A posteriori error estimates for fourth order hyperbolic control problems by mixed finite element methods

Abstract

In this paper, we consider the a posteriori error estimates of the mixed finite element method for the optimal control problems governed by fourth order hyperbolic equations. The state is discretized by the order k Raviart–Thomas mixed elements and control is discretized by piecewise polynomials of degree k. We adopt the mixed elliptic reconstruction to derive the a posteriori error estimates for both the state and the control approximations.

Introduction

The finite element approximation for optimal control problems has an enormously important function in the numerical approach for these problems. Scientists have studied extensively this area; see, for example, [4, 12, 13, 21, 25]. They discussed the a priori error estimates using finite element approximations, such as [1, 16, 23], in which elliptic or parabolic problems are considered by optimal control theory. They studied adaptivity for many optimal control problems; for example, see [4, 11, 17, 20,21,22].

In some optimal control problems, for the objective function containing a gradient of the state variable, we use mixed finite element methods to discretize the state equation, so that the scalar variable and its flux variable can be approximated in the same accuracy; for example, see . Many scientists have addressed the mixed finite element methods for elliptic problems [6,7,8, 14], for the first bi-harmonic equation , for parabolic problems  and for hyperbolic problems [9, 15].

The purpose of this work is to discuss the a posteriori error estimates of the semidiscrete mixed finite element approximation for fourth order hyperbolic optimal control problems. Considering the fourth order hyperbolic equations by the idea of a mixed elliptic reconstruction , we obtain the error estimates for the state and the control approximations. The following is the model we considered:

\begin{aligned}& \min_{u \in K\in U} \biggl\{ \frac{1}{2} \int _{0}^{T} \bigl( \Vert \Delta y \Vert ^{2}+ \Vert y-y_{d} \Vert ^{2}+ \Vert u \Vert ^{2} \bigr)\,dt \biggr\} , \end{aligned}
(1.1)
\begin{aligned}& y_{tt}(x,t)+\Delta ^{2} y(x,t) =f(x,t)+ u(x,t), \quad x\in \varOmega , t \in J, \end{aligned}
(1.2)
\begin{aligned}& y(x,t) =\frac{\partial y}{\partial n}=0, \quad x\in {\partial \varOmega }, t \in J, \end{aligned}
(1.3)
\begin{aligned}& y(x,0) =y_{0}(x), \quad x\in \varOmega , \end{aligned}
(1.4)
\begin{aligned}& y_{t}(x,0) =y_{1}(x), \quad x\in \varOmega , \end{aligned}
(1.5)

where $$\varOmega \subset {\mathbf{R}^{2}}$$ is an open set of polygon with ∂Ω. K is in $$U=L^{2}(J;L^{2}(\varOmega ))$$, a closed convex set, $$J=[0,T]$$, $$f,y_{d}\in L^{2}(J;L^{2}(\varOmega ))$$ and $$y_{0}, y_{1}\in H^{4}(\varOmega )$$. K is defined as follows:

\begin{aligned} K= \biggl\{ u\in U: \int _{0}^{T} \int _{\varOmega }u \,dx\,dt\geq 0 \biggr\} . \end{aligned}
(1.6)

Let $$\tilde{y}=-\Delta y$$, $$\tilde{\boldsymbol{p}}=-\nabla y$$ and $$\boldsymbol{p}=- \nabla \tilde{y}$$, then (1.1)–(1.5) can be written as

\begin{aligned}& \min_{u \in K\in U} \biggl\{ \frac{1}{2} \int _{0}^{T} \bigl( \Vert \tilde{y} \Vert ^{2}+ \Vert y-y_{d} \Vert ^{2}+ \Vert u \Vert ^{2} \bigr)\,dt \biggr\} , \end{aligned}
(1.7)
\begin{aligned}& y_{tt}(x,t)+\operatorname {div}\boldsymbol{p}(x,t) =f(x,t)+ u(x,t), \quad x\in \varOmega , t \in J, \end{aligned}
(1.8)
\begin{aligned}& \boldsymbol{p}(x,t) =-\nabla {\tilde{y}(x,t)}, \quad x\in \varOmega , t \in J, \end{aligned}
(1.9)
\begin{aligned}& \operatorname {div}\tilde{\boldsymbol{p}}(x,t) =\tilde{y}(x,t), \quad x\in \varOmega , t\in J, \end{aligned}
(1.10)
\begin{aligned}& \tilde{\boldsymbol{p}}(x,t) =-\nabla y(x,t), \quad x\in \varOmega , t\in J, \end{aligned}
(1.11)
\begin{aligned}& y(x,t) =-\tilde{p}\cdot n=0, \quad x\in {\partial \varOmega }, t \in J, \end{aligned}
(1.12)
\begin{aligned}& y(x,0) =y_{0}(x), \quad x\in \varOmega , \end{aligned}
(1.13)
\begin{aligned}& y_{t}(x,0) =y_{1}(x), \quad x\in \varOmega . \end{aligned}
(1.14)

In the paper, we adopt the standard notation $$W^{m,p}(\varOmega )$$ for Sobolev space on Ω with a norm $$\Vert v\Vert _{m,p}$$ given by $$\Vert v \Vert _{m,p}^{p}:=\sum_{\vert \alpha \vert \leq m}\Vert D^{\alpha }v\Vert _{L^{p}(\varOmega )}^{p}$$, and a seminorm $$\vert v\vert _{m,p}$$ given by $$\vert v\vert _{m,p}^{p}:=\sum_{\vert \alpha \vert = m}\Vert D^{\alpha }v\Vert _{L^{p}( \varOmega )}^{p}$$. We set $$W_{0}^{m,p}(\varOmega )=\{v\in W^{m,p}(\varOmega ): \gamma (D^{\alpha }v )\vert _{\partial \varOmega }=0, \vert \alpha \vert =m \}$$, where γ is the trace operator. We denote $$W^{m,2}(\varOmega )(W_{0}^{m,2}(\varOmega ))$$ by $$H^{m}(\varOmega )(H_{0}^{m}(\varOmega ))$$.

We denote by $$L^{s}(0,T;W^{m,p}(\varOmega ))$$ the Banach space of all $$L^{s}$$ integrable functions from J into $$W^{m,p}(\varOmega )$$ with norm $$\Vert v\Vert _{L^{s}(J;W^{m,p}(\varOmega ))}= (\int _{0}^{T}\Vert v\Vert _{W^{m,p}( \varOmega )}^{s}\,dt )^{\frac{1}{s}}$$ for $$s\in [1,\infty )$$, and the standard modification for $$s=\infty$$. For simplicity of presentation, we denote $$\Vert v\Vert _{L^{s}(J;W^{m,p}(\varOmega ))}$$ by $$\Vert v\Vert _{L^{s}(W^{m,p})}$$. Similarly, one can define the spaces $$H^{1}(J;W^{m,p}(\varOmega ))$$ and $$C^{k}(J;W^{m,p}(\varOmega ))$$. We can find details in . C is a general positive constant independent of h.

The rest of this paper is as follows: In Sect. 2, we introduce the optimal control problems and its mixed finite element scheme. Section 2 ends with the definition of the mixed elliptic reconstructions, which is useful in deriving the a posteriori estimates for the fourth order hyperbolic optimal control problems in Sect. 3. Finally, we make some concluding remarks in Sect. 4.

Optimal control problems for mixed methods

A semidiscrete approximation of a mixed finite element for the optimal control problems (1.7)–(1.14) will be constructed. We set the state spaces $$\boldsymbol{L}=L^{2}(J;\boldsymbol{V})$$, $$\boldsymbol{L}_{0}=L^{2}(J;\boldsymbol{V}_{0})$$ and $$Q=L^{2}(J;W)$$, $$W=L^{2}(\varOmega )$$, where V, and $$\boldsymbol{V}_{0}$$ are defined as follows:

\begin{aligned}& \boldsymbol{V}=H(\operatorname {div};\varOmega )= \bigl\{ \boldsymbol{v}\in \bigl(L^{2}( \varOmega )\bigr)^{2}, \operatorname {div}\boldsymbol{v}\in L^{2}(\varOmega ) \bigr\} , \\& \boldsymbol{V}_{0}=H_{0}(\operatorname {div};\varOmega )= \bigl\{ \boldsymbol{v}\in H(\operatorname {div}, \varOmega ), \boldsymbol{v}\cdot n\vert _{\partial \varOmega }=0 \bigr\} . \end{aligned}

The space V is a Hilbert space, its norm is defined as follows:

$$\Vert \boldsymbol{v}\Vert _{H(\operatorname {div};\varOmega )}= \bigl( \Vert \boldsymbol{v}\Vert _{0,\varOmega } ^{2}+ \Vert \operatorname {div}\boldsymbol{v}\Vert _{0,\varOmega }^{2} \bigr)^{1/2}.$$

