \Biggl(\Delta t\sum_{i=1}^{n} \bigl( \bigl\Vert \theta ^{i} \bigr\Vert ^{2}+ \bigl\Vert \theta ^{i-1} \bigr\Vert ^{2}\bigr)+Th^{6} \\ &\qquad{} +\bigl((\Delta t)^{4}+h^{8}\bigr) \int _{0}^{t_{n}}\bigl( \Vert u_{t} \Vert _{4}^{2}+ \Vert Du_{t} \Vert ^{4}+ \bigl\Vert D^{2}u_{t} \bigr\Vert ^{2}+ \Vert u_{ttt} \Vert ^{2}\bigr)\,dt \Biggr). \end{aligned}$$
Hence
$$\begin{aligned} (1-C\Delta t) \bigl\Vert \theta ^{n} \bigr\Vert ^{2} \leq (1+C\Delta t) \bigl\Vert \theta ^{0} \bigr\Vert ^{2} +C \Biggl(\Delta t\sum_{i=1}^{n-1} \bigl\Vert \theta ^{i} \bigr\Vert ^{2}+Th^{6}+( \Delta t)^{4}+h^{8} \Biggr). \end{aligned}$$
If Δt is small enough, we have
$$\begin{aligned} \bigl\Vert \theta ^{n} \bigr\Vert ^{2}\leq \frac{1+C\Delta t}{1-C\Delta t} \bigl\Vert \theta ^{0} \bigr\Vert ^{2} +\frac{C}{1-C\Delta t} \Biggl(\Delta t\sum_{i=1}^{n-1} \bigl\Vert \theta ^{i} \bigr\Vert ^{2}+Th^{6}+( \Delta t)^{4}+h^{8} \Biggr). \end{aligned}$$
By discrete Gronwall’s inequality, it gives
$$\begin{aligned} \bigl\Vert \theta ^{n} \bigr\Vert \leq C\bigl((\Delta t)^{2}+h^{3}\bigr). \end{aligned}$$
Using (9) and (50), we get
$$\begin{aligned} \bigl\Vert \theta ^{0} \bigr\Vert \leq \bigl\Vert u(0)-u_{h}(0) \bigr\Vert + \bigl\Vert u(0)-R_{h}u(0) \bigr\Vert \leq Ch^{4} \bigl\Vert u(0) \bigr\Vert _{4}. \end{aligned}$$
Finally, we obtain (51). The proof is completed. □
In the following theorem, we introduce the error estimate in \(H^{2}\) norm.
Theorem 4.3
Let\(u^{n}\)be the solution to (5), \(u_{h}^{n}\)be the solution to the fully discrete problem (36), \(u(0)\in H^{4}(I)\), \(u_{t}\in L^{2}(0,T;H^{4}(I))\cap L^{2}(0,T;W^{2,4}(I)) \), \(u_{ttt}\in L^{2}(0,T;L^{2}(I))\), and\(u_{h}^{0}\in U_{h}\)satisfying
$$\begin{aligned} \bigl\vert u(0)-u_{h}^{0} \bigr\vert _{2}\leq Ch^{2} \bigl\Vert u(0) \bigr\Vert _{4}. \end{aligned}$$
(68)
Then we have the following error estimate:
$$\begin{aligned} \bigl\vert u^{n}-u_{h}^{n} \bigr\vert _{2}\leq C\bigl(\Delta t+h^{2}\bigr). \end{aligned}$$
(69)
Proof
Letting \(v_{h}=\partial _{t} \theta ^{n}\) in (58), we get
