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].
Anzeige
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:
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 \(0< a_{0}\leq a\leq a^{0}\), \(a(x)\in W^{1,\infty}(\Omega)\), \(c(x)\in W^{1,\infty}(\Omega)\), \(\mathbf{b}(x)\in(W^{1,\infty}(\Omega))^{2}\).
$$\begin{aligned}& \min_{u \in K\subset U} \biggl\{ \frac{1}{2}\int _{0}^{T} \bigl(\|\mathbf{p}-\mathbf{p}_{d} \| ^{2}+\| y-y_{d}\|^{2}+\| u\|^{2} \bigr)\, dt \biggr\} , \end{aligned}$$
(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)
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 \(\|\cdot\|_{m}=\|\cdot\|_{m,2}\), \(\|\cdot\|=\|\cdot\|_{0,2}\).
Anzeige
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 which can be written in the form of the first order system and
$$ -z_{t}-\operatorname{div}\bigl(a(\nabla z+\mathbf{p}- \mathbf {p}_{d})\bigr)+\mathbf{b}\cdot(\nabla z+\mathbf{p} - \mathbf{p}_{d})+cz=y-y_{d},\quad x\in \Omega, t\in J, $$
(2.1)
$$ -z_{t}+\operatorname{div}\mathbf{q}-a^{-1} \mathbf{b}\cdot \mathbf{q}+cz=y-y_{d}, \quad \mathbf{q} =-a(\nabla z+ \mathbf{p}-\mathbf{p}_{d}), x\in\Omega, t\in J $$
(2.2)
$$ z(x,t)=0, \quad x\in{\partial\Omega}, t\in J, \qquad z(x,T)=0, \quad x\in\Omega. $$
(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: The Hilbert space V is equipped with the following norm:
$$\mathbf{V}=H(\operatorname{div};\Omega)= \bigl\{ \mathbf{v}\in \bigl(L^{2}(\Omega)\bigr)^{2},\operatorname{div}\mathbf{v}\in L^{2}(\Omega) \bigr\} ,\qquad W=L^{2}(\Omega). $$
$$\|\mathbf{v}\|_{H(\operatorname{div};\Omega)}= \bigl(\|\mathbf{v}\| _{0,\Omega}^{2}+ \|\operatorname{div} \mathbf{v}\|_{0,\Omega}^{2} \bigr)^{1/2}. $$
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:
where \((\cdot,\cdot)\) is the inner product of \(L^{2}(\Omega)\).
$$\begin{aligned}& (\alpha{\mathbf{p}},\mathbf{v})-(y,\operatorname {div} \mathbf{v})+(\boldsymbol{\beta}y,\mathbf{v})=0, \quad \forall \mathbf{v}\in \mathbf{V}, t\in J, \end{aligned}$$
(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)
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
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:
$$\begin{aligned}& \min_{u_{h}(t) \in K_{h}} \biggl\{ \frac{1}{2}\int _{0}^{T} \bigl(\| \mathbf{p}_{h}- \mathbf{p}_{d}\|^{2}+\| y_{h}-y_{d} \|^{2}+\| u_{h}\|^{2} \bigr)\, dt \biggr\} , \end{aligned}$$
(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)
$$\begin{aligned}& (\alpha{\mathbf{p}}_{h},\mathbf {v}_{h})-(y_{h}, \operatorname{div}\mathbf{v}_{h})+(\boldsymbol{\beta }y_{h}, \mathbf{v} _{h})=0, \quad \forall \mathbf{v}_{h}\in \mathbf{V}_{h}, t\in J, \end{aligned}$$
(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
where \(f^{i}=f^{i}(x)=f(x,t_{i})\), \(y_{d}^{i}=y_{d}(x,t_{i})\), and \(\mathbf {p}_{d}^{i}=\mathbf{p}_{d}(x,t_{i})\).
$$\begin{aligned}& \min_{u^{i}_{h} \in K_{h}} \Biggl\{ \frac{1}{2}\sum _{i=1}^{N} \Delta t \bigl(\bigl\Vert \mathbf{p}_{h}^{i}-\mathbf{p}_{d}^{i} \bigr\Vert ^{2}+\bigl\Vert y_{h}^{i}-y_{d}^{i} \bigr\Vert ^{2}+\bigl\Vert u_{h}^{i}\bigr\Vert ^{2} \bigr) \Biggr\} , \end{aligned}$$
(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)
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 \(\mathbf{v}\in\mathbf{V}\),
where \(\mathcal{E}_{h}\) denotes the set of element sides in \(\mathcal{T}_{h}\).