Now we introduce operators: div, , curl and Curl. For any $$\boldsymbol{v}=(\boldsymbol{v}_{1},\boldsymbol{v}_{2})\in (H ^{1}(\varOmega ))^{2}$$ or $$w\in H^{1}(\varOmega )$$,

\begin{aligned}& \operatorname {div}\boldsymbol{v}=\partial _{1} \boldsymbol{v}_{1}+\partial _{2} \boldsymbol{v}_{2}, \quad \quad \nabla w=(\partial _{1}w, \partial _{2}w), \end{aligned}
(2.1)
\begin{aligned}& \operatorname{curl} \boldsymbol{v}=\partial _{1}\boldsymbol{v}_{2}- \partial _{2} \boldsymbol{v}_{1}, \quad\quad \operatorname{Curl} w=(- \partial _{2}w,\partial _{1}w). \end{aligned}
(2.2)

Next, (1.7)–(1.14) can be rewritten into weak form as follows: find $$(\tilde{\boldsymbol{p}},y,\boldsymbol{p},y,u)\in (\boldsymbol{L}_{0}\times Q\times \boldsymbol{L}\times Q\times K)$$, such that

\begin{aligned}& \min_{u \in K\subset U} \biggl\{ \frac{1}{2} \int _{0}^{T} \bigl( \Vert \tilde{y} \Vert ^{2}+ \Vert y-y_{d} \Vert ^{2}+ \Vert u \Vert ^{2} \bigr)\,dt \biggr\} , \end{aligned}
(2.3)
\begin{aligned}& (y_{tt},w)+(\operatorname {div}{\boldsymbol{p}},w) =(f+u,w), \quad \forall w\in W, t\in J, \end{aligned}
(2.4)
\begin{aligned}& (\boldsymbol{p},\boldsymbol{v})-(\tilde{y},\operatorname {div}\boldsymbol{v}) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}_{0}, t\in J, \end{aligned}
(2.5)
\begin{aligned}& (\operatorname {div}\tilde{\boldsymbol{p}},w) =(\tilde{y},w), \quad \forall w\in W, t\in J, \end{aligned}
(2.6)
\begin{aligned}& (\tilde{\boldsymbol{p}},\boldsymbol{v})-(y,\operatorname {div}\boldsymbol{v}) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, t\in J, \end{aligned}
(2.7)
\begin{aligned}& y(x,0) =y_{0}(x), \quad \forall x\in \varOmega , \end{aligned}
(2.8)
\begin{aligned}& y_{t}(x,0) =y_{1}(x), \quad \forall x\in \varOmega . \end{aligned}
(2.9)

From , we know that the above optimal control problem has a unique solution $$(\tilde{\boldsymbol{p}},y,\boldsymbol{p},\tilde{y},u)$$, and that $$(\tilde{\boldsymbol{p}},y,\boldsymbol{p},\tilde{y},u)$$ is the solution of (2.3)–(2.9) if and only if there is a co-state $$(\tilde{\boldsymbol{q}},z,\boldsymbol{q},\tilde{z})\in (\boldsymbol{L}_{0}\times Q\times \boldsymbol{L}\times Q)$$ such that $$(\tilde{\boldsymbol{p}},y,\boldsymbol{p},\tilde{y}, \tilde{\boldsymbol{q}},z,\boldsymbol{q},\tilde{z},u)$$ satisfies the following optimality conditions:

\begin{aligned}& (y_{tt},w)+(\operatorname {div}{\boldsymbol{p}},w) =(f+u,w), \quad \forall w\in W, t\in J, \end{aligned}
(2.10)
\begin{aligned}& (\boldsymbol{p},\boldsymbol{v})-(\tilde{y},\operatorname {div}\boldsymbol{v}) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, t\in J, \end{aligned}
(2.11)
\begin{aligned}& (\operatorname {div}\tilde{\boldsymbol{p}},w) =(\tilde{y},w), \quad \forall w\in W, t\in J, \end{aligned}
(2.12)
\begin{aligned}& (\tilde{\boldsymbol{p}},\boldsymbol{v})-(y,\operatorname {div}\boldsymbol{v}) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, t\in J, \end{aligned}
(2.13)
\begin{aligned}& y(x,0) =y_{0}(x), \quad \forall x\in \varOmega , \end{aligned}
(2.14)
\begin{aligned}& y_{t}(x,0) =y_{1}(x), \quad \forall x\in \varOmega , \end{aligned}
(2.15)
\begin{aligned}& (z_{tt},w)+(\operatorname {div}{\boldsymbol{q}},w) =(y-y_{d},w), \quad \forall w\in W, t\in J, \end{aligned}
(2.16)
\begin{aligned}& (\boldsymbol{q},\boldsymbol{v})-(\tilde{z},\operatorname {div}\boldsymbol{v}) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, t\in J, \end{aligned}
(2.17)
\begin{aligned}& (\operatorname {div}\tilde{\boldsymbol{q}},w) =(\tilde{y}+\tilde{z},w), \quad \forall w\in W, t\in J, \end{aligned}
(2.18)
\begin{aligned}& (\tilde{\boldsymbol{q}},\boldsymbol{v})-(z,\operatorname {div}\boldsymbol{v}) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, t\in J, \end{aligned}
(2.19)
\begin{aligned}& z(x,T) =0, \quad \forall x\in \varOmega , \end{aligned}
(2.20)
\begin{aligned}& z_{t}(x,T) =0, \quad \forall x\in \varOmega , \end{aligned}
(2.21)
\begin{aligned}& \int _{0}^{T}{(u+z, {\tilde{u}}-u)} \,dt \geq 0, \quad \forall \tilde{u} \in K, \end{aligned}
(2.22)

where $$(\cdot ,\cdot )$$ is the inner product of $$L^{2}(\varOmega )$$.

K is a control constraint, so we can get a relationship between u and z. This relationship is important for our result.

Lemma 2.1

Let $$(z,u)$$ be the solution of (2.10)(2.22). Then we have $$u=\max\{0,{\check{z}}\}-z$$, where

$${\check{z}}=\frac{\int _{0}^{T}\int _{\varOmega }z\,dx\,dt}{\int _{0}^{T}\int _{\varOmega }1\,dx\,dt}$$

denotes the integral average on $$\varOmega \times J$$ of the function z.

Let $${\mathcal{T}}_{h}$$ be regular triangulations of Ω, $$h_{\tau }$$ is the diameter of τ and $$h=\max h_{\tau }$$. Furthermore, let $$\mathcal{E}_{h}$$ be the set of element sides of the triangulation $${\mathcal{T}}_{h}$$ with $$\varGamma _{h}=\bigcup \mathcal{E} _{h}$$. Let $$\boldsymbol{V}_{h}\times W_{h}\subset \boldsymbol{V}\times W$$ denote the Raviart–Thomas space  associated with the triangulations $$\mathcal {T}_{h}$$ of Ω. $$P_{k}$$ denotes the space of polynomials of total degree no greater than k ($$k\geq 0$$). Let $$\boldsymbol{V}({\tau })= \{\boldsymbol{v}\in P_{k}^{2}({\tau })+x\cdot P_{k}({\tau })\}$$, $$W({\tau })=P _{k}({\tau })$$. We set

\begin{aligned}& \boldsymbol{V}_{h}:=\bigl\{ \boldsymbol{v}_{h}\in \boldsymbol{V}:\forall {\tau }\in \mathcal {T}_{h}, \boldsymbol{v}_{h}\vert _{\tau }\in \boldsymbol{V}({\tau })\bigr\} , \\& W_{h}:=\bigl\{ w_{h}\in W: \forall {\tau }\in \mathcal {T}_{h},w_{h}\vert _{ \tau }\in W({\tau })\bigr\} , \\& K_{h}:=L^{2}(J;W_{h})\cap K. \end{aligned}

We now discretize (2.3)–(2.9). We calculate $$( \tilde{\boldsymbol{p}}_{h},y_{h},\boldsymbol{p}_{h},\tilde{y}_{h},u_{h})$$ such that

\begin{aligned}& \min_{u_{h} \in K_{h}} \biggl\{ \frac{1}{2} \int _{0}^{T} \bigl( \Vert \tilde{y}_{h} \Vert ^{2}+ \Vert y_{h}-y_{d} \Vert ^{2}+ \Vert u_{h} \Vert ^{2} \bigr)\,dt \biggr\} , \end{aligned}
(2.23)
\begin{aligned}& (y_{htt},w_{h})+(\operatorname {div}{\boldsymbol{p}}_{h},w_{h}) =(f+u_{h},w_{h}), \quad \forall w_{h}\in W_{h}, t\in J, \end{aligned}
(2.24)
\begin{aligned}& (\boldsymbol{p}_{h},\boldsymbol{v}_{h})-( \tilde{y}_{h},\operatorname {div}\boldsymbol{v}_{h}) =0, \quad \forall \boldsymbol{v}_{h}\in \boldsymbol{V}_{h}, t\in J, \end{aligned}
(2.25)
\begin{aligned}& (\operatorname {div}\tilde{\boldsymbol{p}}_{h},w_{h}) =(\tilde{y}_{h},w_{h}), \quad \forall w_{h}\in W_{h}, t\in J, \end{aligned}
(2.26)
\begin{aligned}& (\tilde{\boldsymbol{p}}_{h},\boldsymbol{v}_{h})-(y_{h}, \operatorname {div}\boldsymbol{v}_{h}) =0, \quad \forall \boldsymbol{v}_{h}\in \boldsymbol{V}_{h}, t\in J, \end{aligned}
(2.27)
\begin{aligned}& y_{h}(x,0) =y_{0}^{h}(x), \quad \forall x\in \varOmega , \end{aligned}
(2.28)
\begin{aligned}& y_{ht}(x,0) =y_{1}^{h}(x), \quad \forall x\in \varOmega , \end{aligned}
(2.29)