$$\begin{aligned} & \bigl\Vert \partial _{t}\theta ^{n} \bigr\Vert ^{2}+\frac{1}{2\Delta t} \bigl(\alpha ^{n- \frac{1}{2}}\bigl(D^{2}\theta ^{n}+D^{2}\theta ^{n-1}\bigr),D^{2}\theta ^{n}-D^{2} \theta ^{n-1}\bigr) \\ &\qquad{}+\frac{1}{2} \biggl(\frac{\partial \alpha }{\partial t}\bigl(x,t^{n- \frac{1}{2}}\bigr) \bigl(D^{2}u^{n}-D^{2}u^{n-1} \bigr),D^{2}\theta ^{n}-D^{2}\theta ^{n-1} \biggr) \\ &\quad \leq \bigl\Vert r^{n} \bigr\Vert ^{2}+ \bigl\Vert D F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert ^{2}+ \frac{1}{2} \bigl\Vert \partial _{t} \theta ^{n} \bigr\Vert ^{2}. \end{aligned}$$
By Cauchy’s inequality, we have
$$\begin{aligned} &\bigl(\alpha ^{n-\frac{1}{2}}D^{2} \theta ^{n},D^{2}\theta ^{n}\bigr)-\bigl(\alpha ^{n- \frac{1}{2}}D^{2}\theta ^{n-1},D^{2}\theta ^{n-1}\bigr) \\ &\quad \leq M_{1}\Delta t \bigl\Vert D^{2}u^{n}-D^{2}u^{n-1} \bigr\Vert \bigl\Vert D^{2}\theta ^{n}-D^{2} \theta ^{n-1} \bigr\Vert \\ &\qquad{}+2\Delta t\bigl( \bigl\Vert r^{n} \bigr\Vert ^{2}+ \bigl\Vert D F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert ^{2}\bigr) \\ &\quad\leq \frac{M_{1}^{2}\Delta t}{2} \bigl\Vert D^{2}u^{n}-D^{2}u^{n-1} \bigr\Vert ^{2}+ \Delta t\bigl( \bigl\Vert D^{2}\theta ^{n} \bigr\Vert ^{2}+ \bigl\Vert D^{2}\theta ^{n-1} \bigr\Vert ^{2}\bigr) \\ &\qquad{}+2\Delta t\bigl( \bigl\Vert r^{n} \bigr\Vert ^{2}+ \bigl\Vert D F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert ^{2}\bigr). \end{aligned}$$
(70)
Using the Newton–Leibniz formula and Hölder’s inequality, we obtain
$$\begin{aligned} \bigl\vert D^{2}u^{n}-D^{2}u^{n-1} \bigr\vert ^{2}\leq \biggl\vert \Delta t \int _{t^{n-1}}^{t^{n}} \bigl\vert D^{2}u_{t} \bigr\vert \,dt \biggr\vert \leq \Delta t \int _{t^{n-1}}^{t^{n}} \bigl\vert D^{2}u_{t} \bigr\vert ^{2}\,dt. \end{aligned}$$
Based on (70), we have
$$\begin{aligned} &\bigl(\alpha ^{n-\frac{1}{2}}D^{2} \theta ^{n},D^{2}\theta ^{n}\bigr)-\bigl(\alpha ^{n- \frac{1}{2}}D^{2}\theta ^{n-1},D^{2}\theta ^{n-1}\bigr) \\ &\quad \leq \frac{M_{1}^{2}(\Delta t)^{2}}{2} \int _{t^{n-1}}^{t^{n}} \bigl\Vert D^{2}u_{t} \bigr\Vert ^{2}\,dt+\Delta t\bigl( \bigl\Vert D^{2}\theta ^{n} \bigr\Vert ^{2}+ \bigl\Vert D^{2}\theta ^{n-1} \bigr\Vert ^{2}\bigr) \\ &\qquad{}+2\Delta t\bigl( \bigl\Vert r^{n} \bigr\Vert ^{2}+ \bigl\Vert D F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert ^{2}\bigr). \end{aligned}$$
(71)
There exists \(\xi \in (t^{n-\frac{3}{2}},t^{n-\frac{1}{2}})\) such that
