A posteriori error estimates of the lowest order Raviart-Thomas mixed finite element methods for convective diffusion optimal control problems

Abstract

In this paper, we consider the mixed finite element methods for quadratic optimal control problems governed by convective diffusion equations. The state and the co-state are discretized by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. Using some proper duality problems, we derive a posteriori $L^{2}(0,T;L^{2}(\Omega))$ error estimates for the scalar functions. Such estimates, which are apparently not available in the literature, are an important step toward developing reliable adaptive mixed finite element approximation schemes for the control problem.

1 Introduction

As far as we know, optimal control problems [1] have been extensively utilized in many aspects of the modern life such as social, economic, scientific, and engineering numerical simulation. Thus, they must be solved successfully with efficient numerical methods. Among these numerical methods, finite element method is a good choice. There have been extensive studies in the convergence of finite element approximation of optimal control problems; see [2–6]. A systematic introduction to finite element methods for PDEs and optimal control can be found for example in [7–9].

Recently, the adaptive finite element method has been investigated extensively. It has become one of the most popular methods in the scientific computation and numerical modeling. An adaptive finite element approximation ensures a higher density of nodes in a certain area of the given domain, where the solution is more difficult to approximate, indicated by a posteriori error estimators. Hence it is an important approach to boost the accuracy and efficiency of finite element discretizations. There are lots of works concentrating on the adaptivity of various optimal control problems. See, for example, [10–19].

In many control problems, the objective functional contains the gradient of the state variables. Thus, the accuracy of the gradient is important in numerical discretization of the coupled state equations. Mixed finite element methods are appropriate for the state equations in such cases since both the scalar variable and its flux variable can be approximated to the same accuracy by using such methods; see, for example, [20–23].

We shall use the lowest order Raviart-Thomas mixed finite element to discretize the state and the co-state, and use the piecewise constant space to approximate the control variable. Using some proper duality problems, we derive a posteriori $L^{2}(0,T;L^{2}(\Omega))$ error estimates for the scalar functions. The optimal control problems that we are interested in are as follows:

$$\begin{aligned}& \min_{u \in K\subset U} \biggl\{ \frac{1}{2}\int _{0}^{T} \bigl(\|\mathbf{p}-\mathbf{p}_{d} \| ^{2}+\| y-y_{d}\|^{2}+\| u\|^{2} \bigr)\, dt \biggr\} , \end{aligned}$$

(1.1)

$$\begin{aligned}& y_{t}+\operatorname{div}\mathbf{p}+cy=f+u, \quad x\in \Omega, t \in J, \end{aligned}$$

(1.2)

$$\begin{aligned}& \mathbf{p}=-a(\nabla y+\mathbf{b}y), \quad x\in\Omega, t\in J, \end{aligned}$$

(1.3)

$$\begin{aligned}& y(x,t)=0,\quad x\in{\partial\Omega}, t\in J,\qquad y(x,0)=y_{0}(x), \quad x\in\Omega, \end{aligned}$$

(1.4)

where the bounded open set $\Omega\subset{\mathbf{R}^{2}}$ is a convex polygon with the boundary ∂Ω. $J=[0,T]$. Let K be a closed convex set in the control space $U=L^{2}(J;L^{2}(\Omega ))$, $\mathbf{p}, \mathbf{p}_{d}\in(L^{2}(J;H^{1}(\Omega)))^{2}$, $u, y, y_{d}\in L^{2}(J;H^{1}(\Omega))$, $f\in L^{2}(J;L^{2}(\Omega))$, $y_{0}(x)\in H_{0}^{1}(\Omega)$. Moreover, we assume that $0< a_{0}\leq a\leq a^{0}$, $a(x)\in W^{1,\infty}(\Omega)$, $c(x)\in W^{1,\infty}(\Omega)$, $\mathbf{b}(x)\in(W^{1,\infty}(\Omega))^{2}$.

We assume that the constraint on the control is an obstacle such that

$$ K= \bigl\{ u\in U: u(x,t)\geq0, \mbox{a.e. in } \Omega\times J \bigr\} . $$

In this paper, we adopt the standard notation $W^{m,p}(\Omega)$ for Sobolev spaces on Ω with a norm $\|\cdot\|_{m,p}$ given by $\| v \|_{m,p}^{p}=\sum_{|\alpha|\leq m}\| D^{\alpha}v\|_{L^{p}(\Omega)}^{p}$, a semi-norm $|\cdot|_{m,p}$ given by $| v|_{m,p}^{p}=\sum_{|\alpha|= m}\| D^{\alpha}v\|_{L^{p}(\Omega)}^{p}$. We set $W_{0}^{m,p}(\Omega)=\{v\in W^{m,p}(\Omega): v|_{\partial\Omega}=0\}$. For $p=2$, we denote $H^{m}(\Omega)=W^{m,2}(\Omega)$, $H_{0}^{m}(\Omega)=W_{0}^{m,2}(\Omega)$, and $\|\cdot\|_{m}=\|\cdot\|_{m,2}$, $\|\cdot\|=\|\cdot\|_{0,2}$.

We denote by $L^{s}(0,T;W^{m,p}(\Omega))$ the Banach space of all $L^{s}$ integrable functions from J into $W^{m,p}(\Omega)$ with norm $\| v\|_{L^{s}(J;W^{m,p}(\Omega))}= (\int_{0}^{T}\|v\|_{W^{m,p}(\Omega )}^{s}\, dt )^{\frac{1}{s}}$ for $s\in[1,\infty)$, and the standard modification for $s=\infty$. Similarly, one can define the spaces $H^{l}(J;W^{m,p}(\Omega))$ and $C^{k}(J;W^{m,p}(\Omega))$. In addition C denotes a general positive constant independent of h and Δt, where h is the spatial mesh-size for the control and state discretization and Δt is the time increment.

The plan of this paper is as follows. In next section, we shall give a brief review on the mixed finite element method and the backward Euler discretization, and then we construct the approximation for the optimal control problems (1.1)-(1.4). Then, using two duality problems, we derive a posteriori $L^{2}(0,T;L^{2}(\Omega))$ error estimates for the scalar functions in Section 3. Finally, we give a conclusion and indicate some possible future work.

2 Mixed methods of parabolic optimal control problems

In this section, we shall study the mixed finite element and the backward Euler discretization approximation of convective diffusion optimal control problems (1.1)-(1.4). For the sake of simplicity, we assume that the domain Ω is a convex polygon. Now, we introduce the co-state parabolic equation

$$ -z_{t}-\operatorname{div}\bigl(a(\nabla z+\mathbf{p}- \mathbf {p}_{d})\bigr)+\mathbf{b}\cdot(\nabla z+\mathbf{p} - \mathbf{p}_{d})+cz=y-y_{d},\quad x\in \Omega, t\in J, $$

(2.1)

which can be written in the form of the first order system

$$ -z_{t}+\operatorname{div}\mathbf{q}-a^{-1} \mathbf{b}\cdot \mathbf{q}+cz=y-y_{d}, \quad \mathbf{q} =-a(\nabla z+ \mathbf{p}-\mathbf{p}_{d}), x\in\Omega, t\in J $$

(2.2)

and

$$ z(x,t)=0, \quad x\in{\partial\Omega}, t\in J, \qquad z(x,T)=0, \quad x\in\Omega. $$

(2.3)

To be definite, we shall take the state spaces $\mathbf{L}=L^{2}(J;\mathbf{V})$ and $Q=H^{1}(J;W)$, where V and W are defined as follows:

$$\mathbf{V}=H(\operatorname{div};\Omega)= \bigl\{ \mathbf{v}\in \bigl(L^{2}(\Omega)\bigr)^{2},\operatorname{div}\mathbf{v}\in L^{2}(\Omega) \bigr\} ,\qquad W=L^{2}(\Omega). $$

The Hilbert space V is equipped with the following norm:

$$\|\mathbf{v}\|_{H(\operatorname{div};\Omega)}= \bigl(\|\mathbf{v}\| _{0,\Omega}^{2}+ \|\operatorname{div} \mathbf{v}\|_{0,\Omega}^{2} \bigr)^{1/2}. $$

Let $\alpha=a^{-1}$ and $\boldsymbol{\beta}=\alpha\mathbf{b}$. We recast (1.1)-(1.4) as the following weak form: find $(\mathbf{p},y,u)\in\mathbf{L}\times Q\times K$ such that

$$\begin{aligned}& \min_{u \in K\subset U} \biggl\{ \frac{1}{2}\int _{0}^{T} \bigl(\| \mathbf{p}-\mathbf{p}_{d} \|^{2}+\| y-y_{d}\|^{2}+\| u\|^{2} \bigr) \, dt \biggr\} , \end{aligned}$$

(2.4)

$$\begin{aligned}& (\alpha{\mathbf{p}},\mathbf{v})-(y,\operatorname {div} \mathbf{v})+(\boldsymbol{\beta}y,\mathbf{v})=0,\quad \forall \mathbf{v}\in \mathbf{V}, t\in J, \end{aligned}$$