where $$y_{0}^{h}(x)\in W_{h}$$ and $$y_{1}^{h}(x)\in W_{h}$$ are the mixed elliptic projections of $$y_{0}$$ and $$y_{1}$$. The optimal control problem (2.23)–(2.29) again has an unique solution $$( \tilde{\boldsymbol{p}}_{h},y_{h},\boldsymbol{p}_{h},\tilde{y}_{h},u_{h})$$, and $$(\tilde{\boldsymbol{p}}_{h},y_{h},\boldsymbol{p}_{h},\tilde{y}_{h},u_{h})$$ is the solution of (2.23)–(2.29) if and only if there is a co-state $$(\tilde{\boldsymbol{q}}_{h},z_{h},\boldsymbol{q}_{h},\tilde{z}_{h})$$ such that the following optimality conditions hold:

\begin{aligned}& (y_{htt},w_{h})+(\operatorname {div}{\boldsymbol{p}}_{h},w_{h}) =(f+u_{h},w_{h}), \quad \forall w_{h}\in W_{h}, t\in J, \end{aligned}
(2.30)
\begin{aligned}& (\boldsymbol{p}_{h},\boldsymbol{v}_{h})-( \tilde{y}_{h},\operatorname {div}\boldsymbol{v}_{h}) =0, \quad \forall \boldsymbol{v}_{h}\in \boldsymbol{V}_{h}, t\in J, \end{aligned}
(2.31)
\begin{aligned}& (\operatorname {div}\tilde{\boldsymbol{p}}_{h},w_{h}) =( \tilde{y}_{h},w_{h}), \quad \forall w_{h}\in W_{h}, t\in J, \end{aligned}
(2.32)
\begin{aligned}& (\tilde{\boldsymbol{p}}_{h},\boldsymbol{v}_{h})-(y_{h},\operatorname {div}\boldsymbol{v}_{h}) =0, \quad \forall \boldsymbol{v}_{h}\in \boldsymbol{V}_{h}, t\in J, \end{aligned}
(2.33)
\begin{aligned}& y_{h}(x,0) =y_{0}^{h}(x), \quad \forall x\in \varOmega , \end{aligned}
(2.34)
\begin{aligned}& y_{ht}(x,0) =y_{1}^{h}(x), \quad \forall x\in \varOmega , \end{aligned}
(2.35)
\begin{aligned}& (z_{htt},w_{h})+(\operatorname {div}{\boldsymbol{q}}_{h},w_{h}) =(y_{h}-y_{d},w_{h}), \quad \forall w_{h}\in W_{h}, t\in J, \end{aligned}
(2.36)
\begin{aligned}& (\boldsymbol{q}_{h},\boldsymbol{v}_{h})-( \tilde{z}_{h},\operatorname {div}\boldsymbol{v}_{h}) =0, \quad \forall \boldsymbol{v}_{h}\in \boldsymbol{V}_{h}, t\in J, \end{aligned}
(2.37)
\begin{aligned}& (\operatorname {div}\tilde{\boldsymbol{q}}_{h},w_{h}) =( \tilde{y}_{h}+\tilde{z}_{h},w _{h}), \quad \forall w_{h}\in W_{h}, t\in J, \end{aligned}
(2.38)
\begin{aligned}& (\tilde{\boldsymbol{q}}_{h},\boldsymbol{v}_{h})-(z_{h}, \operatorname {div}\boldsymbol{v}_{h}) =0, \quad \forall \boldsymbol{v}_{h}\in \boldsymbol{V}_{h}, t\in J, \end{aligned}
(2.39)
\begin{aligned}& z_{h}(x,T) =0, \quad \forall x\in \varOmega , \end{aligned}
(2.40)
\begin{aligned}& z_{ht}(x,T) =0, \quad \forall x\in \varOmega , \end{aligned}
(2.41)
\begin{aligned}& \int _{0}^{T}(u_{h}+z_{h}, \tilde{u}_{h}-u_{h}) \,dt \geq 0, \quad \forall \tilde{u}_{h} \in K_{h}. \end{aligned}
(2.42)

For Lemma 2.1, the relationship between $$u_{h}$$ and $$z_{h}$$ is given as follows:

\begin{aligned} u_{h}=\max\{0,{\check{z}_{h} \}}-z_{h}, \end{aligned}
(2.43)

where $${\check{z}_{h}}=\frac{\int _{0}^{T}\int _{\varOmega }z_{h}\,dx\,dt}{\int _{0}^{T}\int _{\varOmega }1\,dx\,dt}$$ is the integral average on $$\varOmega \times J$$ of the function $$z_{h}$$.

Now, we give the local definition of $$\operatorname {div}_{h}$$, $$\operatorname{curl} _{h}:H^{1}(\mathcal {T}_{h})^{2}\rightarrow L^{2}(\varOmega )$$ and $$\nabla _{h}$$, $$\operatorname{Curl}_{h}:H^{1}(\mathcal {T}_{h})\rightarrow L^{2}(\varOmega )^{2}$$, such that for any $$T\in \mathcal {T}_{h}$$

\begin{aligned}& \operatorname {div}_{h}\boldsymbol{v}\vert _{T}:=\operatorname {div}( \boldsymbol{v}\vert _{T}),\quad\quad \operatorname{curl}_{h}\boldsymbol{v}\vert _{T}:= \operatorname{curl}(\boldsymbol{v}\vert _{T}), \end{aligned}
(2.44)
\begin{aligned}& \nabla _{h}\boldsymbol{v}\vert _{T}:=\nabla ( \boldsymbol{v}\vert _{T}),\quad \quad \operatorname{Curl} _{h}\boldsymbol{v}\vert _{T}:=\operatorname{Curl}(\boldsymbol{v}\vert _{T}). \end{aligned}
(2.45)

Set $$P_{h}:W\rightarrow W_{h}$$ to be the orthogonal $$L^{2}(\varOmega )$$-projection into $$W_{h}$$ , which satisfies

\begin{aligned}& (P_{h}w-w,\chi )=0, \quad w\in W, \chi \in W_{h}, \end{aligned}
(2.46)
\begin{aligned}& \Vert P_{h} w-w \Vert _{0,q}\leq C h^{t} \Vert w \Vert _{t,q}, \quad 0\leq t\leq k+1, \text{if } w\in W\cap W^{t,q}(\varOmega ), \end{aligned}
(2.47)
\begin{aligned}& \Vert P_{h} w-w \Vert _{-r}\leq C h^{r+t} \Vert w \Vert _{t},\quad 0\leq r, t\leq k+1, \text{if } w\in H^{t}(\varOmega ). \end{aligned}
(2.48)

Next, introduce the Fortin projection (see  and ) $$\varPi _{h}: \boldsymbol{V}\rightarrow \boldsymbol{V}_{h}$$, which satisfies: for any $$\boldsymbol{q}\in \boldsymbol{V}$$

\begin{aligned}& \bigl(\operatorname {div}(\varPi _{h}\boldsymbol{q}-\boldsymbol{q}),w_{h}\bigr)=0, \quad \forall \boldsymbol{q}\in \boldsymbol{V}, w_{h}\in W_{h}, \end{aligned}
(2.49)
\begin{aligned}& \Vert \boldsymbol{q}-\varPi _{h}\boldsymbol{q}\Vert _{0,q}\leq Ch^{r} \Vert \boldsymbol{q}\Vert _{r,q}, \quad 1/q< r\leq k+1, \forall \boldsymbol{q}\in \boldsymbol{V}\cap \bigl(W^{r,q}( \varOmega )\bigr)^{2}, \end{aligned}
(2.50)
\begin{aligned}& \bigl\Vert \operatorname {div}(\boldsymbol{q}-\varPi _{h}\boldsymbol{q}) \bigr\Vert _{0}\leq Ch^{r} \Vert \operatorname {div}\boldsymbol{q}\Vert _{r}, \quad 0\leq r\leq k+1, \forall \operatorname {div}\boldsymbol{q}\in H^{r}(\varOmega ). \end{aligned}
(2.51)

\begin{aligned} \operatorname {div}\circ \varPi _{h}=P_{h}\circ \operatorname {div}:\boldsymbol{V}\rightarrow W_{h} \quad \text{and}\quad \operatorname {div}(I- \varPi _{h})\boldsymbol{V}\perp W_{h}, \end{aligned}
(2.52)

where I denotes the identity operator.