$$\begin{aligned} &\bigl(\alpha ^{n-\frac{1}{2}}\bigl(D^{2}\theta ^{n}+D^{2}\theta ^{n-1}\bigr),D^{2} \theta ^{n}-D^{2}\theta ^{n-1}\bigr) \\ &\quad=\bigl(\alpha ^{n-\frac{1}{2}}D^{2}\theta ^{n},D^{2} \theta ^{n}\bigr)-\bigl(\alpha ^{n- \frac{3}{2}}D^{2}\theta ^{n-1},D^{2}\theta ^{n-1}\bigr) -\bigl(\bigl(\alpha ^{n-\frac{1}{2}}- \alpha ^{n-\frac{3}{2}}\bigr)D^{2}\theta ^{n-1},D^{2}\theta ^{n-1}\bigr)) \\ &\quad =\bigl(\alpha ^{n-\frac{1}{2}}D^{2}\theta ^{n},D^{2} \theta ^{n}\bigr)-\bigl(\alpha ^{n- \frac{3}{2}}D^{2}\theta ^{n-1},D^{2}\theta ^{n-1}\bigr) -\Delta t \biggl( \frac{\partial \alpha }{\partial t}(x,\xi )D^{2}\theta ^{n-1},D^{2} \theta ^{n-1} \biggr). \end{aligned}$$
Then we have
$$\begin{aligned} &\bigl(\alpha ^{n-\frac{1}{2}}D^{2} \theta ^{n},D^{2}\theta ^{n}\bigr)-\bigl(\alpha ^{n- \frac{3}{2}}D^{2}\theta ^{n-1},D^{2}\theta ^{n-1}\bigr) -\Delta t \biggl( \frac{\partial \alpha }{\partial t}(x,\xi )D^{2} \theta ^{n-1},D^{2} \theta ^{n-1} \biggr) \\ &\quad \leq \frac{M_{1}^{2}(\Delta t)^{2}}{2} \int _{t^{n-1}}^{t^{n}} \bigl\Vert D^{2}u_{t} \bigr\Vert ^{2}\,dt+\Delta t\bigl( \bigl\Vert D^{2}\theta ^{n} \bigr\Vert ^{2}+ \bigl\Vert D^{2}\theta ^{n-1} \bigr\Vert ^{2}\bigr) \\ &\qquad{}+2\Delta t\bigl( \bigl\Vert r^{n} \bigr\Vert ^{2}+ \bigl\Vert D F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert ^{2}\bigr). \end{aligned}$$
(72)
Taking the sum over n and using (4), we can obtain
$$\begin{aligned} &\bigl(\alpha ^{n-\frac{1}{2}}D^{2} \theta ^{n},D^{2}\theta ^{n}\bigr)-\bigl(\alpha ^{ \frac{1}{2}}D^{2}\theta ^{1},D^{2}\theta ^{1}\bigr) \\ &\quad \leq C\Delta t\sum_{j=2}^{n}\bigl( \bigl\Vert D^{2}\theta ^{j-1} \bigr\Vert ^{2}+ \bigl\Vert D^{2} \theta ^{j} \bigr\Vert ^{2}+ \bigl\Vert r^{j} \bigr\Vert ^{2}+ \bigl\Vert D F \bigl(Du^{j},Du^{j-1},Du_{h}^{j},Du_{h}^{j-1} \bigr) \bigr\Vert ^{2}\bigr) \\ &\qquad{}+C(\Delta t)^{2} \int _{0}^{t^{n}} \bigl\Vert D^{2}u_{t} \bigr\Vert ^{2}\,dt. \end{aligned}$$
(73)
It follows from (3) that
$$\begin{aligned} \bigl(\alpha ^{n-\frac{1}{2}}D^{2}\theta ^{n},D^{2} \theta ^{n}\bigr)\geq s \bigl\Vert D^{2} \theta ^{n} \bigr\Vert ^{2},\qquad -\bigl(\alpha ^{\frac{1}{2}}D^{2} \theta ^{1},D^{2} \theta ^{1}\bigr)\geq -S \bigl\Vert D^{2}\theta ^{1} \bigr\Vert ^{2}. \end{aligned}$$
Then one has