$$\begin{aligned}& \int_{E}w_{h}(\mathbf{v}- \Pi_{h}\mathbf{v})\cdot\boldsymbol{\nu }_{E}\, ds=0,\quad w_{h}\in W_{h}, E\in\mathcal{E}_{h}, \end{aligned}$$
(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)
We have the commuting diagram property where I denotes the identity operator.
$$ \operatorname{div}\circ\Pi_{h}=R_{h}\circ \operatorname{div}:\mathbf{V} \rightarrow W_{h} \quad \text{and}\quad \operatorname{div}(I-\Pi _{h})\mathbf{V}\perp W_{h}, $$
(2.56)
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: 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
$$\begin{aligned}& \Omega_{-}=\bigl\{ x\in\Omega:\tilde{Z}_{h}(x)\leq0\bigr\} , \\& \Omega_{0}=\bigl\{ x\in\Omega:\tilde{Z}_{h}(x)>0,U_{h}(x)=0 \bigr\} , \\& \Omega_{+}=\bigl\{ x\in\Omega:\tilde{Z}_{h}(x)>0,U_{h}(x)>0 \bigr\} . \end{aligned}$$
$$\bar{\Omega}=\bar{\Omega}_{-}\cup\bar{\Omega}_{0}\cup\bar {\Omega}_{+}. $$
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
where
$$ \|u-U_{h}\|_{L^{2}(J;L^{2}(\Omega))}^{2}\leq C \eta_{1}^{2} + \bigl\Vert \tilde{Z}_{h}-z(U_{h}) \bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, $$
(3.1)
$$\eta_{1}^{2}=\|U_{h}+\tilde{Z}_{h} \|_{L^{2}(J;L^{2}(\Omega_{-}\cup \Omega_{+}))}^{2}. $$
Proof
It follows from (2.14) that We first estimate \(I_{1}\). Note that It is easy to see that where δ is an arbitrary small positive number, \(C(\delta)\) is dependent on \(\delta^{-1}\). Furthermore, we have It yields Then (3.3)-(3.5) imply that Moreover, it is clear that Now we turn to \(I_{3}\). Note that Then from (2.8)-(2.13) and (2.46)-(2.51), we have Thus, we obtain from (3.2) and (3.6)-(3.8) which proves (3.1). □
$$\begin{aligned}& \|u-U_{h}\|_{L^{2}(J;L^{2}(\Omega))}^{2} \\& \quad = \int _{0}^{T}(u-U_{h},u-U_{h})\, dt \\& \quad = \int_{0}^{T}(u+z,u-U_{h})\, dt+ \int_{0}^{T}(U_{h}+ \tilde{Z}_{h},U_{h}-u)\, dt \\& \qquad {} + \int_{0}^{T}\bigl(\tilde{Z}_{h}-z(U_{h}),u-U_{h} \bigr)\, dt+\int_{0}^{T}\bigl(z(U_{h})-z,u-U_{h} \bigr)\, dt \\& \quad \leq \int_{0}^{T}(U_{h}+ \tilde{Z}_{h},U_{h}-u)\, dt+ \int_{0}^{T} \bigl(\tilde{Z}_{h}-z(U_{h}),u-U_{h}\bigr)\, dt \\& \qquad {} +\int_{0}^{T}\bigl(z(U_{h})-z,u-U_{h} \bigr)\, dt \\& \quad = :I_{1}+I_{2}+I_{3}. \end{aligned}$$
(3.2)
$$\begin{aligned} \begin{aligned}[b] I_{1}&=\int_{0}^{T}(U_{h}+ \tilde{Z}_{h},U_{h}-u)\, dt \\ &=\int_{0}^{T}\int_{\Omega_{-}\cup\Omega _{+}}(U_{h}+ \tilde{Z}_{h}) (U_{h}-u)\, dx\, dt+\int_{0}^{T} \int_{\Omega _{0}}(U_{h}+\tilde{Z}_{h}) (U_{h}-u)\, dx\, dt. \end{aligned} \end{aligned}$$
(3.3)