Next, the intermediate variable $$\tilde{u}\in K$$ is introduced as follows:

\begin{aligned}& \bigl(y_{tt}(\tilde{u}),w\bigr)+\bigl(\operatorname {div}{\boldsymbol{p}}(\tilde{u}),w\bigr) =(f+\tilde{u},w), \quad \forall w\in W, t\in J, \end{aligned}
(2.53)
\begin{aligned}& \bigl(\boldsymbol{p}(\tilde{u}),\boldsymbol{v}\bigr)-\bigl( \tilde{y}(\tilde{u}),\operatorname {div}\boldsymbol{v}\bigr) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, t\in J, \end{aligned}
(2.54)
\begin{aligned}& \bigl(\operatorname {div}\tilde{\boldsymbol{p}}(\tilde{u}),w\bigr) =\bigl( \tilde{y}(\tilde{u}),w\bigr), \quad \forall w\in W, t\in J, \end{aligned}
(2.55)
\begin{aligned}& \bigl(\tilde{\boldsymbol{p}}(\tilde{u}),\boldsymbol{v}\bigr)-\bigl(y( \tilde{u}),\operatorname {div}\boldsymbol{v}\bigr) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, t\in J, \end{aligned}
(2.56)
\begin{aligned}& y(\tilde{u}) (x,0) =y_{0}(x), \quad \forall x\in \varOmega , \end{aligned}
(2.57)
\begin{aligned}& y_{t}(\tilde{u}) (x,0) =y_{1}(x), \quad \forall x\in \varOmega , \end{aligned}
(2.58)
\begin{aligned}& \bigl(z_{tt}(\tilde{u}),w\bigr)+\bigl(\operatorname {div}{\boldsymbol{q}}(\tilde{u}),w\bigr) =\bigl(y(\tilde{u})-y _{d},w\bigr), \quad \forall w\in W, t\in J, \end{aligned}
(2.59)
\begin{aligned}& \bigl(\boldsymbol{q}(\tilde{u}),\boldsymbol{v}\bigr)-\bigl(\tilde{z}( \tilde{u}),\operatorname {div}\boldsymbol{v}\bigr) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, t\in J, \end{aligned}
(2.60)
\begin{aligned}& \bigl(\operatorname {div}\tilde{\boldsymbol{q}}(\tilde{u}),w\bigr) =\bigl( \tilde{y}(\tilde{u})+ \tilde{z}(\tilde{u}),w\bigr), \quad \forall w\in W, t\in J, \end{aligned}
(2.61)
\begin{aligned}& \bigl(\tilde{\boldsymbol{q}}(\tilde{u}),\boldsymbol{v}\bigr)- \bigl(z(\tilde{u}),\operatorname {div}\boldsymbol{v}\bigr) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, t \in J, \end{aligned}
(2.62)
\begin{aligned}& z(\tilde{u}) (x,T) =0, \quad \forall x\in \varOmega , \end{aligned}
(2.63)
\begin{aligned}& z_{t}(\tilde{u}) (x,T) =0, \quad \forall x\in \varOmega . \end{aligned}
(2.64)

Next, we present mixed elliptic constructions $$(\tilde{\bar{\boldsymbol{p}}}, \bar{y}, \bar{\boldsymbol{p}}, \tilde{\bar{y}},\tilde{\bar{\boldsymbol{q}}}, \bar{z},\bar{\boldsymbol{q}}, \tilde{\bar{z}})\in (\boldsymbol{V}\times W)^{4}$$:

\begin{aligned}& (\tilde{\bar{\boldsymbol{p}}},\boldsymbol{v})-(\bar{y},\operatorname {div}\boldsymbol{v}) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(2.65)
\begin{aligned}& (\operatorname {div}\tilde{\bar{\boldsymbol{p}}},w) =(\tilde{y}_{h},w), \quad \forall w\in W, \end{aligned}
(2.66)
\begin{aligned}& (\bar{\boldsymbol{p}},\boldsymbol{v})-(\tilde{\bar{y}},\operatorname {div}\boldsymbol{v}) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(2.67)
\begin{aligned}& (\operatorname {div}\bar{\boldsymbol{p}},w) =(f+u_{h}-y_{htt},w), \quad \forall w\in W, \end{aligned}
(2.68)
\begin{aligned}& (\tilde{\bar{\boldsymbol{q}}},\boldsymbol{v})-(\bar{z},\operatorname {div}\boldsymbol{v}) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(2.69)
\begin{aligned}& (\operatorname {div}\tilde{\bar{\boldsymbol{q}}},w) =(\tilde{z}_{h}+ \tilde{y}_{h},w), \quad \forall w\in W, \end{aligned}
(2.70)
\begin{aligned}& (\bar{\boldsymbol{q}},\boldsymbol{v})-(\tilde{\bar{z}},\operatorname {div}\boldsymbol{v}) =0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(2.71)
\begin{aligned}& (\operatorname {div}\bar{\boldsymbol{q}},w) =(y_{h}-y_{d}-z_{htt},w), \quad \forall w\in W. \end{aligned}
(2.72)

For simplicity of presentation, we resolve the errors in following forms:

\begin{aligned}& \tilde{\boldsymbol{p}}-\tilde{\boldsymbol{p}}_{h}=\tilde{\boldsymbol{p}}-\tilde{\boldsymbol{p}}(u _{h})+\tilde{\boldsymbol{p}}(u_{h})- \tilde{\bar{\boldsymbol{p}}}+ \tilde{\bar{\boldsymbol{p}}} -\tilde{\boldsymbol{p}}_{h}=r_{1}+e_{1}+ \eta _{1}, \end{aligned}
(2.73)
\begin{aligned}& y-y_{h}=y-y(u_{h})+y(u_{h})- \bar{y}+\bar{y} -y_{h}=r_{2}+e_{2}+\eta _{2}, \end{aligned}
(2.74)
\begin{aligned}& \boldsymbol{p}-\boldsymbol{p}_{h}=\boldsymbol{p}-\boldsymbol{p}(u_{h})+\boldsymbol{p}(u_{h})- \bar{\boldsymbol{p}}+\bar{\boldsymbol{p}}-\boldsymbol{p}_{h}=r_{3}+e_{3}+ \eta _{3}, \end{aligned}
(2.75)
\begin{aligned}& \tilde{y}-\tilde{y}_{h}=\tilde{y}-\tilde{y}(u_{h})+ \tilde{y}(u_{h})- \tilde{\bar{y}}+\tilde{\bar{y}} - \tilde{y}_{h}=r_{4}+e_{4}+\eta _{4}, \end{aligned}
(2.76)
\begin{aligned}& \tilde{\boldsymbol{q}}-\tilde{\boldsymbol{q}}_{h}=\tilde{\boldsymbol{q}}-\tilde{\boldsymbol{q}}(u _{h})+\tilde{\boldsymbol{q}}(u_{h})- \tilde{\bar{\boldsymbol{q}}}+ \tilde{\bar{\boldsymbol{q}}} -\tilde{\boldsymbol{q}}_{h}=r_{5}+e_{5}+ \eta _{5}, \end{aligned}
(2.77)
\begin{aligned}& z-z_{h}=z-z(u_{h})+z(u_{h})- \bar{z}+\bar{z} -z_{h}=r_{6}+e_{6}+\eta _{6}, \end{aligned}
(2.78)
\begin{aligned}& \boldsymbol{q}-\boldsymbol{q}_{h}=\boldsymbol{q}-\boldsymbol{q}(u_{h})+\boldsymbol{q}(u_{h})- \bar{\boldsymbol{q}}+\bar{\boldsymbol{q}}-\boldsymbol{q}_{h}=r_{7}+e_{7}+ \eta _{7}, \end{aligned}
(2.79)
\begin{aligned}& \tilde{z}-\tilde{z}_{h}=\tilde{z}-\tilde{z}(u_{h})+ \tilde{z}(u_{h})- \tilde{\bar{z}}+\tilde{\bar{z}} - \tilde{z}_{h}=r_{8}+e_{8}+\eta _{8}. \end{aligned}
(2.80)

From mixed elliptic reconstructions , we derive the error estimates as below.

Lemma 2.2

([8, 14])

For Raviart–Thomas elements, there exists a positive constant C which depends the domain Ω, the shape regularity of the elements and polynomial degree k such that