$$\begin{aligned} &s \bigl\Vert D^{2}\theta ^{n} \bigr\Vert ^{2}-S \bigl\Vert D^{2}\theta ^{1} \bigr\Vert ^{2} \\ &\quad \leq C\Delta t\sum_{j=2}^{n}\bigl( \bigl\Vert D^{2}\theta ^{j} \bigr\Vert ^{2}+ \bigl\Vert r^{j} \bigr\Vert ^{2}+ \bigl\Vert D F \bigl(Du^{j},Du^{j-1},Du_{h}^{j},Du_{h}^{j-1} \bigr) \bigr\Vert ^{2}\bigr) \\ &\qquad{}+C(\Delta t)^{2} \int _{0}^{t^{n}} \bigl\Vert D^{2}u_{t} \bigr\Vert ^{2}\,dt. \end{aligned}$$
(74)
Introducing some symbols \(I_{1}\), \(I_{2}\), \(I_{3}\), then a direct calculation gives
$$\begin{aligned} & \bigl\Vert D F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert \\ &\quad = \biggl\Vert D \biggl(\frac{1}{2}\bigl(\bigl(Du^{n} \bigr)^{3}+\bigl(Du^{n-1}\bigr)^{3}\bigr)- \frac{1}{2}\bigl(Du^{n}+Du^{n-1}\bigr) - \frac{H(Du_{h}^{n})-H(Du_{h}^{n-1})}{Du_{h}^{n}-Du_{h}^{n-1}} \biggr) \biggr\Vert \\ &\quad= \biggl\Vert D \biggl(\frac{1}{2}\bigl(\bigl(Du^{n} \bigr)^{3}+\bigl(Du^{n-1}\bigr)^{3}\bigr)- \frac{1}{4}\bigl(Du^{n}+Du^{n-1}\bigr) \bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr) \biggr) \\ &\qquad{}+\frac{1}{4}D\bigl(\bigl(Du^{n}+Du^{n-1}\bigr) \bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr) -\bigl(Du_{h}^{n}+Du_{h}^{n-1} \bigr) \bigl( \bigl\vert Du_{h}^{n} \bigr\vert ^{2}+ \bigl\vert Du_{h}^{n-1} \bigr\vert ^{2}\bigr)\bigr) \\ &\qquad{}-\frac{1}{2}D\bigl(\bigl(Du^{n}+Du^{n-1}\bigr)- \bigl(Du_{h}^{n}+Du_{h}^{n-1}\bigr) \bigr) \biggr\Vert \\ &\quad =\Vert I_{1}+I_{2}+I_{3} \Vert . \end{aligned}$$
It is obvious that
$$\begin{aligned} \bigl\Vert D F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert \leq \Vert I_{1} \Vert + \Vert I_{2} \Vert + \Vert I_{3} \Vert . \end{aligned}$$
First, applying the triangle inequality to \(\|I_{1}\|\), we get
$$\begin{aligned} \Vert I_{1} \Vert ={}&\frac{1}{4} \bigl\Vert D\bigl(\bigl(Du^{n}\bigr)^{3}- \bigl\vert Du^{n} \bigr\vert ^{2}Du^{n-1}-Du^{n} \bigl\vert Du^{n-1} \bigr\vert ^{2}+\bigl(Du^{n-1} \bigr)^{3}\bigr) \bigr\Vert ^{2} \\ ={}&\frac{1}{4} \bigl\Vert D\bigl(\bigl(Du^{n}+Du^{n-1} \bigr) \bigl(Du^{n}-Du^{n-1}\bigr)^{2}\bigr) \bigr\Vert ^{2} \\ ={}&\frac{1}{4} \bigl\Vert \bigl(D^{2}u^{n}+D^{2}u^{n-1} \bigr) \bigl(Du^{n}-Du^{n-1}\bigr)^{2} \\ &{}+2\bigl(Du^{n}+Du^{n-1}\bigr) \bigl(Du^{n}-Du^{n-1} \bigr) \bigl(D^{2}u^{n}-D^{2}u^{n-1}\bigr) \bigr\Vert ^{2} \\ \leq {}&\frac{1}{4}\bigl( \bigl\Vert D^{2}u^{n} \bigr\Vert ^{2} + \bigl\Vert D^{2}u^{n-1} \bigr\Vert ^{2} \bigr) \bigl\Vert \bigl(Du^{n}-Du^{n-1} \bigr)^{2} \bigr\Vert ^{2} \\ &{}+\frac{1}{2}\bigl( \bigl\vert Du^{n} \bigr\vert _{\infty }^{2} + \bigl\vert Du^{n-1} \bigr\vert _{\infty }\bigr)^{2} \bigl\Vert Du^{n}-Du^{n-1} \bigr\Vert ^{2} \bigl\Vert \bigl(D^{2}u^{n}-D^{2}u^{n-1} \bigr) \bigr\Vert ^{2}. \end{aligned}$$