(2.5)

$$\begin{aligned}& (y_{t},w)+(\operatorname{div} {\mathbf{p}},w)+(cy,w)=(f+u,w), \quad \forall w\in W, t\in J, \end{aligned}$$

(2.6)

$$\begin{aligned}& y(x,0)=y_{0}(x), \quad \forall x\in\Omega. \end{aligned}$$

(2.7)

It follows from [1] and [16] that the optimal control problem (2.4)-(2.7) has a unique solution $(\mathbf{p},y,u)$, and that a triplet $(\mathbf{p},y,u)$ is the solution of (2.4)-(2.7) if and only if there is a co-state $(\mathbf{q},z)\in\mathbf{L}\times Q$ such that $({\mathbf {p}},y,{\mathbf{q}},z,u)$ satisfies the following optimality conditions:

$$\begin{aligned}& (\alpha{\mathbf{p}},\mathbf{v})-(y,\operatorname {div} \mathbf{v})+(\boldsymbol{\beta}y,\mathbf{v})=0, \quad \forall \mathbf{v}\in \mathbf{V}, t\in J, \end{aligned}$$

(2.8)

$$\begin{aligned}& (y_{t},w)+(\operatorname{div} {\mathbf {p}},w)+(cy,w)=(f+u,w),\quad \forall w\in W, t\in J, \end{aligned}$$

(2.9)

$$\begin{aligned}& y(x,0)=y_{0}(x),\quad \forall x\in\Omega, \end{aligned}$$

(2.10)

$$\begin{aligned}& (\alpha{\mathbf{q}},\mathbf{v})-(z,\operatorname {div} \mathbf{v})=-(\mathbf{p}-\mathbf{p}_{d},\mathbf{v}), \quad \forall \mathbf{v}\in\mathbf{V}, t\in J, \end{aligned}$$

(2.11)

$$\begin{aligned}& -(z_{t},w)+(\operatorname{div} {\mathbf {q}},w)-( \boldsymbol{\beta}\cdot\mathbf{q} ,w)+(cz,w)=(y-y_{d},w),\quad \forall w\in W, t\in J, \end{aligned}$$

(2.12)

$$\begin{aligned}& z(x,T)=0,\quad \forall x\in\Omega, \end{aligned}$$

(2.13)

$$\begin{aligned}& \int_{0}^{T}{(u+z, {\tilde{u}}-u)} \, dt\geq0,\quad \forall \tilde{u} \in K, \end{aligned}$$

(2.14)

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

Let ${\mathcal{T}}_{h}$ be regular triangulations of Ω. $h_{\tau}$ is the diameter of τ and $h=\max h_{\tau}$. Let $\mathbf{V}_{h}\times W_{h}\subset\mathbf{V}\times W$ denote the lowest order Raviart-Thomas space [24] associated with the triangulations ${\mathcal{T}}_{h}$ of Ω. $P_{k}$ denotes the space of polynomials of total degree of at most k ($k\geq0$). Let $\mathbf{V}({\tau})=\{\mathbf{v}\in P_{0}^{2}({\tau})+x\cdot P_{0}({\tau })\}$, $W({\tau})=P_{0}({\tau})$. We define

$$\begin{aligned}& \mathbf{V}_{h}:=\bigl\{ \mathbf{v}_{h}\in\mathbf{V}:\forall { \tau}\in {\mathcal{T}}_{h}, \mathbf{v} _{h}|_{\tau}\in \mathbf{V}({\tau})\bigr\} , \\& W_{h}:=\bigl\{ w_{h}\in W: \forall {\tau}\in{ \mathcal{T}}_{h},w_{h}|_{\tau }\in W({\tau})\bigr\} , \\& K_{h}:=K\cap W_{h}. \end{aligned}$$

The mixed finite element discretization of (2.4)-(2.7) is as follows: compute $(\mathbf{p}_{h},y_{h},u_{h})\in L^{2}(J;\mathbf {V}_{h})\times H^{1}(J;W_{h})\times K_{h}$ such that

$$\begin{aligned}& \min_{u_{h}(t) \in K_{h}} \biggl\{ \frac{1}{2}\int _{0}^{T} \bigl(\| \mathbf{p}_{h}- \mathbf{p}_{d}\|^{2}+\| y_{h}-y_{d} \|^{2}+\| u_{h}\|^{2} \bigr)\, dt \biggr\} , \end{aligned}$$

(2.15)

$$\begin{aligned}& (\alpha{\mathbf{p}}_{h},\mathbf{v}_{h})-(y_{h}, \operatorname{div}\mathbf {v}_{h})+(\boldsymbol{\beta}y_{h}, \mathbf{v}_{h})=0,\quad \forall \mathbf{v}_{h}\in \mathbf{V}_{h}, t\in J, \end{aligned}$$

(2.16)

$$\begin{aligned}& (y_{ht},w_{h})+(\operatorname{div}\mathbf {p}_{h},w_{h})+(cy_{h},w_{h})=(f+u_{h},w_{h}), \quad \forall w_{h}\in W_{h}, t\in J, \end{aligned}$$

(2.17)

$$\begin{aligned}& y_{h}(x,0)=y_{0}^{h}(x), \quad \forall x\in\Omega, \end{aligned}$$

(2.18)

where $y_{0}^{h}(x)\in W_{h}$ is an approximation of $y_{0}$. The optimal control problem (2.15)-(2.18) again has a unique solution $(\mathbf{p}_{h},y_{h},u_{h})$, and that a triplet $({\mathbf {p}}_{h},y_{h},u_{h})$ is the solution of (2.15)-(2.18) if and only if there is a co-state $(\mathbf{q}_{h},z_{h})\in L^{2}(J;\mathbf{V}_{h})\times H^{1}(J;W_{h})$ such that $(\mathbf{p}_{h},y_{h},\mathbf{q}_{h},z_{h},u_{h})$ satisfies the following optimality conditions:

$$\begin{aligned}& (\alpha{\mathbf{p}}_{h},\mathbf {v}_{h})-(y_{h}, \operatorname{div}\mathbf{v}_{h})+(\boldsymbol{\beta }y_{h}, \mathbf{v} _{h})=0, \quad \forall \mathbf{v}_{h}\in \mathbf{V}_{h}, t\in J, \end{aligned}$$

(2.19)

$$\begin{aligned}& (y_{ht},w_{h})+(\operatorname{div}\mathbf {p}_{h},w_{h})+(cy_{h},w_{h})=(f+u_{h},w_{h}), \quad \forall w_{h}\in W_{h}, t\in J, \end{aligned}$$

(2.20)

$$\begin{aligned}& y_{h}(x,0)=y_{0}^{h}(x), \quad \forall x\in\Omega, \end{aligned}$$

(2.21)

$$\begin{aligned}& (\alpha{\mathbf{q}}_{h},\mathbf {v}_{h})-(z_{h}, \operatorname{div}\mathbf{v}_{h})=-(\mathbf{p} _{h}-p_{d}, \mathbf{v}_{h}), \quad \forall \mathbf{v}_{h}\in \mathbf{V}_{h}, t\in J, \end{aligned}$$

(2.22)

$$\begin{aligned}& -(z_{ht},w_{h})+(\operatorname{div} {\mathbf {q}}_{h},w_{h})-(\boldsymbol{\beta}\cdot\mathbf{q} _{h},w_{h})+(cz_{h},w_{h}) \\& \quad =(y_{h}-y_{d},w_{h}), \quad \forall w_{h}\in W_{h}, t\in J, \end{aligned}$$

(2.23)

$$\begin{aligned}& z_{h}(x,T)=0, \quad \forall x\in\Omega, \end{aligned}$$

(2.24)

$$\begin{aligned}& \int_{0}^{T}(u_{h}+z_{h}, {\tilde{u}_{h}}-u_{h})\, dt \geq0,\quad \forall \tilde{u}_{h} \in K_{h}. \end{aligned}$$

(2.25)

We now consider the fully discrete approximation for the above semidiscrete problem. Let $\Delta t>0$, $N=T/\Delta t\in \mathbb{Z}$, and $t_{i}=i \Delta t$, $i\in\mathbb{Z}$. Also, let

$$d_{t}\psi^{i}=\frac{\psi^{i}-\psi^{i-1}}{\Delta t}. $$

We address the fully discrete approximation scheme to find $(\mathbf{p}_{h}^{i},y_{h}^{i},u_{h}^{i})\in\mathbf{V}_{h}\times W_{h}\times K_{h}$, $i=1, 2, \ldots, N$, such that

$$\begin{aligned}& \min_{u^{i}_{h} \in K_{h}} \Biggl\{ \frac{1}{2}\sum _{i=1}^{N} \Delta t \bigl(\bigl\Vert \mathbf{p}_{h}^{i}-\mathbf{p}_{d}^{i} \bigr\Vert ^{2}+\bigl\Vert y_{h}^{i}-y_{d}^{i} \bigr\Vert ^{2}+\bigl\Vert u_{h}^{i}\bigr\Vert ^{2} \bigr) \Biggr\} , \end{aligned}$$

(2.26)

$$\begin{aligned}& \bigl(\alpha{\mathbf{p}}_{h}^{i}, \mathbf{v}_{h}\bigr)-\bigl(y_{h}^{i}, \operatorname{div} \mathbf{v}_{h}\bigr)+\bigl(\boldsymbol{ \beta}y_{h}^{i},\mathbf{v}_{h}\bigr)=0,\quad \forall \mathbf{v}_{h}\in \mathbf{V}_{h}, \end{aligned}$$

(2.27)

$$\begin{aligned}& \bigl(d_{t}{y_{h}^{i}},w_{h} \bigr)+\bigl(\operatorname{div} {\mathbf{p}}_{h}^{i},w_{h} \bigr)+\bigl(cy_{h}^{i},w_{h}\bigr)= \bigl(f^{i}+u_{h}^{i},w_{h}\bigr),\quad \forall w_{h}\in W_{h}, \end{aligned}$$

(2.28)

$$\begin{aligned}& y_{h}^{0}(x)=y_{0}^{h}(x), \quad \forall x\in \Omega, \end{aligned}$$

(2.29)

where $f^{i}=f^{i}(x)=f(x,t_{i})$, $y_{d}^{i}=y_{d}(x,t_{i})$, and $\mathbf {p}_{d}^{i}=\mathbf{p}_{d}(x,t_{i})$.