\begin{aligned}& \Vert \eta _{2} \Vert \leq C \Bigl( \bigl\Vert h^{1+\min \{1,k\}}(\operatorname{div} \tilde{\boldsymbol{p}}_{h}- \tilde{y}_{h}) \bigr\Vert +\min_{w_{h}\in W_{h}} \bigl\Vert h( \tilde{\boldsymbol{p}}_{h}-\nabla _{h}w_{h}) \bigr\Vert \Bigr), \end{aligned}
(2.81)
\begin{aligned}& \Vert \eta _{2t} \Vert \leq C \Bigl( \bigl\Vert h^{1+\min \{1,k\}}(\operatorname{div} \tilde{\boldsymbol{p}}_{ht}- \tilde{y}_{ht}) \bigr\Vert +\min_{w_{h}\in W_{h}} \bigl\Vert h( \tilde{\boldsymbol{p}}_{ht}-\nabla _{h}w_{h}) \bigr\Vert \Bigr), \end{aligned}
(2.82)
\begin{aligned}& \Vert \eta _{2tt} \Vert \leq C \Bigl( \bigl\Vert h^{1+\min \{1,k\}}(\operatorname{div} \tilde{\boldsymbol{p}}_{htt}- \tilde{y}_{htt}) \bigr\Vert +\min_{w_{h}\in W_{h}} \bigl\Vert h( \tilde{\boldsymbol{p}}_{htt}-\nabla _{h}w_{h}) \bigr\Vert \Bigr), \end{aligned}
(2.83)
\begin{aligned}& \Vert \eta _{2ttt} \Vert \leq C \Bigl( \bigl\Vert h^{1+\min \{1,k\}}(\operatorname{div} \tilde{\boldsymbol{p}}_{httt}- \tilde{y}_{httt}) \bigr\Vert +\min_{w_{h}\in W _{h}} \bigl\Vert h( \tilde{\boldsymbol{p}}_{httt}-\nabla _{h}w_{h}) \bigr\Vert \Bigr), \end{aligned}
(2.84)
\begin{aligned}& \Vert \eta _{4} \Vert \leq C \Bigl( \bigl\Vert h^{1+\min \{1,k\}}(y_{htt}+\operatorname{div} \boldsymbol{p}_{h}-f-u_{h}) \bigr\Vert +\min_{w_{h}\in W_{h}} \bigl\Vert h(\boldsymbol{p}_{h}- \nabla _{h}w_{h}) \bigr\Vert \Bigr), \end{aligned}
(2.85)
\begin{aligned}& \Vert \eta _{4t} \Vert \leq C \Bigl( \bigl\Vert h^{1+\min \{1,k\}}(y_{htt}+\operatorname{div} \boldsymbol{p}_{h}-f-u_{h})_{t} \bigr\Vert +\min_{w_{h}\in W_{h}} \bigl\Vert h(\boldsymbol{p}_{ht}- \nabla _{h}w_{h}) \bigr\Vert \Bigr), \end{aligned}
(2.86)
\begin{aligned}& \Vert \eta _{4tt} \Vert \leq C \Bigl( \bigl\Vert h^{1+\min \{1,k\}}(y_{htt}+\operatorname{div} \boldsymbol{p}_{h}-f-u_{h})_{tt} \bigr\Vert +\min_{w_{h}\in W_{h}} \bigl\Vert h(\boldsymbol{p}_{htt}- \nabla _{h}w_{h}) \bigr\Vert \Bigr), \end{aligned}
(2.87)
\begin{aligned}& \Vert \eta _{6} \Vert \leq C \Bigl( \bigl\Vert h^{1+\min \{1,k\}}(\operatorname{div} \tilde{\boldsymbol{q}}_{h}- \tilde{y}_{h}-\tilde{z}_{h}) \bigr\Vert +\min _{w_{h}\in W_{h}} \bigl\Vert h( \tilde{\boldsymbol{q}}_{h}-\nabla _{h}w_{h}) \bigr\Vert \Bigr), \end{aligned}
(2.88)
\begin{aligned}& \Vert \eta _{6t} \Vert \leq C \Bigl( \bigl\Vert h^{1+\min \{1,k\}}(\operatorname{div} \tilde{\boldsymbol{q}}_{ht}- \tilde{y}_{ht}-\tilde{z}_{ht}) \bigr\Vert +\min _{w_{h}\in W_{h}} \bigl\Vert h( \tilde{\boldsymbol{q}}_{ht}-\nabla _{h}w_{h}) \bigr\Vert \Bigr), \end{aligned}
(2.89)
\begin{aligned}& \Vert \eta _{6tt} \Vert \leq C \Bigl( \bigl\Vert h^{1+\min \{1,k\}}(\operatorname{div} \tilde{\boldsymbol{q}}_{htt}- \tilde{y}_{htt}-\tilde{z}_{htt}) \bigr\Vert +\min _{w_{h}\in W_{h}} \bigl\Vert h( \tilde{\boldsymbol{q}}_{htt}-\nabla _{h}w_{h}) \bigr\Vert \Bigr), \end{aligned}
(2.90)
\begin{aligned}& \Vert \eta _{8} \Vert \leq C \Bigl( \bigl\Vert h^{1+\min \{1,k\}}(z_{htt}+\operatorname{div} \boldsymbol{q}_{h}-y_{h}+y_{d}) \bigr\Vert +\min_{w_{h}\in W_{h}} \bigl\Vert h(\boldsymbol{q}_{h}- \nabla _{h}w_{h}) \bigr\Vert \Bigr), \end{aligned}
(2.91)
\begin{aligned}& \Vert \eta _{8t} \Vert \leq C \Bigl( \bigl\Vert h^{1+\min \{1,k\}}(z_{htt}+\operatorname{div} \boldsymbol{q}_{h}-y_{h}+y_{d})_{t} \bigr\Vert +\min_{w_{h}\in W_{h}} \bigl\Vert h(\boldsymbol{q}_{ht}- \nabla _{h}w_{h}) \bigr\Vert \Bigr), \end{aligned}
(2.92)
\begin{aligned}& \Vert \eta _{1} \Vert \leq C \bigl( \bigl\Vert h^{\frac{1}{2}}J(\tilde{\boldsymbol{p}}_{h} \cdot \boldsymbol{t}) \bigr\Vert _{0,\varGamma _{h}} + \Vert h \operatorname{curl}_{h} \tilde{ \boldsymbol{p}}_{h} \Vert + \bigl\Vert h(\operatorname{div}\tilde{\boldsymbol{p}}_{h}-\tilde{y}_{h}) \bigr\Vert \bigr), \end{aligned}
(2.93)
\begin{aligned}& \Vert \eta _{3} \Vert \leq C \bigl( \bigl\Vert h^{\frac{1}{2}}J(\boldsymbol{p}_{h}\cdot \boldsymbol{t}) \bigr\Vert _{0,\varGamma _{h}}+ \Vert h \operatorname{curl}_{h}\boldsymbol{p}_{h} \Vert + \bigl\Vert h(y _{htt}+\operatorname{div} \boldsymbol{p}_{h}-f-u_{h}) \bigr\Vert \bigr), \end{aligned}
(2.94)
\begin{aligned}& \Vert \eta _{5} \Vert \leq C \bigl( \bigl\Vert h^{\frac{1}{2}}J(\tilde{\boldsymbol{q}}_{h} \cdot \boldsymbol{t}) \bigr\Vert _{0,\varGamma _{h}} + \Vert h \operatorname{curl}_{h} \tilde{ \boldsymbol{q}}_{h} \Vert + \bigl\Vert h(\operatorname{div}\tilde{\boldsymbol{q}}_{h}-\tilde{y}_{h}- \tilde{y}_{h}) \bigr\Vert \bigr), \end{aligned}
(2.95)
\begin{aligned}& \Vert \eta _{7} \Vert \leq C \bigl( \bigl\Vert h^{\frac{1}{2}}J(\boldsymbol{q}_{h}\cdot \boldsymbol{t}) \bigr\Vert _{0,\varGamma _{h}} + \Vert h \operatorname{curl}_{h}\boldsymbol{q}_{h} \Vert + \bigl\Vert h(z _{htt}+\operatorname{div} \boldsymbol{q}_{h}-y_{h}+y_{d}) \bigr\Vert \bigr), \end{aligned}
(2.96)
\begin{aligned}& \Vert \operatorname{div}\eta _{1} \Vert \leq C \Vert \operatorname{div}\tilde{\boldsymbol{p}}_{h}-\tilde{y} _{h} \Vert , \Vert \operatorname{div} \eta _{3} \Vert \leq C \Vert y_{htt}+\operatorname{div} \boldsymbol{p}_{h}-f-u _{h} \Vert , \end{aligned}
(2.97)
\begin{aligned}& \Vert \operatorname{div} \eta _{5} \Vert \leq C \Vert \operatorname{div}\tilde{\boldsymbol{q}}_{h}- \tilde{y}_{h}- \tilde{z}_{h} \Vert , \Vert \operatorname{div}\eta _{7} \Vert \leq C \Vert z_{htt}+ \operatorname{div} \boldsymbol{q}_{h}-y_{h}+y_{d} \Vert , \end{aligned}
(2.98)

where $$\nabla _{h}$$ and $$\operatorname{curl}_{h}$$ have been defined in (2.44)(2.45), $$J(\boldsymbol{v}\cdot \boldsymbol{t})$$ denotes the jump of $$\boldsymbol{v}\cdot \boldsymbol{t}$$ across the element edge E for all $$\boldsymbol{v}\in \boldsymbol{V}$$ with t being the tangential unit vector along the edge $$E\in \varGamma _{h}$$.