$$\begin{aligned}& \int_{0}^{T}\int _{\Omega_{-}\cup\Omega _{+}}(U_{h}+\tilde{Z}_{h}) (U_{h}-u)\, dx\, dt \\& \quad \leq C(\delta)\|U_{h}+\tilde{Z}_{h}\| _{L^{2}(J;L^{2}(\Omega_{-}\cup\Omega_{+}))}^{2}+\delta\|u-U_{h}\| _{L^{2}(J;L^{2}(\Omega_{-}\cup\Omega_{+}))}^{2} \\& \quad = C(\delta)\eta_{1}^{2}+\delta\|u-U_{h}\| _{L^{2}(J;L^{2}(\Omega))}^{2}, \end{aligned}$$
(3.4)
$$U_{h}+\tilde{Z}_{h}\geq\tilde{Z}_{h}>0, \qquad U_{h}-u=0-u\leq0 \quad \text{on } \Omega_{0}. $$
$$ \int_{0}^{T}\int _{\Omega_{0}}(U_{h}+\tilde{Z}_{h}) (U_{h}-u)\, dx\, dt\leq0. $$
(3.5)
$$ I_{1}\leq C(\delta)\eta_{1}^{2}+ \delta \|u-U_{h}\|_{L^{2}(J;L^{2}(\Omega))}^{2}. $$
(3.6)
$$\begin{aligned} I_{2}&= \int_{0}^{T} \bigl(\tilde{Z}_{h}-z(U_{h}),u-U_{h}\bigr)\, dt \\ &\leq C(\delta)\bigl\Vert \tilde{Z}_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega ))}^{2}+\delta \|u-U_{h} \|_{L^{2}(J;L^{2}(\Omega))}^{2}. \end{aligned}$$
(3.7)
$$y(x,0)=y(U_{h}) (x,0)=y_{0}(x)\quad \text{and} \quad z(x,T)=z(U_{h}) (x,T)=0. $$
$$\begin{aligned} I_{3} =&\int_{0}^{T} \bigl(z(U_{h})-z,u-U_{h}\bigr)\,dt=\int_{0}^{T} \bigl(u-U_{h},z(U_{h})-z\bigr)\,dt \\ =&\int_{0}^{T} \bigl(\bigl( \bigl(y-y(U_{h})\bigr)_{t},z(U_{h})-z\bigr)+ \bigl(\operatorname{div}\bigl(\mathbf{p} -\mathbf{p}(U_{h}) \bigr),z(U_{h})-z\bigr) \bigr)\,dt \\ &{}+\int_{0}^{T} \bigl(\bigl(c \bigl(y-y(U_{h})\bigr),z(U_{h})-z\bigr)-\bigl(\boldsymbol { \beta} \bigl(y-y(U_{h})\bigr),\mathbf{q}(U_{h})-\mathbf{q} \bigr) \bigr)\,dt \\ &{}-\int_{0}^{T} \bigl(\bigl(\alpha\bigl( \mathbf{p}-\mathbf {p}(U_{h})\bigr),\mathbf{q}(U_{h})- \mathbf{q} \bigr)-\bigl(y-y(U_{h}),\operatorname{div}\bigl( \mathbf{q}(U_{h})-\mathbf{q}\bigr)\bigr) \bigr)\,dt \\ =&\int_{0}^{T} \bigl(-\bigl( \bigl(z(U_{h})-z\bigr)_{t},y-y(U_{h})\bigr)+ \bigl(\operatorname{div}\bigl(\mathbf{q} (U_{h})-\mathbf{q} \bigr),y-y(U_{h})\bigr) \bigr)\,dt \\ &{}+\int_{0}^{T} \bigl(\bigl(c \bigl(z(U_{h})-z\bigr),y-y(U_{h})\bigr)-\bigl(\boldsymbol { \beta}\cdot \bigl(\mathbf{q}(U_{h})-\mathbf{q}\bigr),y-y(U_{h}) \bigr) \bigr)\,dt \\ &{}-\int_{0}^{T} \bigl(\bigl(\alpha\bigl( \mathbf{q}(U_{h})-\mathbf {q}\bigr),\mathbf{p}-\mathbf{p} (U_{h})\bigr)-\bigl(z(U_{h})-z,\operatorname{div}\bigl( \mathbf{p}-\mathbf {p}(U_{h})\bigr)\bigr) \bigr)\,dt \\ =&\int_{0}^{T} \bigl(\bigl(y(U_{h})-y,y-y(U_{h}) \bigr)+\bigl(\mathbf{p} (U_{h})-\mathbf{p},\mathbf{p}- \mathbf{p}(U_{h})\bigr) \bigr)\,dt\leq0. \end{aligned}$$
(3.8)
$$ \|u-U_{h}\|_{L^{2}(J;L^{2}(\Omega ))}^{2}\leq C \eta_{1}^{2} + \bigl\Vert \tilde{Z}_{h}-z(U_{h}) \bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, $$
(3.9)
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: and