It follows that the control problem (2.26)-(2.29) has a unique solution $(\mathbf{p}_{h}^{i},y_{h}^{i},u_{h}^{i})$, $i=1, 2, \ldots, N$, and that a triplet $(\mathbf{p}_{h}^{i},y_{h}^{i},u_{h}^{i})\in\mathbf {V}_{h}\times W_{h}\times K_{h}$, $i=1, 2, \ldots, N$, is the solution of (2.26)-(2.29) if and only if there is a co-state $({\mathbf{q}}_{h}^{i-1},z_{h}^{i-1})\in\mathbf{V}_{h}\times W_{h}$ such that $({\mathbf{p}}_{h}^{i},y_{h}^{i},{\mathbf{q}}_{h}^{i-1},z_{h}^{i-1},u_{h}^{i})\in (\mathbf{V}_{h}\times W_{h})^{2}\times K_{h}$ satisfies the following optimality conditions:

$$\begin{aligned}& \bigl(\alpha\mathbf{p}_{h}^{i},\mathbf {v}_{h}\bigr)-\bigl(y_{h}^{i},\operatorname{div} \mathbf{v}_{h}\bigr)+\bigl(\boldsymbol{\beta}y_{h}^{i}, \mathbf{v}_{h}\bigr)=0, \quad \forall \mathbf{v}_{h}\in \mathbf{V}_{h}, \end{aligned}$$

(2.30)

$$\begin{aligned}& \bigl(d_{t} y_{h}^{i},w_{h} \bigr)+\bigl(\operatorname{div}\mathbf{p} _{h}^{i},w_{h} \bigr)+\bigl(cy_{h}^{i},w_{h}\bigr)= \bigl(f^{i}+u_{h}^{i},w_{h}\bigr), \quad \forall w_{h}\in W_{h}, \end{aligned}$$

(2.31)

$$\begin{aligned}& y_{h}^{0}(x)=y_{0}^{h}(x), \quad \forall x\in\Omega, \end{aligned}$$

(2.32)

$$\begin{aligned}& \bigl(\alpha\mathbf{q}_{h}^{i-1},\mathbf {v}_{h}\bigr)-\bigl(z_{h}^{i-1},\operatorname{div} \mathbf{v}_{h}\bigr)=-\bigl(\mathbf{p} _{h}^{i}- \mathbf{p}_{d}^{i},\mathbf{v}_{h}\bigr), \quad \forall \mathbf{v}_{h}\in \mathbf{V}_{h}, \end{aligned}$$

(2.33)

$$\begin{aligned}& -\bigl(d_{t} z_{h}^{i},w_{h} \bigr)+\bigl(\operatorname{div}\mathbf {q}_{h}^{i-1},w_{h} \bigr)-\bigl(\boldsymbol{\beta}\cdot\mathbf{q} _{h}^{i-1},w_{h} \bigr)+\bigl(cz_{h}^{i-1},w_{h}\bigr) \\& \quad = \bigl(y_{h}^{i}-y_{d}^{i},w_{h} \bigr), \quad \forall w_{h}\in W_{h}, \end{aligned}$$

(2.34)

$$\begin{aligned}& z_{h}^{N}(x)=0, \quad \forall x\in\Omega, \end{aligned}$$

(2.35)

$$\begin{aligned}& \bigl(u_{h}^{i}+z_{h}^{i-1}, \tilde{u}_{h}-u_{h}^{i}\bigr)\geq0, \quad \forall \tilde{u}_{h} \in K_{h}. \end{aligned}$$

(2.36)

For $i=0$ and $i=N$, we let

$$\begin{aligned}& \bigl(\alpha\mathbf{p}_{h}^{0},\mathbf {v}_{h}\bigr)-\bigl(y_{h}^{0},\operatorname{div} \mathbf{v}_{h}\bigr)+\bigl(\boldsymbol{\beta}y_{h}^{0}, \mathbf{v}_{h}\bigr)=0, \quad \forall \mathbf{v}_{h}\in \mathbf{V}_{h}, \end{aligned}$$

(2.37)

$$\begin{aligned}& \bigl(\alpha\mathbf{q}_{h}^{N},\mathbf {v}_{h}\bigr)-\bigl(z_{h}^{N},\operatorname{div} \mathbf{v}_{h}\bigr)=-\bigl(\mathbf{p} _{h}^{N}- \mathbf{p}_{d}^{N},\mathbf{v}_{h}\bigr),\quad \forall \mathbf{v}_{h}\in \mathbf{V}_{h}. \end{aligned}$$

(2.38)

For $i=1, 2,\ldots, N$, let

$$\begin{aligned}& Y_{h}|_{(t_{i-1},t_{i}]} = \bigl((t_{i}-t)y_{h}^{i-1}+(t-t_{i-1})y_{h}^{i} \bigr)/{\Delta t}, \\& Z_{h}|_{(t_{i-1},t_{i}]} = \bigl((t_{i}-t)z_{h}^{i-1}+(t-t_{i-1})z_{h}^{i} \bigr)/{\Delta t}, \\& {P}_{h}|_{(t_{i-1},t_{i}]} = \bigl((t_{i}-t)\mathbf {p}_{h}^{i-1}+(t-t_{i-1})\mathbf{p} _{h}^{i} \bigr)/{\Delta t}, \\& {Q}_{h}|_{(t_{i-1},t_{i}]} = \bigl((t_{i}-t)\mathbf {q}_{h}^{i-1}+(t-t_{i-1})\mathbf{q} _{h}^{i} \bigr)/{\Delta t}, \\& U_{h}|_{(t_{i-1},t_{i}]} =u_{h}^{i}. \end{aligned}$$

For any function $w\in C(J;L^{2}(\Omega))$, let

$$\hat{w}(x,t)|_{t\in(t_{i-1},t_{i}]}=w(x,t_{i}),\qquad \tilde{w}(x,t)|_{t\in(t_{i-1},t_{i}]}=w(x,t_{i-1}). $$

Moreover, we let

$$\begin{aligned}& \bar{\mathbf{p}}_{d}|_{(t_{i-1},t_{i}]}= \bigl((t_{i}-t) \mathbf {p}_{d}^{i}+(t-t_{i-1})\mathbf{p} _{d}^{i+1} \bigr)/{\Delta t},\quad i=1,2,\ldots,N-1,\qquad \bar{ \mathbf{p} }_{d}|_{(t_{N-1},t_{N}]}=\mathbf{p}_{d}^{N}, \\& \bar{P}_{h}|_{(t_{i-1},t_{i}]}= \bigl((t_{i}-t)\mathbf {p}_{h}^{i}+(t-t_{i-1})\mathbf{p} _{h}^{i+1} \bigr)/{\Delta t}, \quad i=1,2,\ldots,N-1, \qquad \bar {P}_{h}|_{(t_{N-1},t_{N}]}=\mathbf{p}_{h}^{N}. \end{aligned}$$

Then the optimality conditions (2.30)-(2.36) satisfy

$$\begin{aligned}& (\alpha\hat{P}_{h},\mathbf{v}_{h})-(\hat {Y}_{h},\operatorname{div}\mathbf{v}_{h})+(\boldsymbol{\beta} \hat{Y}_{h},\mathbf{v}_{h})=0, \quad \forall \mathbf{v}_{h}\in\mathbf{V}_{h}, \end{aligned}$$

(2.39)

$$\begin{aligned}& ({ Y_{ht}},w_{h})+(\operatorname{div}\hat {P}_{h},w_{h})+(c\hat {Y}_{h},w_{h})=( \hat{f}+U_{h},w_{h}), \quad \forall w_{h}\in W_{h}, \end{aligned}$$

(2.40)

$$\begin{aligned}& Y_{h}(x,0)=y_{0}^{h}(x), \quad \forall x\in\Omega, \end{aligned}$$