Based on Sobolev’s embedding theorem, we have
$$\begin{aligned} \Vert I_{1} \Vert \leq{} &C\bigl( \bigl\Vert \bigl(Du^{n}-Du^{n-1}\bigr)^{2} \bigr\Vert ^{2}+ \bigl\Vert Du^{n}-Du^{n-1} \bigr\Vert ^{2} \bigl\Vert \bigl(D^{2}u^{n}-D^{2}u^{n-1} \bigr) \bigr\Vert ^{2}\bigr) \\ \leq {}&C(\Delta t)^{2} \biggl( \biggl( \int _{t_{n-1}}^{t_{n}} \bigl\Vert Du_{t}(t) \bigr\Vert ^{2}\,dt \biggr)^{2} + \int _{t_{n-1}}^{t_{n}} \bigl\Vert Du_{t}(t) \bigr\Vert ^{2}\,dt \int _{t_{n-1}}^{t_{n}} \bigl\Vert D^{2}u_{t}(t) \bigr\Vert ^{2}\,dt \biggr) \\ \leq {}&C(\Delta t)^{2} \biggl( \biggl( \int _{t_{n-1}}^{t_{n}} \bigl\Vert Du_{t}(t) \bigr\Vert ^{2}\,dt \biggr)^{2} + \biggl( \int _{t_{n-1}}^{t_{n}} \bigl\Vert D^{2}u_{t}(t) \bigr\Vert ^{2}\,dt \biggr)^{2} \biggr). \end{aligned}$$
Further, Hölder’s inequality yields
$$\begin{aligned} \Vert I_{1} \Vert \leq C(\Delta t)^{3} \biggl( \int _{t_{n-1}}^{t_{n}} \bigl\Vert Du_{t}(t) \bigr\Vert ^{4}\,dt + \int _{t_{n-1}}^{t_{n}} \bigl\Vert D^{2}u_{t}(t) \bigr\Vert ^{4}\,dt \biggr). \end{aligned}$$
(75)
Second, we analyze \(\|I_{2}\|\). A direct calculation gives
$$\begin{aligned} I_{2}={}&D\bigl(\bigl(Du^{n}+Du^{n-1} \bigr) \bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr)-\bigl(Du_{h}^{n}+Du_{h}^{n-1} \bigr) \bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr) \\ &{}+\bigl(Du_{h}^{n}+Du_{h}^{n-1}\bigr) \bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr)-\bigl(Du_{h}^{n}+Du_{h}^{n-1} \bigr) \bigl( \bigl\vert Du_{h}^{n} \bigr\vert ^{2}+ \bigl\vert Du_{h}^{n-1} \bigr\vert ^{2}\bigr)\bigr) \\ ={}&D\bigl(\bigl(Du^{n}+Du^{n-1}\bigr)-\bigl(Du_{h}^{n}+Du_{h}^{n-1} \bigr)\bigr) \bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr) \\ &{}+\bigl(\bigl(Du^{n}+Du^{n-1}\bigr)-\bigl(Du_{h}^{n}+Du_{h}^{n-1} \bigr)\bigr)D\bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr) \\ &{}+D\bigl(Du_{h}^{n}+Du_{h}^{n-1}\bigr) \bigl(\bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr)-\bigl( \bigl\vert Du_{h}^{n} \bigr\vert ^{2}+ \bigl\vert Du_{h}^{n-1} \bigr\vert ^{2}\bigr)\bigr) \\ &{}+\bigl(Du_{h}^{n}+Du_{h}^{n-1}\bigr)D \bigl(\bigl( \bigl\vert Du^{n} \bigr\vert ^{2}+ \bigl\vert Du^{n-1} \bigr\vert ^{2}\bigr)-\bigl( \bigl\vert Du_{h}^{n} \bigr\vert ^{2}+ \bigl\vert Du_{h}^{n-1} \bigr\vert ^{2}\bigr)\bigr) . \end{aligned}$$