$$ \left \{\textstyle\begin{array}{l@{\quad}l} \phi_{t}-\operatorname{div}(a\nabla{\phi}+\mathbf{b}\phi)+c\phi=F, & x\in\Omega, t\in J, \\ \phi|_{\partial\Omega}=0, & t\in J, \\ \phi(x,0)=0, & x\in\Omega \end{array}\displaystyle \right . $$
(3.10)
$$ \left \{\textstyle\begin{array}{l@{\quad}l} -\psi_{t}-\operatorname{div}(a\nabla{\psi})+\mathbf{b}\cdot\nabla \psi+c\psi=F, & x\in \Omega, t\in J, \\ \psi|_{\partial\Omega}=0, & t\in J, \\ \psi(x,T)=0, & x\in\Omega. \end{array}\displaystyle \right . $$
(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
\(\varphi=\psi\),
where
\(|D^{2}\varphi|=\max\{|\partial^{2}\varphi/{\partial x_{i}\, \partial x_{j}}|, 1\leq i,j\leq2\}\).
$$\begin{aligned}& \int_{\Omega}\bigl\vert \varphi(x,t)\bigr\vert ^{2}\, dx \leq C\|F\|_{L^{2}(J;L^{2}(\Omega ))}^{2}, \quad \forall t \in J, \\& \int_{0}^{T}\int_{\Omega}|\nabla \varphi|^{2}\, dx\, dt\leq C\|F\| _{L^{2}(J;L^{2}(\Omega))}^{2}, \\& \int_{0}^{T}\int_{\Omega} \bigl\vert D^{2}\varphi\bigr\vert ^{2}\, dx\, dt\leq C\|F\| _{L^{2}(J;L^{2}(\Omega))}^{2}, \\& \int_{0}^{T}\int_{\Omega} | \varphi_{t}|^{2}\, dx\, dt \leq C\|F\| _{L^{2}(J;L^{2}(\Omega))}^{2}, \end{aligned}$$
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
\(t\in J\),
then
\(f(t)\leq e^{t}g(t)\).
$$ f(t)\leq g(t)+\int_{0}^{t}f(s)\, d s, $$
(3.12)
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
where
$$ \bigl\Vert Y_{h}-y(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}\leq C\sum_{i=2}^{7} \eta_{i}^{2}, $$
(3.13)
$$\begin{aligned}& \eta_{2}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{2} \int _{\tau } ( Y_{ht}+\operatorname{div} \hat{P}_{h}+c \hat{Y}_{h}-\hat{f}-U_{h} )^{2} \, dx\, dt; \\ & \eta_{3}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{2}\int _{\tau} (\alpha P_{h}+\boldsymbol{ \beta}Y_{h} )^{2} \, dx\, dt; \qquad \eta_{4}^{2}= \| \hat{P}_{h}-P_{h}\| _{L^{2}(J;L^{2}(\Omega))}^{2}; \\ & \eta_{5}^{2}=\|\hat{f}-f\|_{L^{2}(J;L^{2}(\Omega))}^{2}; \qquad \eta_{6}^{2}=\| \hat{Y}_{h}-Y_{h} \|_{L^{2}(J;L^{2}(\Omega))}^{2};\qquad \eta_{7}^{2}=\bigl\Vert y_{0}^{h}(x)-y_{0}(x)\bigr\Vert _{L^{2}(\Omega)}^{2}. \end{aligned}$$
Proof
From (2.30) and (2.37), we get the equality 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 Using (2.52), (2.40), the Cauchy inequality, and Lemma 3.1, we have Similarly, using the Cauchy inequality and Lemma 3.1, we have
Hence, using (3.15)-(3.19), we get This proves (3.13). □
$$ (\alpha{P}_{h},\mathbf{v}_{h})-(Y_{h}, \operatorname{div}\mathbf {v}_{h})+(\boldsymbol{\beta}Y_{h}, \mathbf{v}_{h})=0,\quad \forall \mathbf{v}_{h}\in \mathbf{V}_{h}. $$
(3.14)