(2.41)

$$\begin{aligned}& (\alpha\tilde{Q}_{h},\mathbf{v}_{h})-(\tilde {Z}_{h},\operatorname{div}\mathbf{v} _{h})=-( \hat{P}_{h}-\hat{\mathbf{p}}_{d},\mathbf{v}_{h}), \quad \forall \mathbf{v}_{h}\in\mathbf{V}_{h}, \end{aligned}$$

(2.42)

$$\begin{aligned}& -(Z_{ht},w_{h})+(\operatorname{div}\tilde {Q}_{h},w_{h})-(\boldsymbol{\beta}\cdot\tilde {Q}_{h},w_{h})+(c\tilde{Z}_{h},w_{h})=( \hat{Y}_{h}-\hat{y}_{d},w_{h}),\quad \forall w_{h}\in W_{h}, \end{aligned}$$

(2.43)

$$\begin{aligned}& Z_{h}(x,T)=0, \quad \forall x\in\Omega, \end{aligned}$$

(2.44)

$$\begin{aligned}& (U_{h}+\tilde{Z}_{h}, {\tilde{u}_{h}}-U_{h}) \geq0,\quad \forall \tilde{u}_{h} \in K_{h}. \end{aligned}$$

(2.45)

In the rest of the paper, we shall use some intermediate variables. For any control function $U_{h}\in K_{h}$, we first define the state solution $(\mathbf{p}(U_{h}),y(U_{h}),\mathbf{q}(U_{h}),z(U_{h}))$ to satisfy

$$\begin{aligned}& \bigl(\alpha\mathbf{p}(U_{h}),\mathbf {v}\bigr)- \bigl(y(U_{h}),\operatorname{div}\mathbf{v}\bigr)+\bigl(\boldsymbol{ \beta }y(U_{h}),\mathbf{v} \bigr)=0, \quad \forall \mathbf{v}\in \mathbf{V}, \end{aligned}$$

(2.46)

$$\begin{aligned}& \bigl(y_{t}(U_{h}),w\bigr)+\bigl( \operatorname{div}\mathbf {p}(U_{h}),w\bigr)+\bigl(cy(U_{h}),w \bigr)=(f+U_{h},w), \quad \forall w\in W, \end{aligned}$$

(2.47)

$$\begin{aligned}& y(U_{h}) (x,0)=y_{0}(x), \quad \forall x\in \Omega, \end{aligned}$$

(2.48)

$$\begin{aligned}& \bigl(\alpha\mathbf{q}(U_{h}),\mathbf {v}\bigr)- \bigl(z(U_{h}),\operatorname{div}\mathbf{v}\bigr)=-\bigl( \mathbf{p}(U_{h})-\mathbf{p} _{d},\mathbf{v}\bigr), \quad \forall \mathbf{v}\in\mathbf{V}, \end{aligned}$$

(2.49)

$$\begin{aligned}& -\bigl(z_{t}(U_{h}),w\bigr)+\bigl( \operatorname{div}\mathbf {q}(U_{h}),w\bigr)-\bigl(\boldsymbol{\beta} \cdot\mathbf{q} (U_{h}),w\bigr)+\bigl(cz(U_{h}),w\bigr) \\& \quad = \bigl(y(U_{h})-y_{d},w\bigr), \quad \forall w\in W, \end{aligned}$$

(2.50)

$$\begin{aligned}& z(U_{h}) (x,T)=0, \quad \forall x\in\Omega. \end{aligned}$$

(2.51)

Let $R_{h}:W\rightarrow W_{h}$ be the orthogonal $L^{2}(\Omega)$-projection into $W_{h}$ [25], which satisfies

$$\begin{aligned}& (R_{h}w-w,\chi)=0, \quad w\in W, \chi\in W_{h}, \end{aligned}$$

(2.52)

$$\begin{aligned}& \| R_{h} w-w\|_{0,q}\leq Ch\|w \|_{1,q}, \quad \text{if } w\in W\cap W^{1,q}(\Omega). \end{aligned}$$

(2.53)

Let $\Pi_{h}:\mathbf{V}\rightarrow\mathbf{V}_{h}$ be the Raviart-Thomas projection operator [26], which satisfies: for any $\mathbf{v}\in\mathbf{V}$,

$$\begin{aligned}& \int_{E}w_{h}(\mathbf{v}- \Pi_{h}\mathbf{v})\cdot\boldsymbol{\nu }_{E}\, ds=0,\quad w_{h}\in W_{h}, E\in\mathcal{E}_{h}, \end{aligned}$$

(2.54)

$$\begin{aligned}& \int_{\tau}(\mathbf{v}-\Pi_{h} \mathbf{v})\cdot \mathbf{v} _{h}\, dx\, dy=0,\quad \mathbf{v}_{h}\in\mathbf{V}_{h}, \tau\in \mathcal{T}_{h}, \end{aligned}$$

(2.55)

where $\mathcal{E}_{h}$ denotes the set of element sides in $\mathcal{T}_{h}$.

We have the commuting diagram property

$$ \operatorname{div}\circ\Pi_{h}=R_{h}\circ \operatorname{div}:\mathbf{V} \rightarrow W_{h} \quad \text{and}\quad \operatorname{div}(I-\Pi _{h})\mathbf{V}\perp W_{h}, $$

(2.56)

where I denotes the identity operator.

Further, the interpolation operator $\Pi_{h}$ satisfies a local error estimate:

$$ \|\mathbf{v}-\Pi_{h}\mathbf{v}\|_{0,\Omega}\leq Ch |\mathbf{v}|_{1,\mathcal{T}_{h}}, \quad \mathbf{v}\in\mathbf{V}\cap H^{1}( \mathcal {T}_{h}). $$

(2.57)

3 A posteriori error estimates

In this section we study a posteriori error estimates for the mixed finite element approximation to the parabolic optimal control problems.

For the following analysis, we divide the domain Ω into three parts:

$$\begin{aligned}& \Omega_{-}=\bigl\{ x\in\Omega:\tilde{Z}_{h}(x)\leq0\bigr\} , \\& \Omega_{0}=\bigl\{ x\in\Omega:\tilde{Z}_{h}(x)>0,U_{h}(x)=0 \bigr\} , \\& \Omega_{+}=\bigl\{ x\in\Omega:\tilde{Z}_{h}(x)>0,U_{h}(x)>0 \bigr\} . \end{aligned}$$

It is easy to see that the partition of the above three subsets is dependent on t. For all t, the three subsets are not intersected each other, and

$$\bar{\Omega}=\bar{\Omega}_{-}\cup\bar{\Omega}_{0}\cup\bar {\Omega}_{+}. $$

Firstly, let us derive the a posteriori error estimates for the control u.

Theorem 3.1

Let $(y,\mathbf{p},z,\mathbf{q},u)$ and $(Y_{h},P_{h},Z_{h},Q_{h},U_{h})$ be the solutions of (2.8)-(2.14) and (2.39)-(2.45), respectively. Then we have

$$ \|u-U_{h}\|_{L^{2}(J;L^{2}(\Omega))}^{2}\leq C \eta_{1}^{2} + \bigl\Vert \tilde{Z}_{h}-z(U_{h}) \bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, $$

(3.1)

where

$$\eta_{1}^{2}=\|U_{h}+\tilde{Z}_{h} \|_{L^{2}(J;L^{2}(\Omega_{-}\cup \Omega_{+}))}^{2}. $$

Proof

It follows from (2.14) that

$$\begin{aligned}& \|u-U_{h}\|_{L^{2}(J;L^{2}(\Omega))}^{2} \\& \quad = \int _{0}^{T}(u-U_{h},u-U_{h})\, dt \\& \quad = \int_{0}^{T}(u+z,u-U_{h})\, dt+ \int_{0}^{T}(U_{h}+ \tilde{Z}_{h},U_{h}-u)\, dt \\& \qquad {} + \int_{0}^{T}\bigl(\tilde{Z}_{h}-z(U_{h}),u-U_{h} \bigr)\, dt+\int_{0}^{T}\bigl(z(U_{h})-z,u-U_{h} \bigr)\, dt \\& \quad \leq \int_{0}^{T}(U_{h}+ \tilde{Z}_{h},U_{h}-u)\, dt+ \int_{0}^{T} \bigl(\tilde{Z}_{h}-z(U_{h}),u-U_{h}\bigr)\, dt \\& \qquad {} +\int_{0}^{T}\bigl(z(U_{h})-z,u-U_{h} \bigr)\, dt \\& \quad = :I_{1}+I_{2}+I_{3}. \end{aligned}$$

(3.2)

We first estimate $I_{1}$. Note that

$$\begin{aligned} \begin{aligned}[b] I_{1}&=\int_{0}^{T}(U_{h}+ \tilde{Z}_{h},U_{h}-u)\, dt \\ &=\int_{0}^{T}\int_{\Omega_{-}\cup\Omega _{+}}(U_{h}+ \tilde{Z}_{h}) (U_{h}-u)\, dx\, dt+\int_{0}^{T} \int_{\Omega _{0}}(U_{h}+\tilde{Z}_{h}) (U_{h}-u)\, dx\, dt. \end{aligned} \end{aligned}$$

(3.3)