Error estimation of optimal control

In this part, a posteriori error estimation of optimal control problems shall be given. From (2.53)–(2.56) and (2.65)–(2.68), we obtain the error equations

\begin{aligned}& (e_{1},\boldsymbol{v})-(e_{2},\operatorname {div}\boldsymbol{v})=0, \quad \forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(3.1)
\begin{aligned}& (\operatorname {div}e_{1},w)=(e_{4}+\eta _{4},w),\quad\forall w\in W, \end{aligned}
(3.2)
\begin{aligned}& (e_{3},\boldsymbol{v})-(e_{4},\operatorname {div}\boldsymbol{v})=0,\quad\forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(3.3)
\begin{aligned}& (e_{2tt},w)+(\operatorname {div}e_{3},w)=-(\eta _{2tt},w),\quad\forall w\in W. \end{aligned}
(3.4)

Lemma 3.1

Let $$e_{1}$$$$e_{4}$$ satisfy (3.1)(3.4). Then we have

\begin{aligned} & \Vert e_{2} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))} + \Vert e_{1} \Vert _{L^{\infty }(J;H( \operatorname{div};\varOmega ))}+ \Vert e_{4} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert e _{3} \Vert _{L^{\infty }(J;H(\operatorname{div};\varOmega ))} \\ &\quad \leq C\bigl( \Vert \eta _{4t} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \bigl\Vert y_{1}-y_{1}^{h} \bigr\Vert + \bigl\Vert \eta _{2t}(0) \bigr\Vert + \bigl\Vert \Delta y_{0}+\tilde{y}_{h}(0) \bigr\Vert + \Vert \eta _{4} \Vert _{L ^{\infty }(J;L^{2}(\varOmega ))} \\ &\quad\quad{} + \Vert \eta _{2tt} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert \eta _{2ttt} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{4tt} \Vert _{L^{2}(J;L^{2}(\varOmega ))} + \bigl\Vert y_{0}-y_{0}^{h} \bigr\Vert \\ &\quad\quad{}+ \bigl\Vert \operatorname{div} {\eta _{3}}(0) \bigr\Vert + \bigl\Vert \Delta ^{2}y_{0}- \operatorname{div} p_{h}(0) \bigr\Vert \bigr) \\ &\quad\quad{}+ \bigl\Vert \Delta y_{1}+ \tilde{y}_{ht}(0) \bigr\Vert + \bigl\Vert \eta _{4t}(0) \bigr\Vert )+ \bigl\Vert \eta _{2}(0) \bigr\Vert ). \end{aligned}
(3.5)

Proof

Differentiating (3.1)–(3.2) with respect to t, we get

\begin{aligned}& (e_{1t},\boldsymbol{v})-(e_{2t},\operatorname {div}\boldsymbol{v}) =0,\quad\forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(3.6)
\begin{aligned}& (\operatorname {div}e_{1t},w) =(e_{4t}+\eta _{4t},w),\quad\forall w\in W. \end{aligned}
(3.7)

We choose $$\boldsymbol{v}=e_{3}$$ in (3.6), $$w=-e_{4}$$ in (3.7), $$\boldsymbol{v}=-e_{1t}$$ in (3.3) and $$w=e_{2t}$$ in (3.4), separately, then add up the four equations and obtain

\begin{aligned} (e_{2t},e_{2tt})+(e_{4t},e_{4})=-( \eta _{4t},e_{4})-(\eta _{2tt},e_{2t}). \end{aligned}
(3.8)

We integrate (3.8) from 0 to t, use Gronwall’s inequality and the Cauchy inequality, then we obtain

\begin{aligned}& \Vert e_{4} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert e_{2t} \Vert _{L^{\infty }(J;L ^{2}(\varOmega ))} \\& \quad \leq C\bigl( \Vert \eta _{4t} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{2tt} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \bigl\Vert e_{2t}(0) \bigr\Vert + \bigl\Vert e_{4}(0) \bigr\Vert \bigr), \end{aligned}
(3.9)

where

\begin{aligned}& \bigl\Vert e_{2t}(0) \bigr\Vert \leq \bigl\Vert y_{1}-y_{1}^{h} \bigr\Vert + \bigl\Vert \eta _{2t}(0) \bigr\Vert , \end{aligned}
(3.10)
\begin{aligned}& \bigl\Vert e_{4}(0) \bigr\Vert \leq \bigl\Vert \Delta y_{0}+\tilde{y}_{h}(0) \bigr\Vert + \bigl\Vert \eta _{4}(0) \bigr\Vert . \end{aligned}
(3.11)

Note that

\begin{aligned} { \int }_{0}^{t}e_{2s}\,dt=e_{2}(t)-e_{2}(0), \end{aligned}

then we have

\begin{aligned} \begin{aligned}[b] \Vert e_{2} \Vert &\leq C\bigl( \Vert e_{2t} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \bigl\Vert e_{2}(0) \bigr\Vert \bigr)\\&\leq C\bigl( \Vert e_{2t} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \bigl\Vert y_{0}-y_{0} ^{h} \bigr\Vert + \bigl\Vert \eta _{2}(0) \bigr\Vert \bigr).\end{aligned} \end{aligned}
(3.12)

Letting $$\boldsymbol{v}=e_{1}$$ in (3.1), $$w=e_{2}$$ in (3.2), $$\boldsymbol{v}=e_{3}$$ in (3.3) and $$w=e_{4}$$ in (3.4), respectively. We get

\begin{aligned} & \Vert e_{1} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}\leq \Vert \eta _{4} \Vert _{L^{ \infty }(J;L^{2}(\varOmega ))} + \Vert e_{4} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert e_{2} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}, \end{aligned}
(3.13)
\begin{aligned} & \Vert e_{3} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}\leq \Vert e_{2tt} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert \eta _{2tt} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert e_{4} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}. \end{aligned}
(3.14)

Differentiating (3.3)–(3.4) and (3.6)–(3.7) with respect to t, we get

\begin{aligned}& (e_{3t},\boldsymbol{v})-(e_{4t},\operatorname {div}\boldsymbol{v}) =0,\quad\forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(3.15)
\begin{aligned}& (e_{2ttt},w)+(\operatorname {div}e_{3t},w) =-(\eta _{2ttt},w),\quad\forall w\in W, \end{aligned}
(3.16)
\begin{aligned}& (e_{1tt},\boldsymbol{v})-(e_{2tt},\operatorname {div}\boldsymbol{v}) =0,\quad\forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(3.17)
\begin{aligned}& (\operatorname {div}e_{1tt},w) =(e_{4tt}+\eta _{4tt},w),\quad\forall w\in W. \end{aligned}
(3.18)

We choose $$v=-e_{1tt}$$ in (3.15), $$w=e_{2tt}$$ in (3.16), $$v=e_{3t}$$ in (3.17), $$w=-e_{4t}$$ in (3.18), separately. We derive the following after addition for four equations:

\begin{aligned} (e_{2ttt},e_{2tt})+(e_{4tt},e_{4t})=-( \eta _{4tt},e_{4t})-(\eta _{2ttt},e _{2tt}). \end{aligned}
(3.19)

Similar to (3.9), we derive

\begin{aligned}& \Vert e_{2tt} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert e_{4t} \Vert _{L^{\infty }(J;L ^{2}(\varOmega ))} \\& \quad \leq C\bigl( \Vert \eta _{2ttt} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{4tt} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \bigl\Vert e_{2tt}(0) \bigr\Vert + \bigl\Vert e_{4t}(0) \bigr\Vert \bigr). \end{aligned}
(3.20)

Taking $$t=0$$ and $$w=e_{2tt}(0)$$ in (3.4) leads to

\begin{aligned} \bigl\Vert e_{2tt}(0) \bigr\Vert &\leq \bigl\Vert \operatorname {div}e_{3}(0) \bigr\Vert + \bigl\Vert \eta _{2tt}(0) \bigr\Vert \\ &\leq \bigl\Vert \operatorname {div}{\eta _{3}}(0) \bigr\Vert + \bigl\Vert \eta _{2tt}(0) \bigr\Vert + \bigl\Vert \Delta ^{2}y_{0}-\operatorname {div}p_{h}(0) \bigr\Vert . \end{aligned}
(3.21)

Note that

\begin{aligned} \bigl\Vert e_{4t}(0) \bigr\Vert &\leq \bigl\Vert \tilde{y}_{t}(u_{h}) (0)-\tilde{y}_{ht}(0) \bigr\Vert + \bigl\Vert \eta _{4t}(0) \bigr\Vert \\ &\leq \bigl\Vert \Delta y_{1}+\tilde{y}_{ht}(0) \bigr\Vert + \bigl\Vert \eta _{4t}(0) \bigr\Vert . \end{aligned}
(3.22)

At last, setting $$w=\operatorname {div}e_{1}$$ and $$w=\operatorname {div}e_{3}$$ in (3.2) and (3.4), we get

\begin{aligned}& \Vert \operatorname {div}e_{1} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}\leq \Vert \eta _{4} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert e_{4} \Vert _{L^{\infty }(J;L^{2}( \varOmega ))}, \end{aligned}
(3.23)
\begin{aligned}& \Vert \operatorname {div}e_{3} \Vert _{L^{\infty }(J;L^{\infty }(\varOmega ))}\leq \Vert e_{2tt} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert \eta _{2tt} \Vert _{L^{\infty }(J;L ^{2}(\varOmega ))}. \end{aligned}
(3.24)

Thus, using (3.9)–(3.14) and (3.20)–(3.24), we complete the proof. □

By (2.59)–(2.62) and (2.69)–(2.72), we obtain the error equations

\begin{aligned}& (e_{5},\boldsymbol{v})-(e_{6},\operatorname {div}\boldsymbol{v}) =0,\quad\forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(3.25)
\begin{aligned}& (\operatorname {div}e_{5},w) =(e_{4}+e_{8}+ \eta _{4}+\eta _{8},w),\quad\forall w \in W, \end{aligned}
(3.26)
\begin{aligned}& (e_{7},\boldsymbol{v})-(e_{8},\operatorname {div}\boldsymbol{v}) =0,\quad\forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(3.27)
\begin{aligned}& (e_{6tt},w)+(\operatorname {div}e_{7},w) =(\eta _{6tt}+\eta _{2}+e_{2},w),\quad\forall w\in W. \end{aligned}
(3.28)