$$\begin{aligned}& \bigl\Vert Y_{h}-y(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2} \\& \quad =\int_{0}^{T}\bigl(Y_{h}-y(U_{h}),F \bigr)\,dt \\& \quad = \int_{0}^{T}\bigl(Y_{h}-y(U_{h}),- \psi_{t}-\operatorname{div}(a\nabla \psi)+\mathbf{b} \cdot\nabla\psi+c \psi\bigr)\,dt \\& \quad = \int_{0}^{T} \bigl(\bigl( \bigl(Y_{h}-y(U_{h})\bigr)_{t},\psi\bigr)- \bigl(Y_{h},\operatorname {div}\bigl(\Pi _{h}(a\nabla\psi) \bigr)\bigr)+\bigl(\mathbf{p}(U_{h}),\nabla\psi\bigr) \bigr)\,dt \\& \qquad {}+\int_{0}^{T} \bigl(( \mathbf{b}Y_{h},\nabla\psi )+\bigl(c\bigl(Y_{h}-y(U_{h}) \bigr),\psi \bigr) \bigr)\,dt+\bigl(\bigl(Y_{h}-y(U_{h})\bigr) (x,0), \psi(x,0)\bigr) \\& \quad = \int_{0}^{T} \bigl(\bigl( \bigl(Y_{h}-y(U_{h})\bigr)_{t},\psi\bigr)-\bigl( \alpha P_{h},\Pi _{h}(a\nabla\psi)\bigr) \\& \qquad {}-\bigl(\boldsymbol{ \beta}Y_{h},\Pi_{h}(a\nabla\psi )\bigr)-\bigl( \operatorname{div}\mathbf{p}(U_{h}),\psi \bigr) \bigr)\,dt \\& \qquad {} +\int_{0}^{T} \bigl((\boldsymbol{ \beta}Y_{h},a\nabla\psi )+\bigl(c\bigl(Y_{h}-y(U_{h}) \bigr),\psi\bigr) \bigr)\,dt+\bigl(y_{0}^{h}(x)-y_{0}(x), \psi(x,0)\bigr) \\& \quad = \int_{0}^{T} \bigl((Y_{ht}, \psi)+\bigl(\alpha P_{h},a\nabla\psi -\Pi_{h}(a\nabla\psi) \bigr)-(\hat{P}_{h}-P_{h},\nabla\psi)-(\operatorname {div} \hat{P}_{h} ,\psi) \bigr)\,dt \\& \qquad {} +\int_{0}^{T} \bigl(\bigl(\boldsymbol{ \beta}Y_{h},a\nabla\psi-\Pi _{h}(a\nabla\psi ) \bigr)+(cY_{h}-f-U_{h},\psi) \bigr)\,dt+\bigl(y_{0}^{h}(x)-y_{0}(x), \psi(x,0)\bigr) \\& \quad = \int_{0}^{T}(Y_{ht}+ \operatorname{div}\hat{P}_{h}+c\hat {Y}_{h}-\hat {f}-U_{h},\psi)\,dt+\int_{0}^{T}\bigl( \alpha P_{h}+\boldsymbol{\beta}Y_{h},a\nabla \psi-\Pi _{h}(a\nabla\psi)\bigr)\,dt \\& \qquad {} +\int_{0}^{T} \bigl((\hat{f}-f,\psi)+ \bigl(c(Y_{h}-\hat{Y}_{h}),\psi \bigr)+(\hat{P}_{h}-P_{h}, \nabla\psi) \bigr)\,dt+\bigl(y_{0}^{h}(x)-y_{0}(x), \psi(x,0)\bigr) \\& \quad =: L_{1}+L_{2}+L_{3}+L_{4}. \end{aligned}$$
(3.15)
$$\begin{aligned} L_{1}&= \int_{0}^{T}(Y_{ht}+ \operatorname{div}\hat{P}_{h}+c\hat {Y}_{h}-\hat {f}-U_{h},\psi-P_{h} \psi)\, dt \\ &\leq C(\delta)\eta_{2}^{2}+\delta\|\psi\| _{L^{2}(J;H^{1}(\Omega))}^{2} \\ &\leq C\eta_{2}^{2}+ \frac{1}{5}\bigl\Vert Y_{h}-y(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}. \end{aligned}$$
(3.16)
$$\begin{aligned}& L_{2}\leq C\eta_{3}^{2}+ \frac{1}{5}\bigl\Vert Y_{h}-y(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, \end{aligned}$$
(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)
$$ \bigl\Vert Y_{h}-y(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}\leq C\sum_{i=2}^{7} \eta_{i}^{2}. $$
(3.20)
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:
where
\(\eta_{3}\)
and
\(\eta_{6}\)
are defined in Theorem
3.2.