It is easy to see that

$$\begin{aligned}& \int_{0}^{T}\int _{\Omega_{-}\cup\Omega _{+}}(U_{h}+\tilde{Z}_{h}) (U_{h}-u)\, dx\, dt \\& \quad \leq C(\delta)\|U_{h}+\tilde{Z}_{h}\| _{L^{2}(J;L^{2}(\Omega_{-}\cup\Omega_{+}))}^{2}+\delta\|u-U_{h}\| _{L^{2}(J;L^{2}(\Omega_{-}\cup\Omega_{+}))}^{2} \\& \quad = C(\delta)\eta_{1}^{2}+\delta\|u-U_{h}\| _{L^{2}(J;L^{2}(\Omega))}^{2}, \end{aligned}$$

(3.4)

where δ is an arbitrary small positive number, $C(\delta)$ is dependent on $\delta^{-1}$. Furthermore, we have

$$U_{h}+\tilde{Z}_{h}\geq\tilde{Z}_{h}>0, \qquad U_{h}-u=0-u\leq0 \quad \text{on } \Omega_{0}. $$

It yields

$$ \int_{0}^{T}\int _{\Omega_{0}}(U_{h}+\tilde{Z}_{h}) (U_{h}-u)\, dx\, dt\leq0. $$

(3.5)

Then (3.3)-(3.5) imply that

$$ I_{1}\leq C(\delta)\eta_{1}^{2}+ \delta \|u-U_{h}\|_{L^{2}(J;L^{2}(\Omega))}^{2}. $$

(3.6)

Moreover, it is clear that

$$\begin{aligned} I_{2}&= \int_{0}^{T} \bigl(\tilde{Z}_{h}-z(U_{h}),u-U_{h}\bigr)\, dt \\ &\leq C(\delta)\bigl\Vert \tilde{Z}_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega ))}^{2}+\delta \|u-U_{h} \|_{L^{2}(J;L^{2}(\Omega))}^{2}. \end{aligned}$$

(3.7)

Now we turn to $I_{3}$. Note that

$$y(x,0)=y(U_{h}) (x,0)=y_{0}(x)\quad \text{and} \quad z(x,T)=z(U_{h}) (x,T)=0. $$

Then from (2.8)-(2.13) and (2.46)-(2.51), we have

$$\begin{aligned} I_{3} =&\int_{0}^{T} \bigl(z(U_{h})-z,u-U_{h}\bigr)\,dt=\int_{0}^{T} \bigl(u-U_{h},z(U_{h})-z\bigr)\,dt \\ =&\int_{0}^{T} \bigl(\bigl( \bigl(y-y(U_{h})\bigr)_{t},z(U_{h})-z\bigr)+ \bigl(\operatorname{div}\bigl(\mathbf{p} -\mathbf{p}(U_{h}) \bigr),z(U_{h})-z\bigr) \bigr)\,dt \\ &{}+\int_{0}^{T} \bigl(\bigl(c \bigl(y-y(U_{h})\bigr),z(U_{h})-z\bigr)-\bigl(\boldsymbol { \beta} \bigl(y-y(U_{h})\bigr),\mathbf{q}(U_{h})-\mathbf{q} \bigr) \bigr)\,dt \\ &{}-\int_{0}^{T} \bigl(\bigl(\alpha\bigl( \mathbf{p}-\mathbf {p}(U_{h})\bigr),\mathbf{q}(U_{h})- \mathbf{q} \bigr)-\bigl(y-y(U_{h}),\operatorname{div}\bigl( \mathbf{q}(U_{h})-\mathbf{q}\bigr)\bigr) \bigr)\,dt \\ =&\int_{0}^{T} \bigl(-\bigl( \bigl(z(U_{h})-z\bigr)_{t},y-y(U_{h})\bigr)+ \bigl(\operatorname{div}\bigl(\mathbf{q} (U_{h})-\mathbf{q} \bigr),y-y(U_{h})\bigr) \bigr)\,dt \\ &{}+\int_{0}^{T} \bigl(\bigl(c \bigl(z(U_{h})-z\bigr),y-y(U_{h})\bigr)-\bigl(\boldsymbol { \beta}\cdot \bigl(\mathbf{q}(U_{h})-\mathbf{q}\bigr),y-y(U_{h}) \bigr) \bigr)\,dt \\ &{}-\int_{0}^{T} \bigl(\bigl(\alpha\bigl( \mathbf{q}(U_{h})-\mathbf {q}\bigr),\mathbf{p}-\mathbf{p} (U_{h})\bigr)-\bigl(z(U_{h})-z,\operatorname{div}\bigl( \mathbf{p}-\mathbf {p}(U_{h})\bigr)\bigr) \bigr)\,dt \\ =&\int_{0}^{T} \bigl(\bigl(y(U_{h})-y,y-y(U_{h}) \bigr)+\bigl(\mathbf{p} (U_{h})-\mathbf{p},\mathbf{p}- \mathbf{p}(U_{h})\bigr) \bigr)\,dt\leq0. \end{aligned}$$

(3.8)

Thus, we obtain from (3.2) and (3.6)-(3.8)

$$ \|u-U_{h}\|_{L^{2}(J;L^{2}(\Omega ))}^{2}\leq C \eta_{1}^{2} + \bigl\Vert \tilde{Z}_{h}-z(U_{h}) \bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, $$

(3.9)

which proves (3.1). □

In order to estimate the error $\|\tilde{Z}_{h}-z(U_{h})\|_{L^{2}(J;L^{2}(\Omega))}^{2}$, we need the following well-known stability results (see [27, 28] for the details) for the following dual equations:

$$ \left \{\textstyle\begin{array}{l@{\quad}l} \phi_{t}-\operatorname{div}(a\nabla{\phi}+\mathbf{b}\phi)+c\phi=F, & x\in\Omega, t\in J, \\ \phi|_{\partial\Omega}=0, & t\in J, \\ \phi(x,0)=0, & x\in\Omega \end{array}\displaystyle \right . $$

(3.10)

and

$$ \left \{\textstyle\begin{array}{l@{\quad}l} -\psi_{t}-\operatorname{div}(a\nabla{\psi})+\mathbf{b}\cdot\nabla \psi+c\psi=F, & x\in \Omega, t\in J, \\ \psi|_{\partial\Omega}=0, & t\in J, \\ \psi(x,T)=0, & x\in\Omega. \end{array}\displaystyle \right . $$

(3.11)

Lemma 3.1

[28]

Let ϕ and ψ be the solutions of (3.10) and (3.11), respectively. Let Ω be a convex domain. Then, for $\varphi=\phi$ or $\varphi=\psi$,

$$\begin{aligned}& \int_{\Omega}\bigl\vert \varphi(x,t)\bigr\vert ^{2}\, dx \leq C\|F\|_{L^{2}(J;L^{2}(\Omega ))}^{2}, \quad \forall t \in J, \\& \int_{0}^{T}\int_{\Omega}|\nabla \varphi|^{2}\, dx\, dt\leq C\|F\| _{L^{2}(J;L^{2}(\Omega))}^{2}, \\& \int_{0}^{T}\int_{\Omega} \bigl\vert D^{2}\varphi\bigr\vert ^{2}\, dx\, dt\leq C\|F\| _{L^{2}(J;L^{2}(\Omega))}^{2}, \\& \int_{0}^{T}\int_{\Omega} | \varphi_{t}|^{2}\, dx\, dt \leq C\|F\| _{L^{2}(J;L^{2}(\Omega))}^{2}, \end{aligned}$$

where $|D^{2}\varphi|=\max\{|\partial^{2}\varphi/{\partial x_{i}\, \partial x_{j}}|, 1\leq i,j\leq2\}$.

We also need the following Gronwall lemma.

Lemma 3.2

[29]

Let f and g be piecewise continuous nonnegative functions defined on $0\leq t \leq T$, g being non-decreasing. If, for each $t\in J$,

$$ f(t)\leq g(t)+\int_{0}^{t}f(s)\, d s, $$

(3.12)

then $f(t)\leq e^{t}g(t)$.

In the following two theorems, we shall estimate the error $\|\tilde {Z}_{h}-z(U_{h})\|_{L^{2}(J;L^{2}(\Omega))}$.