Lemma 3.2

Let $$e_{5}$$$$e_{8}$$ satisfy (3.25)(3.28). Then we get

\begin{aligned} & \Vert e_{6} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert e_{5} \Vert _{L^{\infty }(J;H( \operatorname{div};\varOmega ))}+ \Vert e_{8} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert e _{7} \Vert _{L^{2}(J;H(\operatorname{div};\varOmega ))} \\ &\quad \leq C\bigl( \Vert \eta _{4} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert \eta _{4t} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{4tt} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{2} \Vert _{L^{2}(J;L^{2}(\varOmega ))} \\ &\quad\quad{} + \Vert \eta _{6tt} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{8} \Vert _{L^{\infty }(J;L ^{2}(\varOmega ))} + \Vert \eta _{8t} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{2ttt} \Vert _{L^{2}(J;L^{2}(\varOmega ))} \\ &\quad\quad{} + \Vert e_{4} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert e_{2} \Vert _{L^{2}(J;L^{2}( \varOmega ))}+ \bigl\Vert \Delta y_{1}+\tilde{y}_{ht}(0) \bigr\Vert + \bigl\Vert \eta _{4t}(0) \bigr\Vert \\ &\quad\quad{} + \bigl\Vert \operatorname{div} \eta _{3}(0) \bigr\Vert + \bigl\Vert \eta _{2tt}(0) \bigr\Vert + \bigl\Vert \Delta ^{2} y_{0}- \operatorname{div} \boldsymbol{p}_{h}(0) \bigr\Vert \bigr). \end{aligned}
(3.29)

Proof

At first, setting $$t=T$$ in (2.69)–(2.70) and (3.25)–(3.26), we derive

\begin{aligned} e_{6}(T)=e_{6t}(T)=0 \end{aligned}
(3.30)

and

\begin{aligned} \bigl\Vert e_{8}(T) \bigr\Vert \leq C\bigl( \bigl\Vert e_{4}(T) \bigr\Vert + \bigl\Vert \eta _{4}(T) \bigr\Vert + \bigl\Vert \eta _{8}(T) \bigr\Vert \bigr). \end{aligned}
(3.31)

Differentiating (3.25)–(3.26) with respect to t, we obtain

\begin{aligned}& (e_{5t},\boldsymbol{v})-(e_{6t},\operatorname {div}\boldsymbol{v}) =0,\quad\forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(3.32)
\begin{aligned}& (\operatorname {div}e_{5t},w) =(e_{4t}+e_{8t}+ \eta _{4t}+\eta _{8t},w),\quad\forall w\in W. \end{aligned}
(3.33)

Let $$\boldsymbol{v}=-e_{7}$$ in (3.32), $$w=e_{8}$$ in (3.33), $$\boldsymbol{v}=e_{5t}$$ in (3.27) and $$w=-e_{6t}$$ in (3.28), separately. After adding up the new equations, we have

\begin{aligned} -(e_{6tt},e_{6t})-(e_{8t},e_{8})=(e_{4t}+ \eta _{4t}+\eta _{8t},e_{8})-( \eta _{6tt}+\eta _{2}+e_{2},e_{6t}). \end{aligned}
(3.34)

Integrating (3.34) from t to T, from (3.30), Gronwall’s inequality and the Cauchy inequality, it is easy to see that

\begin{aligned} & \Vert e_{6t} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert e_{8} \Vert _{L^{\infty }(J;L ^{2}(\varOmega ))} \\ &\quad \leq C\bigl( \Vert \eta _{4t} \Vert _{L^{2}(J;L^{2}(\varOmega ))} + \Vert e _{4t} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{8t} \Vert _{L^{2}(J;L^{2}(\varOmega ))} \\ &\quad \quad{} + \Vert e_{2} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{2} \Vert _{L^{2}(J;L^{2}( \varOmega ))} + \Vert \eta _{6tt} \Vert _{L^{2}(J;L^{2}(\varOmega ))}\bigr)+ \bigl\Vert e_{8}(T) \bigr\Vert . \end{aligned}
(3.35)

Letting $$\boldsymbol{v}=e_{7}$$ in (3.27), we get

\begin{aligned} (e_{7},e_{7})=(e_{8},\operatorname {div}e_{7}). \end{aligned}
(3.36)

Next, for (3.25), we differentiate two times with respect to t, and set $$\boldsymbol{v}=e_{7}$$. for (3.26), we also differentiate two times with respect to t, and set $$w=e_{8}$$. For (3.27), we set $$\boldsymbol{v}=e_{5tt}$$. For (3.28), we set $$w=\operatorname {div}e_{7}$$. Combining the new four equalities, we derive

\begin{aligned} \Vert \operatorname {div}e_{7} \Vert _{L^{2}(J;L^{2}(\varOmega ))}&\leq C\bigl( \Vert e_{2} \Vert _{L^{2}(J;L ^{2}(\varOmega ))}+ \Vert \eta _{6tt} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{2} \Vert _{L^{2}(J;L^{2}(\varOmega ))} + \Vert \eta _{4t} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))} \\ & \quad{} + \Vert \eta _{8t} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))} \bigr) + \Vert e _{4} \Vert _{L^{2}(J;L^{2}(\varOmega ))})+ \Vert e_{8t} \Vert _{L^{2}(J;L^{2}(\varOmega ))}). \end{aligned}
(3.37)

At last, similar to (3.13)–(3.14), (3.23)–(3.24) and (3.36), we have

\begin{aligned}& \begin{aligned}[b] \Vert e_{5} \Vert _{L^{\infty }(J;H(\operatorname {div};\varOmega ))}&\leq C\bigl( \Vert e_{4} \Vert _{L ^{\infty }(J;L^{2}(\varOmega ))}+ \Vert e_{8} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))} \\ & \quad {}+ \Vert \eta _{4} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert \eta _{8} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}\bigr) + \Vert e_{6} \Vert _{L^{\infty }(J;L ^{2}(\varOmega ))}), \end{aligned} \end{aligned}
(3.38)
\begin{aligned}& \begin{aligned}[b] & \Vert e_{7} \Vert _{L^{2}(J;H(\operatorname {div};\varOmega ))} \\ &\quad \leq C\bigl( \Vert e_{2} \Vert _{L^{2}(J;L ^{2}(\varOmega ))}+ \Vert \eta _{6tt} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{2} \Vert _{L^{2}(J;L^{2}(\varOmega ))} + \Vert \eta _{4t} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))} \\ &\quad \quad {}+ \Vert \eta _{8t} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))} + \Vert e _{4} \Vert _{L^{2}(J;L^{2}(\varOmega ))}\bigr)+ \Vert e_{8t} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert e_{8} \Vert _{L^{2}(J;L^{2}(\varOmega ))}). \end{aligned} \end{aligned}
(3.39)

Thus, combining Lemma 3.1, (3.31), (3.35)–(3.40) and (3.5), we complete the proof. □

Choosing $$\tilde{u}=u$$ and $$\tilde{u}=u_{h}$$ in (2.53)–(2.64), respectively, we have the error equations

\begin{aligned} &(r_{1},\boldsymbol{v})-(r_{2},\operatorname {div}\boldsymbol{v})=0,\quad\forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(3.40)
\begin{aligned} & (\operatorname {div}r_{1},w)=(r_{4},w),\quad\forall w\in W, \end{aligned}
(3.41)
\begin{aligned} &(r_{3},\boldsymbol{v})-(r_{4},\operatorname {div}\boldsymbol{v})=0,\quad\forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(3.42)
\begin{aligned} &(r_{2tt},w)+(\operatorname {div}r_{3},w)=(u-u_{h},w), \quad\forall w\in W, \end{aligned}
(3.43)
\begin{aligned} &(r_{5},\boldsymbol{v})-(r_{6},\operatorname {div}\boldsymbol{v})=0,\quad\forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(3.44)
\begin{aligned} &(\operatorname {div}r_{5},w)=(r_{4}+r_{8},w), \quad\forall w\in W, \end{aligned}
(3.45)
\begin{aligned} &(r_{7},\boldsymbol{v})-(r_{8},\operatorname {div}\boldsymbol{v})=0,\quad\forall \boldsymbol{v}\in \boldsymbol{V}, \end{aligned}
(3.46)
\begin{aligned} &(r_{6tt},w)+(\operatorname {div}r_{7},w)=(r_{2},w), \quad\forall w\in W. \end{aligned}
(3.47)

Similar to Lemmas 3.1 and 3.2, Lemma 3.3 is given below.