$$ \bigl\Vert \tilde{Z}_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2} \leq C\sum_{i=3,6,8-14} \eta_{i}^{2}+C\bigl\Vert Y_{h}-y(U_{h}) \bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, $$
(3.21)
$$\begin{aligned}& \eta_{8}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{2}\int _{\tau } ( - Z_{ht}+\operatorname{div} \tilde{Q}_{h}-\boldsymbol{\beta}\cdot \tilde {Q}_{h}+c \tilde{Z}_{h}-\hat{Y}_{h}+\hat{y}_{d} )^{2}\, dx\, dt; \\& \eta_{9}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{2}\int _{\tau} (\alpha Q_{h}+\bar{P}_{h}-\bar{ \mathbf{p}}_{d} )^{2} \, dx \, dt; \qquad \eta _{10}^{2}=\|\tilde{Q}_{h}-Q_{h} \|_{L^{2}(J;L^{2}(\Omega))}^{2}; \\& \eta_{11}^{2}=\|\bar{P}_{h}-P_{h} \|_{L^{2}(J;L^{2}(\Omega))}^{2};\qquad \eta _{12}^{2}=\| \tilde{Z}_{h}-Z_{h}\|_{L^{2}(J;L^{2}(\Omega))}^{2}; \\& \eta_{13}^{2}=\|\bar{\mathbf{p}}_{d}- \mathbf{p}_{d}\| _{L^{2}(J;L^{2}(\Omega))}^{2}; \qquad \eta_{14}^{2}=\|\hat{y}_{d}-y_{d} \|_{L^{2}(J;L^{2}(\Omega))}^{2}, \end{aligned}$$
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 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 First, using the same estimates as (3.16)-(3.19), we have
For \(J_{5}\), using the Cauchy inequality and Lemma 3.1, we have Finally, for \(J_{6}\), using (2.39), (2.46), the Cauchy inequality, and Lemma 3.1, we derive Therefore, it follows from the above estimates that The triangle inequality and (3.30) yield (3.21). □
$$ (\alpha Q_{h},\mathbf{v}_{h})-(Z_{h}, \operatorname{div} \mathbf{v}_{h})=-(\bar{P}_{h}-\bar{ \mathbf{p}}_{d},\mathbf{v}_{h}),\quad \forall \mathbf{v}_{h}\in \mathbf{V}_{h}. $$
(3.22)
$$\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)
$$\begin{aligned}& J_{1}\leq C\eta_{8}^{2}+ \frac{1}{8}\bigl\Vert Z_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, \end{aligned}$$
(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)
$$\begin{aligned} J_{5}&=\int_{0}^{T}\bigl( \hat{Y}_{h}-Y_{h}+Y_{h}-y(U_{h})+y_{d}- \hat {y}_{d},\phi\bigr)\, dt \\ &\leq C\bigl(\eta_{6}^{2}+ \eta_{14}^{2}\bigr)+C\bigl\Vert Y_{h}-y(U_{h}) \bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}+\frac{1}{8}\bigl\Vert Z_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}. \end{aligned}$$
(3.28)