Theorem 3.2

Let $(Y_{h},P_{h},Z_{h},Q_{h},U_{h})$ and $(y(U_{h}),\mathbf{p}(U_{h}),z(U_{h}),\mathbf {q}(U_{h}),U_{h})$ be the solutions of (2.39)-(2.45) and (2.46)-(2.51), respectively. Then we have

$$ \bigl\Vert Y_{h}-y(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}\leq C\sum_{i=2}^{7} \eta_{i}^{2}, $$

(3.13)

where

$$\begin{aligned}& \eta_{2}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{2} \int _{\tau } ( Y_{ht}+\operatorname{div} \hat{P}_{h}+c \hat{Y}_{h}-\hat{f}-U_{h} )^{2} \, dx\, dt; \\ & \eta_{3}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{2}\int _{\tau} (\alpha P_{h}+\boldsymbol{ \beta}Y_{h} )^{2} \, dx\, dt; \qquad \eta_{4}^{2}= \| \hat{P}_{h}-P_{h}\| _{L^{2}(J;L^{2}(\Omega))}^{2}; \\ & \eta_{5}^{2}=\|\hat{f}-f\|_{L^{2}(J;L^{2}(\Omega))}^{2}; \qquad \eta_{6}^{2}=\| \hat{Y}_{h}-Y_{h} \|_{L^{2}(J;L^{2}(\Omega))}^{2};\qquad \eta_{7}^{2}=\bigl\Vert y_{0}^{h}(x)-y_{0}(x)\bigr\Vert _{L^{2}(\Omega)}^{2}. \end{aligned}$$

Proof

From (2.30) and (2.37), we get the equality

$$ (\alpha{P}_{h},\mathbf{v}_{h})-(Y_{h}, \operatorname{div}\mathbf {v}_{h})+(\boldsymbol{\beta}Y_{h}, \mathbf{v}_{h})=0,\quad \forall \mathbf{v}_{h}\in \mathbf{V}_{h}. $$

(3.14)

Let ψ be the solution of (3.11) with $F=Y_{h}-y(U_{h})$, using (2.39)-(2.41), (2.46)-(2.48), and (2.54)-(2.56), we infer that

$$\begin{aligned}& \bigl\Vert Y_{h}-y(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2} \\& \quad =\int_{0}^{T}\bigl(Y_{h}-y(U_{h}),F \bigr)\,dt \\& \quad = \int_{0}^{T}\bigl(Y_{h}-y(U_{h}),- \psi_{t}-\operatorname{div}(a\nabla \psi)+\mathbf{b} \cdot\nabla\psi+c \psi\bigr)\,dt \\& \quad = \int_{0}^{T} \bigl(\bigl( \bigl(Y_{h}-y(U_{h})\bigr)_{t},\psi\bigr)- \bigl(Y_{h},\operatorname {div}\bigl(\Pi _{h}(a\nabla\psi) \bigr)\bigr)+\bigl(\mathbf{p}(U_{h}),\nabla\psi\bigr) \bigr)\,dt \\& \qquad {}+\int_{0}^{T} \bigl(( \mathbf{b}Y_{h},\nabla\psi )+\bigl(c\bigl(Y_{h}-y(U_{h}) \bigr),\psi \bigr) \bigr)\,dt+\bigl(\bigl(Y_{h}-y(U_{h})\bigr) (x,0), \psi(x,0)\bigr) \\& \quad = \int_{0}^{T} \bigl(\bigl( \bigl(Y_{h}-y(U_{h})\bigr)_{t},\psi\bigr)-\bigl( \alpha P_{h},\Pi _{h}(a\nabla\psi)\bigr) \\& \qquad {}-\bigl(\boldsymbol{ \beta}Y_{h},\Pi_{h}(a\nabla\psi )\bigr)-\bigl( \operatorname{div}\mathbf{p}(U_{h}),\psi \bigr) \bigr)\,dt \\& \qquad {} +\int_{0}^{T} \bigl((\boldsymbol{ \beta}Y_{h},a\nabla\psi )+\bigl(c\bigl(Y_{h}-y(U_{h}) \bigr),\psi\bigr) \bigr)\,dt+\bigl(y_{0}^{h}(x)-y_{0}(x), \psi(x,0)\bigr) \\& \quad = \int_{0}^{T} \bigl((Y_{ht}, \psi)+\bigl(\alpha P_{h},a\nabla\psi -\Pi_{h}(a\nabla\psi) \bigr)-(\hat{P}_{h}-P_{h},\nabla\psi)-(\operatorname {div} \hat{P}_{h} ,\psi) \bigr)\,dt \\& \qquad {} +\int_{0}^{T} \bigl(\bigl(\boldsymbol{ \beta}Y_{h},a\nabla\psi-\Pi _{h}(a\nabla\psi ) \bigr)+(cY_{h}-f-U_{h},\psi) \bigr)\,dt+\bigl(y_{0}^{h}(x)-y_{0}(x), \psi(x,0)\bigr) \\& \quad = \int_{0}^{T}(Y_{ht}+ \operatorname{div}\hat{P}_{h}+c\hat {Y}_{h}-\hat {f}-U_{h},\psi)\,dt+\int_{0}^{T}\bigl( \alpha P_{h}+\boldsymbol{\beta}Y_{h},a\nabla \psi-\Pi _{h}(a\nabla\psi)\bigr)\,dt \\& \qquad {} +\int_{0}^{T} \bigl((\hat{f}-f,\psi)+ \bigl(c(Y_{h}-\hat{Y}_{h}),\psi \bigr)+(\hat{P}_{h}-P_{h}, \nabla\psi) \bigr)\,dt+\bigl(y_{0}^{h}(x)-y_{0}(x), \psi(x,0)\bigr) \\& \quad =: L_{1}+L_{2}+L_{3}+L_{4}. \end{aligned}$$

(3.15)

Using (2.52), (2.40), the Cauchy inequality, and Lemma 3.1, we have

$$\begin{aligned} L_{1}&= \int_{0}^{T}(Y_{ht}+ \operatorname{div}\hat{P}_{h}+c\hat {Y}_{h}-\hat {f}-U_{h},\psi-P_{h} \psi)\, dt \\ &\leq C(\delta)\eta_{2}^{2}+\delta\|\psi\| _{L^{2}(J;H^{1}(\Omega))}^{2} \\ &\leq C\eta_{2}^{2}+ \frac{1}{5}\bigl\Vert Y_{h}-y(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}. \end{aligned}$$

(3.16)

Similarly, using the Cauchy inequality and Lemma 3.1, we have

$$\begin{aligned}& L_{2}\leq C\eta_{3}^{2}+ \frac{1}{5}\bigl\Vert Y_{h}-y(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, \end{aligned}$$

(3.17)

$$\begin{aligned}& L_{3}\leq C\bigl(\eta_{4}^{2}+ \eta_{5}^{2}+\eta_{6}^{2}\bigr)+ \frac{1}{5}\bigl\Vert Y_{h}-y(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, \end{aligned}$$

(3.18)

$$\begin{aligned}& L_{4}\leq C\eta_{7}^{2}+ \frac{1}{5}\bigl\Vert Y_{h}-y(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}. \end{aligned}$$

(3.19)

Hence, using (3.15)-(3.19), we get

$$ \bigl\Vert Y_{h}-y(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}\leq C\sum_{i=2}^{7} \eta_{i}^{2}. $$

(3.20)

This proves (3.13). □

Theorem 3.3

Let $(y,\mathbf{p},z,\mathbf{q},u)$ and $(Y_{h},P_{h},Z_{h},Q_{h},U_{h})$ be the solutions of (2.8)-(2.14) and (2.39)-(2.45), respectively. Let $(y(U_{h}),\mathbf{p} (U_{h}),z(U_{h}),\mathbf{q}(U_{h}),U_{h})$ be defined as in (2.46)-(2.51). Then we have the following error estimate:

$$ \bigl\Vert \tilde{Z}_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2} \leq C\sum_{i=3,6,8-14} \eta_{i}^{2}+C\bigl\Vert Y_{h}-y(U_{h}) \bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, $$

(3.21)

where

$$\begin{aligned}& \eta_{8}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{2}\int _{\tau } ( - Z_{ht}+\operatorname{div} \tilde{Q}_{h}-\boldsymbol{\beta}\cdot \tilde {Q}_{h}+c \tilde{Z}_{h}-\hat{Y}_{h}+\hat{y}_{d} )^{2}\, dx\, dt; \\& \eta_{9}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{2}\int _{\tau} (\alpha Q_{h}+\bar{P}_{h}-\bar{ \mathbf{p}}_{d} )^{2} \, dx \, dt; \qquad \eta _{10}^{2}=\|\tilde{Q}_{h}-Q_{h} \|_{L^{2}(J;L^{2}(\Omega))}^{2}; \\& \eta_{11}^{2}=\|\bar{P}_{h}-P_{h} \|_{L^{2}(J;L^{2}(\Omega))}^{2};\qquad \eta _{12}^{2}=\| \tilde{Z}_{h}-Z_{h}\|_{L^{2}(J;L^{2}(\Omega))}^{2}; \\& \eta_{13}^{2}=\|\bar{\mathbf{p}}_{d}- \mathbf{p}_{d}\| _{L^{2}(J;L^{2}(\Omega))}^{2}; \qquad \eta_{14}^{2}=\|\hat{y}_{d}-y_{d} \|_{L^{2}(J;L^{2}(\Omega))}^{2}, \end{aligned}$$

$\eta_{3}$ and $\eta_{6}$ are defined in Theorem  3.2.