With the help of Sobolev’s embedding theorem, we can obtain
$$\begin{aligned} \Vert I_{2} \Vert \leq {}&\bigl( \bigl\vert Du^{n} \bigr\vert _{\infty }^{2}+ \bigl\vert Du^{n-1} \bigr\vert _{\infty }^{2}\bigr) \bigl\Vert \bigl(D^{2}u^{n}+D^{2}u^{n-1} \bigr)-\bigl(D^{2}u_{h}^{n}+D^{2}u_{h}^{n-1} \bigr) \bigr\Vert \\ &{}+2\bigl( \bigl\vert Du^{n} \bigr\vert _{\infty } \bigl\Vert D^{2}u^{n} \bigr\Vert + \bigl\vert Du^{n-1} \bigr\vert _{\infty } \bigl\Vert D^{2}u^{n-1} \bigr\Vert \bigr) \\ &{}\times \bigl\Vert \bigl(Du^{n}+Du^{n-1}\bigr)- \bigl(Du_{h}^{n}+Du_{h}^{n-1}\bigr) \bigr\Vert \\ &{}+\bigl( \bigl\Vert D^{2}u_{h}^{n} \bigr\Vert + \bigl\Vert D^{2}u_{h}^{n-1} \bigr\Vert \bigr) \\ &{}\times \bigl\Vert \bigl(Du^{n}+Du_{h}^{n} \bigr) \bigl(Du^{n}-Du_{h}^{n}\bigr)+ \bigl(Du^{n-1}+Du_{h}^{n-1}\bigr) \bigl(Du^{n-1}-Du_{h}^{n-1}\bigr) \bigr\Vert \\ &{}+\bigl( \bigl\vert Du_{h}^{n} \bigr\vert _{\infty }+ \bigl\vert Du_{h}^{n-1} \bigr\vert _{\infty }\bigr) \\ &{}\times \bigl\Vert D\bigl(\bigl(Du^{n}+Du_{h}^{n} \bigr) \bigl(Du^{n}-Du_{h}^{n}\bigr)+ \bigl(Du^{n-1}+Du_{h}^{n-1}\bigr) \bigl(Du^{n-1}-Du_{h}^{n-1}\bigr)\bigr) \bigr\Vert \\ \leq {}& C\bigl( \bigl\Vert D\theta ^{n} \bigr\Vert + \bigl\Vert D\theta ^{n-1} \bigr\Vert + \bigl\Vert D\rho ^{n} \bigr\Vert + \bigl\Vert D\rho ^{n-1} \bigr\Vert \\ &{}+ \bigl\Vert D^{2}\theta ^{n} \bigr\Vert + \bigl\Vert D^{2}\theta ^{n-1} \bigr\Vert + \bigl\Vert D^{2}\rho ^{n} \bigr\Vert + \bigl\Vert D^{2} \rho ^{n-1} \bigr\Vert \bigr). \end{aligned}$$
Then
$$\begin{aligned} \Vert I_{2} \Vert \leq C\bigl( \bigl\Vert \theta ^{n} \bigr\Vert + \bigl\Vert \theta ^{n-1} \bigr\Vert + \bigl\Vert D^{2}\theta ^{n} \bigr\Vert + \bigl\Vert D^{2}\theta ^{n-1} \bigr\Vert +h^{2}\bigr). \end{aligned}$$
(76)
For \(\|I_{3}\|\), by the triangle inequality and (9), one can have