Lemma 3.3

Let $$r_{1}$$$$r_{8}$$ satisfy (3.40)(3.47). Then we have

\begin{aligned}& \begin{aligned}[b] & \Vert r_{2t} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert r_{4} \Vert _{L^{\infty }(J;L ^{2}(\varOmega ))}+ \Vert r_{1} \Vert _{L^{\infty }(J;H(\operatorname{div};\varOmega ))}+ \Vert r _{3} \Vert _{L^{2}(J;H(\operatorname{div};\varOmega ))} \\ & \quad \leq C \Vert u-u_{h} \Vert _{L^{2}(J;L^{2}(\varOmega ))},\end{aligned} \end{aligned}
(3.48)
\begin{aligned}& \begin{aligned}[b] & \Vert r_{4t} \Vert _{L^{\infty }(J;L^{2}\varOmega ))}+ \Vert r_{6t} \Vert _{L^{\infty }(J;L ^{2}\varOmega ))}+ \Vert r_{8} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))} \\ &\quad \leq \epsilon \bigl\Vert u(0)-u_{h}(0) \bigr\Vert \\ &\quad \quad{} +C \Vert u-u_{h} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+\epsilon \bigl\Vert (u-u_{h})_{t} \bigr\Vert _{L^{2}(J;L^{2}(\varOmega ))}, \end{aligned} \end{aligned}
(3.49)
\begin{aligned}& \begin{aligned}[b] & \Vert r_{5} \Vert _{L^{\infty }(J;H(\operatorname{div};\varOmega ))}+ \Vert r_{7} \Vert _{L^{2}(J;H( \operatorname{div};\varOmega ))} \\ &\quad \leq \epsilon \bigl\Vert u(0)-u_{h}(0) \bigr\Vert \\ &\quad\quad {} +C \Vert u-u_{h} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+\epsilon \bigl\Vert (u-u_{h})_{t} \bigr\Vert _{L^{2}(J;L^{2}(\varOmega ))}, \end{aligned} \end{aligned}
(3.50)

where ϵ is an arbitrary small positive constant.

Lemma 3.4

 Let $$(\tilde{\boldsymbol{p}},y,\boldsymbol{p},\tilde{y}, \tilde{\boldsymbol{q}},z,\boldsymbol{q},\tilde{z},u)$$ and $$(\tilde{\boldsymbol{p}}_{h},y _{h},\boldsymbol{p}_{h},\tilde{y}_{h},\tilde{\boldsymbol{q}}_{h},z_{h},\boldsymbol{q}_{h},u _{h})$$ be the solutions of (2.10)(2.22) and (2.30)(2.42), respectively. Suppose that $$(u_{h}+z_{h})\vert _{ \tau }\in H^{1}(\tau )$$ and that there exists $$w\in K_{h}$$ such that

\begin{aligned} \Vert u-u_{h} \Vert _{L^{2}(J;L^{2}(\varOmega ))} \leq C(\theta + \bigl\Vert z_{h}-z(u_{h}) \bigr\Vert _{L^{2}(J;L^{2}(\varOmega ))}, \end{aligned}
(3.51)

where

\begin{aligned} \theta = \biggl({ \int }_{0}^{T}\sum_{\tau }h_{\tau }^{2} \vert u_{h}+z_{h} \vert ^{2} _{H^{1}({\tau })} \,dt \biggr)^{\frac{1}{2}}. \end{aligned}

Now, by Lemmas 3.13.3, the important result of this paper is given as follows.

Theorem 3.1

Let $$(y,\tilde{\boldsymbol{p}},\tilde{y},\boldsymbol{p},z,\tilde{\boldsymbol{q}}, \tilde{z},\boldsymbol{q},u)$$ and $$(y_{h},\tilde{\boldsymbol{p}}_{h},\tilde{y}_{h},\boldsymbol{p}_{h},z_{h},\tilde{\boldsymbol{q}}_{h}, \tilde{z}_{h},\boldsymbol{q}_{h},u_{h})$$ be the solutions of (2.9)(2.19) and (2.26)(2.36), respectively. Then we have

\begin{aligned}& \Vert u- u_{h} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert y-y_{h} \Vert _{L^{\infty }(J;L ^{2}(\varOmega ))}+ \Vert \tilde{y}-\tilde{y}_{h} \Vert _{L^{\infty }(J;L^{2}( \varOmega ))} \\& \quad \quad {} + \Vert \tilde{\boldsymbol{p}}-\tilde{\boldsymbol{p}}_{h} \Vert _{L^{\infty }(J;H(\operatorname{div}; \varOmega ))}+ \Vert \boldsymbol{p}-\boldsymbol{p}_{h} \Vert _{L^{\infty }(J;H(\operatorname{div};\varOmega ))}+ \Vert z-z_{h} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))} \\& \quad \quad {} + \Vert \tilde{z}-\tilde{z}_{h} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert \tilde{\boldsymbol{q}}-\tilde{\boldsymbol{q}}_{h} \Vert _{L^{\infty }(J;H(\operatorname{div}; \varOmega ))}+ \Vert \boldsymbol{q}-\boldsymbol{q}_{h} \Vert _{L^{2}(J;H(\operatorname{div};\varOmega ))} \\& \quad \leq C\bigl( \Vert \eta _{1} \Vert _{L^{\infty }(J;H(\operatorname{div};\varOmega ))}+ \Vert \eta _{3} \Vert _{L^{\infty }(J;H(\operatorname{div};\varOmega ))}+ \Vert \eta _{5} \Vert _{L^{\infty }(J;H( \operatorname{div};\varOmega ))}+ \Vert \eta _{7} \Vert _{L^{2}(J;H(\operatorname{div};\varOmega ))} \\& \quad \quad {} + \Vert \eta _{4} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert \eta _{4t} \Vert _{L^{2}(J;L ^{2}(\varOmega ))}+ \Vert \eta _{4tt} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{2} \Vert _{L^{2}(J;L^{2}(\varOmega ))} \\& \quad \quad {} + \Vert \eta _{2tt} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert \eta _{2ttt} \Vert _{L ^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{6} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}+ \Vert \eta _{6t} \Vert _{L^{2}(J;L^{2}(\varOmega ))} \\& \quad \quad {} + \Vert \eta _{6tt} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert \eta _{8} \Vert _{L^{\infty }(J;L ^{2}(\varOmega ))}+ \Vert \eta _{8t} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \bigl\Vert y_{0}-y_{0} ^{h} \bigr\Vert \\& \quad \quad {} + \bigl\Vert \Delta y_{0}+\tilde{y}_{h}(0) \bigr\Vert + \bigl\Vert y_{1}-y_{1}^{h} \bigr\Vert + \bigl\Vert \operatorname{div} \eta _{3}(0) \bigr\Vert + \bigl\Vert \Delta ^{2} y_{0}-\operatorname{div} \boldsymbol{p}_{h}(0) \bigr\Vert \bigr) \\& \quad \quad {} + \bigl\Vert \Delta y_{1}+ \tilde{y}_{ht}(0) \bigr\Vert + \bigl\Vert \eta _{2}(0) \bigr\Vert + \bigl\Vert \eta _{2t}(0) \bigr\Vert + \bigl\Vert \eta _{4t}(0) \bigr\Vert . \end{aligned}
(3.52)

Proof

From Lemma 2.1 and (2.43), we have

\begin{aligned}& \bigl\Vert u(0)-u_{h}(0) \bigr\Vert \leq \Vert u-u_{h} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))} \leq C \Vert z-z_{h} \Vert _{L^{\infty }(J;L^{2}(\varOmega ))}, \end{aligned}
(3.53)
\begin{aligned}& \begin{aligned}[b] \bigl\Vert ( u-u_{h})_{t} \bigr\Vert _{L^{2}(J;L^{2}(\varOmega ))}&= \bigl\Vert ( z-z_{h})_{t} \bigr\Vert _{L ^{2}(J;L^{2}(\varOmega ))} \\ &\leq \Vert e_{6t} \Vert _{L^{2}(J;L^{2}(\varOmega ))} + \Vert \eta _{6t} \Vert _{L^{2}(J;L^{2}(\varOmega ))}+ \Vert r_{6t} \Vert _{L^{2}(J;L^{2}(\varOmega ))}. \end{aligned} \end{aligned}
(3.54)

For sufficiently small ϵ, using Lemmas 3.13.3 and (3.35), (3.53)–(3.54), we complete the proof. □

Conclusion and future work

In the article, using semidiscrete Raviart–Thomas mixed finite element methods, we studied fourth order hyperbolic equations of quadratic problems for optimal control, and then got the posteriori error estimates. In subsequent work, an a posteriori estimation will be considered by a fully discrete approximation of the mixed finite element. Of course, the error estimates of the same problems certainly also can be discussed with nonlinear fourth order hyperbolic equations.

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Acknowledgements

The authors express their thanks to the referees for their helpful suggestions, which led to improvements of the presentation.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Funding

This work is supported by Youth Innovative Talents Project (Natural Science) of research on humanities and social sciences in Guangdong normal university (2017KQNCX265), The issue for the 13th Five-Year plan for the development of philosophy and social sciences in Guangzhou of 2018 (2018GZGJ168), School projects of Huashang College Guangdong University of Finance and Economics.

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CH, ZG and LG participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.

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Correspondence to Lianhong Guo.

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