Proof

Similar to (3.14), using (2.33), (2.38), and the definitions of $Z_{h}$, $Q_{h}$, $\bar{P}_{h}$, and $\bar{\mathbf{p}}_{d}$, we get

$$ (\alpha Q_{h},\mathbf{v}_{h})-(Z_{h}, \operatorname{div} \mathbf{v}_{h})=-(\bar{P}_{h}-\bar{ \mathbf{p}}_{d},\mathbf{v}_{h}),\quad \forall \mathbf{v}_{h}\in \mathbf{V}_{h}. $$

(3.22)

Let ϕ be the solution of (3.10) with $F=Z_{h}-z(U_{h})$. Then it follows from (2.42)-(2.44), (2.49)-(2.51), and (2.54)-(2.56) that

$$\begin{aligned}& \bigl\Vert Z_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2} \\& \quad =\int_{0}^{T}\bigl(Z_{h}-z(U_{h}),F \bigr)\,dt \\& \quad = \int_{0}^{T} \bigl(Z_{h}-z(U_{h}), \phi_{t}-\operatorname {div}(a\nabla\phi +\mathbf{b}\phi)+c\phi \bigr) \,dt \\& \quad = \int_{0}^{T} \bigl(\bigl(- \bigl(Z_{h}-z(U_{h})\bigr)_{t},\phi \bigr)- \bigl(Z_{h},\operatorname{div}\bigl(\Pi _{h}(a\nabla\phi+ \mathbf{b}\phi)\bigr)\bigr) \bigr)\,dt \\& \qquad {} +\int_{0}^{T}\bigl(\alpha \mathbf{q}(U_{h})+\mathbf {p}(U_{h})-\mathbf{p}_{d},a \nabla\phi +\mathbf{b}\phi\bigr)\,dt+\int_{0}^{T} \bigl(c\bigl(Z_{h}-z(U_{h})\bigr),\phi\bigr)\,dt \\& \quad = \int_{0}^{T} \bigl(\bigl(- \bigl(Z_{h}-z(U_{h})\bigr)_{t},\phi\bigr)-\bigl( \alpha Q_{h}+\bar{P}_{h}-\bar{\mathbf{p}}_{d}, \Pi_{h}(a\nabla\phi+\mathbf {b}\phi )\bigr) \bigr)\,dt \\& \qquad {} +\int_{0}^{T} \bigl(\bigl( \mathbf{p}(U_{h})-\mathbf{p}_{d},a\nabla \phi+\mathbf{b}\phi \bigr)-\bigl(\operatorname{div}\mathbf{q}(U_{h}),\phi\bigr)+\bigl( \boldsymbol{\beta }\cdot\mathbf{q}(U_{h}),\phi\bigr) \bigr)\,dt \\& \qquad {}+\int_{0}^{T}\bigl(c \bigl(Z_{h}-z(U_{h})\bigr),\phi\bigr)\,dt \\& \quad = \int_{0}^{T} \bigl(\bigl(- \bigl(Z_{h}-z(U_{h})\bigr)_{t},\phi\bigr)+\bigl( \alpha Q_{h}+\bar{P}_{h}-\bar{\mathbf{p}}_{d},a \nabla\phi+\mathbf{b}\phi -\Pi _{h}(a\nabla\phi+\mathbf{b}\phi)\bigr) \bigr)\,dt \\& \qquad {}+\int_{0}^{T} \bigl(\bigl(\alpha( \tilde{Q}_{h}-Q_{h})-\alpha\tilde {Q}_{h},a\nabla \phi+\mathbf{b}\phi\bigr)-\bigl(\operatorname{div}\mathbf {q}(U_{h}), \phi\bigr)+\bigl(\boldsymbol{\beta}\cdot\mathbf{q} (U_{h}),\phi\bigr) \bigr)\,dt \\& \qquad {}+\int_{0}^{T}\bigl(c \bigl(Z_{h}-z(U_{h}),\phi\bigr)\bigr)\,dt+\int _{0}^{T}\bigl(\mathbf{p} (U_{h})- \bar{P}_{h}+\bar{\mathbf{p}}_{d}-\mathbf{p}_{d},a \nabla\phi +\mathbf{b}\phi\bigr)\,dt \\& \quad = \int_{0}^{T}(- Z_{ht}+ \operatorname{div}\tilde{Q}_{h}-\boldsymbol {\beta}\cdot \tilde{Q}_{h}+c\tilde{Z}_{h}-\hat{Y}_{h}+ \hat{y}_{d},\phi)\,dt+\int_{0}^{T} \bigl(c(Z_{h}-\tilde{Z}_{h}),\phi\bigr)\,dt \\& \qquad {}+\int_{0}^{T}\bigl(\alpha Q_{h}+\bar{P}_{h}-\bar{\mathbf {p}}_{d},a\nabla \phi+\mathbf{b}\phi-\Pi_{h}(a\nabla\phi+\mathbf{b}\phi)\bigr)\,dt \\& \qquad {}+\int _{0}^{T}\bigl(\alpha (\tilde{Q}_{h}-Q_{h}),a \nabla\phi+\mathbf{b}\phi\bigr)\,dt +\int_{0}^{T}\bigl(y_{d}- \hat{y}_{d}+\hat{Y}_{h}-y(U_{h}),\phi \bigr)\,dt \\& \qquad {}+ \int_{0}^{T}\bigl(\mathbf{p}(U_{h})- \bar{P}_{h}+\bar{\mathbf {p}}_{d}-\mathbf{p}_{d},a \nabla \phi+\mathbf{b}\phi\bigr)\,dt \\& \quad =: J_{1}+J_{2}+\cdots+J_{6}. \end{aligned}$$

(3.23)

First, using the same estimates as (3.16)-(3.19), we have

$$\begin{aligned}& J_{1}\leq C\eta_{8}^{2}+ \frac{1}{8}\bigl\Vert Z_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, \end{aligned}$$

(3.24)

$$\begin{aligned}& J_{2}\leq C\eta_{12}^{2}+ \frac{1}{8}\bigl\Vert Z_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, \end{aligned}$$

(3.25)

$$\begin{aligned}& J_{3}\leq C\eta_{9}^{2}+ \frac{1}{8}\bigl\Vert Z_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, \end{aligned}$$

(3.26)

$$\begin{aligned}& J_{4}\leq C\eta_{10}^{2}+ \frac{1}{8}\bigl\Vert Z_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}. \end{aligned}$$

(3.27)

For $J_{5}$, using the Cauchy inequality and Lemma 3.1, we have

$$\begin{aligned} J_{5}&=\int_{0}^{T}\bigl( \hat{Y}_{h}-Y_{h}+Y_{h}-y(U_{h})+y_{d}- \hat {y}_{d},\phi\bigr)\, dt \\ &\leq C\bigl(\eta_{6}^{2}+ \eta_{14}^{2}\bigr)+C\bigl\Vert Y_{h}-y(U_{h}) \bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}+\frac{1}{8}\bigl\Vert Z_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}. \end{aligned}$$

(3.28)

Finally, for $J_{6}$, using (2.39), (2.46), the Cauchy inequality, and Lemma 3.1, we derive

$$\begin{aligned} J_{6} = &\int_{0}^{T}\bigl( \mathbf{p}(U_{h})-P_{h}+P_{h}-\bar {P}_{h}+\bar{\mathbf{p} }_{d}-\mathbf{p}_{d},a \nabla\phi+\mathbf{b}\phi\bigr)\,dt \\ =& \int_{0}^{T}\bigl(\alpha\bigl( \mathbf{p}(U_{h})-P_{h}\bigr),a^{2}\nabla\phi +a \mathbf{b} \phi\bigr)\,dt \\ &{}+\int_{0}^{T}(P_{h}- \bar{P}_{h}+\bar{\mathbf{p}}_{d}-\mathbf {p}_{d},a\nabla \phi+\mathbf{b}\phi)\,dt \\ =& \int_{0}^{T} \bigl(\bigl(y(U_{h}), \operatorname{div}\bigl(a^{2}\nabla \phi+a\mathbf{b}\phi \bigr)\bigr)- \bigl(\boldsymbol{\beta}y(U_{h}),a^{2}\nabla\phi+a\mathbf{b} \phi\bigr) \bigr)\,dt \\ &{} +\int_{0}^{T}\bigl(\alpha P_{h}, \Pi_{h}\bigl(a^{2}\nabla\phi+a\mathbf {b}\phi \bigr)-a^{2}\nabla\phi-a\mathbf{b}\phi\bigr)\,dt \\ &{} +\int_{0}^{T} \bigl(\bigl(\boldsymbol{ \beta}Y_{h},\Pi _{h}\bigl(a^{2}\nabla\phi+a \mathbf{b} \phi\bigr)\bigr)-\bigl(Y_{h},\operatorname{div}\bigl( \Pi_{h}\bigl(a^{2}\nabla\phi+a\mathbf {b}\phi\bigr)\bigr) \bigr) \bigr)\,dt \\ &{} +\int_{0}^{T}(P_{h}- \bar{P}_{h}+\bar{\mathbf {p}}_{d}-\mathbf{p} _{d},a\nabla\phi+\mathbf{b}\phi)\,dt \\ =& \int_{0}^{T} \bigl(\bigl(y(U_{h})-Y_{h}, \operatorname {div}\bigl(a^{2}\nabla\phi+a\mathbf{b} \phi\bigr)\bigr)+ \bigl(\boldsymbol{\beta}\bigl(Y_{h}-y(U_{h}) \bigr),a^{2}\nabla\phi+a\mathbf {b}\phi\bigr) \bigr)\,dt \\ &{} +\int_{0}^{T}\bigl(\alpha P_{h}+ \boldsymbol{\beta}Y_{h},\Pi _{h}\bigl(a^{2}\nabla \phi +a\mathbf{b}\phi\bigr)-a^{2}\nabla\phi-a\mathbf{b}\phi\bigr)\,dt \\ &{} +\int_{0}^{T}(P_{h}- \bar{P}_{h}+\bar{\mathbf {p}}_{d}-\mathbf{p} _{d},a\nabla\phi+\mathbf{b}\phi)\,dt \\ \leq& C\bigl(\eta_{3}^{2}+\eta_{11}^{2}+ \eta_{13}^{2}\bigr)+C\bigl\Vert Y_{h}-y(U_{h}) \bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}+\frac{1}{8}\bigl\Vert Z_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}. \end{aligned}$$