$$\begin{aligned} \Vert I_{3} \Vert ={}& \bigl\Vert D^{2}\theta ^{n}+D^{2}\rho ^{n}+D^{2}\theta ^{n-1}+D^{2} \rho ^{n-1} \bigr\Vert \\ \leq {}& \bigl\Vert D^{2}\theta ^{n} \bigr\Vert + \bigl\Vert D^{2}\theta ^{n-1} \bigr\Vert + \bigl\Vert D^{2}\rho ^{n} \bigr\Vert + \bigl\Vert D^{2} \rho ^{n-1} \bigr\Vert \\ \leq {}& \bigl\Vert D^{2}\theta ^{n} \bigr\Vert + \bigl\Vert D^{2}\theta ^{n-1} \bigr\Vert +Ch^{2}. \end{aligned}$$
(77)
By (75)–(77), we get
$$\begin{aligned} & \bigl\Vert D F\bigl(Du^{n},Du^{n-1},Du_{h}^{n},Du_{h}^{n-1} \bigr) \bigr\Vert ^{2} \\ &\quad \leq C\bigl( \bigl\Vert \theta ^{n} \bigr\Vert ^{2}+ \bigl\Vert \theta ^{n-1} \bigr\Vert ^{2}+ \bigl\Vert D^{2}\theta ^{n} \bigr\Vert ^{2}+ \bigl\Vert D^{2}\theta ^{n-1} \bigr\Vert ^{2}+h^{4} \bigr) \\ &\qquad{}+C(\Delta t)^{3} \biggl( \int _{t_{n-1}}^{t_{n}} \bigl\Vert Du_{t}(t) \bigr\Vert ^{4}\,dt + \int _{t_{n-1}}^{t_{n}} \bigl\Vert D^{2}u_{t}(t) \bigr\Vert ^{4}\,dt \biggr). \end{aligned}$$
(78)
Substituting (66) and (78) into (74), we obtain
$$\begin{aligned} &s \bigl\Vert D^{2}\theta ^{n} \bigr\Vert ^{2}-S \bigl\Vert D^{2}\theta ^{1} \bigr\Vert ^{2} \\ &\quad \leq C\Delta t\sum_{j=1}^{n}\bigl( \bigl\Vert D^{2}\theta ^{j} \bigr\Vert ^{2}+ \bigl\Vert \theta ^{j} \bigr\Vert ^{2}\bigr) \\ &\qquad{}+C\bigl((\Delta t)^{2}+h^{4}\bigr) \int _{0}^{t_{n}}\bigl( \Vert u_{t} \Vert _{4}^{2}+ \Vert u_{ttt} \Vert ^{2}+ \bigl\Vert Du_{t}(t) \bigr\Vert ^{4}+ \bigl\Vert D^{2}u_{t}(t) \bigr\Vert ^{4}\bigr)\,dt. \end{aligned}$$
(79)
Letting \(n=1\) in (71), based on (66) and (78), we have
$$\begin{aligned} \bigl\Vert D^{2}\theta ^{1} \bigr\Vert \leq C \bigl\Vert D^{2}\theta ^{0} \bigr\Vert +O(\Delta t). \end{aligned}$$
(80)
By (79) and (80), we get
$$\begin{aligned} \bigl\Vert D^{2}\theta ^{n} \bigr\Vert ^{2} \leq C( \bigl\Vert D^{2}\theta ^{0} \bigr\Vert ^{2}+(\Delta t)^{2}+h^{4}+ \Delta t\sum _{j=1}^{n-1}\bigl( \bigl\Vert D^{2}\theta ^{j} \bigr\Vert ^{2}+ \bigl\Vert \theta ^{j} \bigr\Vert ^{2}\bigr). \end{aligned}$$
Using (51), we have
$$\begin{aligned} \bigl\Vert D^{2}\theta ^{n} \bigr\Vert ^{2} \leq C\Biggl( \bigl\Vert D^{2}\theta ^{0} \bigr\Vert ^{2}+(\Delta t)^{2}+h^{4}+ \Delta t\sum _{j=1}^{n-1} \bigl\Vert D^{2}\theta ^{j} \bigr\Vert ^{2}\Biggr). \end{aligned}$$
If Δt is sufficiently small, discrete Gronwall’s inequality yields
$$\begin{aligned} \bigl\Vert D^{2}\theta ^{n} \bigr\Vert \leq C\bigl( \Delta t+h^{2}\bigr). \end{aligned}$$
This completes the proof. □