(3.29)

Therefore, it follows from the above estimates that

$$ \bigl\Vert Z_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2} \leq C\sum_{i=3,6,8-14} \eta_{i}^{2}+C\bigl\Vert Y_{h}-y(U_{h}) \bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}. $$

(3.30)

The triangle inequality and (3.30) yield (3.21). □

Remark 3.1

If we use the higher order RT mixed finite elements to approximate the state variables and the co-state variables, then the estimators $\eta _{2}^{2}$, $\eta_{3}^{2}$, $\eta_{8}^{2}$, and $\eta_{9}^{2}$ in Theorem 3.2 and Theorem 3.3 can be improved by

$$\begin{aligned}& \eta_{2}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{4} \int _{\tau } ( Y_{ht}+\operatorname{div} \hat{P}_{h}+c \hat{Y}_{h}-\hat{f}-U_{h} )^{2} \, dx\, dt; \\& \eta_{3}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{2}\int _{\tau} (\alpha P_{h}+\nabla_{h} Y_{h}+\boldsymbol{\beta}Y_{h} )^{2} \, dx\, dt; \\& \eta_{8}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{4}\int _{\tau } ( - Z_{ht}+\operatorname{div} \tilde{Q}_{h}-\boldsymbol{\beta}\cdot \tilde {Q}_{h}+c \tilde{Z}_{h}-\hat{Y}_{h}+\hat{y}_{d} )^{2}\, dx\, dt; \\& \eta_{9}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{2}\int _{\tau} (\alpha Q_{h}+\nabla_{h} Z_{h}+\bar{P}_{h}-\bar{\mathbf{p}}_{d} )^{2} \, dx\, dt, \end{aligned}$$

where $\nabla_{h} \chi|_{\tau}=\nabla(\chi|_{\tau})$.

Let $({\mathbf{p}},y,{\mathbf{q}},z,u)$ and $(P_{h},Y_{h},Q_{h},Z_{h},U_{h})$ be the solutions of (2.8)-(2.14) and (2.39)-(2.45), respectively. We decompose the errors as follows:

$$\begin{aligned}& {\mathbf{p}}-P_{h} ={\mathbf{p}}-{\mathbf {p}}(U_{h})+{ \mathbf{p}}(U_{h})-P_{h}:=\epsilon _{1}+ \varepsilon_{1}, \\& y-Y_{h} =y-y(U_{h})+y(U_{h})-Y_{h}:=r_{1}+e_{1}, \\& {\mathbf{q}}-Q_{h} ={\mathbf{q}}-{\mathbf{q}}(U_{h})+{ \mathbf {q}}(U_{h})-Q_{h}:=\epsilon_{2}+ \varepsilon_{2}, \\& z-Z_{h} =z-z(U_{h})+z(U_{h})-Z_{h}:=r_{2}+e_{2}. \end{aligned}$$

From (2.8)-(2.13) and (2.46)-(2.51), we derive the error equations:

$$\begin{aligned}& (\alpha\epsilon_{1},\mathbf{v})-(r_{1}, \operatorname {div}\mathbf{v})+(\boldsymbol{\beta}r_{1},\mathbf{v} )=0, \quad \forall \mathbf{v}\in\mathbf{V}, \end{aligned}$$

(3.31)

$$\begin{aligned}& ( r_{1t},w)+(\operatorname{div}\epsilon_{1},w)+(c r_{1},w)=(u-U_{h},w),\quad \forall w\in W, \end{aligned}$$

(3.32)

$$\begin{aligned}& (\alpha\epsilon_{2},\mathbf{v})-(r_{2}, \operatorname {div}\mathbf{v})=-(\epsilon _{1},\mathbf{v}), \quad \forall \mathbf{v}\in\mathbf{V}, \end{aligned}$$

(3.33)

$$\begin{aligned}& -(r_{2t},w)+(\operatorname{div}\epsilon _{2},w)-(\boldsymbol{\beta}\cdot\epsilon _{2},w)+(c r_{2},w)=(r_{1},w), \quad \forall w\in W. \end{aligned}$$

(3.34)

Theorem 3.4

There is a constant $C>0$, independent of h, such that

$$\begin{aligned}& \|\epsilon_{1}\|_{L^{2}(J;L^{2}(\Omega))}+\| r_{1}\| _{L^{2}(J;L^{2}(\Omega))}\leq C\| u-U_{h}\|_{L^{2}(J;L^{2}(\Omega))}, \end{aligned}$$

(3.35)

$$\begin{aligned}& \|\epsilon_{2}\|_{L^{2}(J;L^{2}(\Omega))}+\| r_{2}\| _{L^{2}(J;L^{2}(\Omega))}\leq C\| u-U_{h}\|_{L^{2}(J;L^{2}(\Omega))}. \end{aligned}$$

(3.36)

Proof

Choosing $\mathbf{v}=\epsilon_{1}$ and $w=r_{1}$ as the test functions and add the two relations of (3.31)-(3.32), we have

$$ (\alpha\epsilon_{1},\epsilon_{1})+(r_{1t},r_{1})=(u-U_{h},r_{1})-( \boldsymbol {\beta}r_{1}, \epsilon_{1})-(cr_{1},r_{1}). $$

(3.37)

Then, using the ϵ-Cauchy inequality, we can find an estimate as follows:

$$ (a\epsilon_{1},\epsilon_{1})+(r_{1t},r_{1}) \leq C \bigl(\| r_{1}\|_{L^{2}(\Omega)}^{2}+ \|u-U_{h}\|_{L^{2}(\Omega)}^{2} \bigr)+\frac {1}{2}(a \epsilon_{1},\epsilon_{1}). $$

(3.38)

Note that

$$({r_{1t}},r_{1})=\frac{1}{2}\frac{\partial}{\partial t}\| r_{1}\|_{L^{2}(\Omega)}^{2}, $$

then, using the assumption on a, we can obtain

$$ \frac{1}{2}a_{0}\|\epsilon_{1} \|_{L^{2}(\Omega)}^{2}+\frac{1}{2}\frac {\partial}{\partial t}\| r_{1}\|_{L^{2}(\Omega)}^{2} \leq C \bigl(\| r_{1} \|_{L^{2}(\Omega)}^{2}+\| u-U_{h}\|_{L^{2}(\Omega)}^{2} \bigr). $$

(3.39)

Integrating (3.39) in time, and since $r_{1}(0)=0$, using Lemma 3.2 to get

$$ \|\epsilon_{1}\|_{L^{2}(J;L^{2}(\Omega))}^{2}+\| r_{1}\|_{L^{\infty}(J;L^{2}(\Omega))}^{2} \leq C\| u-U_{h} \|_{L^{2}(J;L^{2}(\Omega))}^{2}, $$

(3.40)

implies (3.35).

Similarly, we can obtain

$$ \|\epsilon_{2}\|_{L^{2}(J;L^{2}(\Omega))}^{2}+ \|r_{2}\|_{L^{\infty }(J;L^{2}(\Omega))}^{2} \leq C\bigl(\| \epsilon_{1}\|_{L^{2}(J;L^{2}(\Omega))}^{2}+\| r_{1} \|_{L^{2}(J;L^{2}(\Omega))}^{2}\bigr). $$

(3.41)

Using (3.41) and (3.35), we complete the proof of Theorem 3.4. □

Collecting Theorems 3.1-3.4, we can derive the following results.

Theorem 3.5

Let $(\mathbf{p},y,\mathbf{q},z,u)$ and $(P_{h},Y_{h}, Q_{h},Z_{h},U_{h})$ be the solutions of (2.8)-(2.14) and (2.39)-(2.45), respectively. Then we have

$$ \| u-U_{h}\|_{L^{2}(J;L^{2}(\Omega))}^{2}+\| y-Y_{h}\| _{L^{2}(J;L^{2}(\Omega))}^{2}+\|z-Z_{h} \|_{L^{2}(J;L^{2}(\Omega))}^{2}\leq C\sum_{i=1}^{14} \eta_{i}^{2}, $$

(3.42)

where $\eta_{1}$ is defined in Theorem  3.1, $\eta_{2},\ldots,\eta_{7}$ are defined in Theorem  3.2, and $\eta_{8},\ldots,\eta_{14}$ are defined in Theorems 3.3, respectively.