$$\begin{aligned} J_{6} = &\int_{0}^{T}\bigl( \mathbf{p}(U_{h})-P_{h}+P_{h}-\bar {P}_{h}+\bar{\mathbf{p} }_{d}-\mathbf{p}_{d},a \nabla\phi+\mathbf{b}\phi\bigr)\,dt \\ =& \int_{0}^{T}\bigl(\alpha\bigl( \mathbf{p}(U_{h})-P_{h}\bigr),a^{2}\nabla\phi +a \mathbf{b} \phi\bigr)\,dt \\ &{}+\int_{0}^{T}(P_{h}- \bar{P}_{h}+\bar{\mathbf{p}}_{d}-\mathbf {p}_{d},a\nabla \phi+\mathbf{b}\phi)\,dt \\ =& \int_{0}^{T} \bigl(\bigl(y(U_{h}), \operatorname{div}\bigl(a^{2}\nabla \phi+a\mathbf{b}\phi \bigr)\bigr)- \bigl(\boldsymbol{\beta}y(U_{h}),a^{2}\nabla\phi+a\mathbf{b} \phi\bigr) \bigr)\,dt \\ &{} +\int_{0}^{T}\bigl(\alpha P_{h}, \Pi_{h}\bigl(a^{2}\nabla\phi+a\mathbf {b}\phi \bigr)-a^{2}\nabla\phi-a\mathbf{b}\phi\bigr)\,dt \\ &{} +\int_{0}^{T} \bigl(\bigl(\boldsymbol{ \beta}Y_{h},\Pi _{h}\bigl(a^{2}\nabla\phi+a \mathbf{b} \phi\bigr)\bigr)-\bigl(Y_{h},\operatorname{div}\bigl( \Pi_{h}\bigl(a^{2}\nabla\phi+a\mathbf {b}\phi\bigr)\bigr) \bigr) \bigr)\,dt \\ &{} +\int_{0}^{T}(P_{h}- \bar{P}_{h}+\bar{\mathbf {p}}_{d}-\mathbf{p} _{d},a\nabla\phi+\mathbf{b}\phi)\,dt \\ =& \int_{0}^{T} \bigl(\bigl(y(U_{h})-Y_{h}, \operatorname {div}\bigl(a^{2}\nabla\phi+a\mathbf{b} \phi\bigr)\bigr)+ \bigl(\boldsymbol{\beta}\bigl(Y_{h}-y(U_{h}) \bigr),a^{2}\nabla\phi+a\mathbf {b}\phi\bigr) \bigr)\,dt \\ &{} +\int_{0}^{T}\bigl(\alpha P_{h}+ \boldsymbol{\beta}Y_{h},\Pi _{h}\bigl(a^{2}\nabla \phi +a\mathbf{b}\phi\bigr)-a^{2}\nabla\phi-a\mathbf{b}\phi\bigr)\,dt \\ &{} +\int_{0}^{T}(P_{h}- \bar{P}_{h}+\bar{\mathbf {p}}_{d}-\mathbf{p} _{d},a\nabla\phi+\mathbf{b}\phi)\,dt \\ \leq& C\bigl(\eta_{3}^{2}+\eta_{11}^{2}+ \eta_{13}^{2}\bigr)+C\bigl\Vert Y_{h}-y(U_{h}) \bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}+\frac{1}{8}\bigl\Vert Z_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}. \end{aligned}$$
(3.29)
$$ \bigl\Vert Z_{h}-z(U_{h})\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2} \leq C\sum_{i=3,6,8-14} \eta_{i}^{2}+C\bigl\Vert Y_{h}-y(U_{h}) \bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}. $$
(3.30)
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 where \(\nabla_{h} \chi|_{\tau}=\nabla(\chi|_{\tau})\).
$$\begin{aligned}& \eta_{2}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{4} \int _{\tau } ( Y_{ht}+\operatorname{div} \hat{P}_{h}+c \hat{Y}_{h}-\hat{f}-U_{h} )^{2} \, dx\, dt; \\& \eta_{3}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{2}\int _{\tau} (\alpha P_{h}+\nabla_{h} Y_{h}+\boldsymbol{\beta}Y_{h} )^{2} \, dx\, dt; \\& \eta_{8}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{4}\int _{\tau } ( - Z_{ht}+\operatorname{div} \tilde{Q}_{h}-\boldsymbol{\beta}\cdot \tilde {Q}_{h}+c \tilde{Z}_{h}-\hat{Y}_{h}+\hat{y}_{d} )^{2}\, dx\, dt; \\& \eta_{9}^{2}=\int_{0}^{T} \sum_{\tau}h_{\tau}^{2}\int _{\tau} (\alpha Q_{h}+\nabla_{h} Z_{h}+\bar{P}_{h}-\bar{\mathbf{p}}_{d} )^{2} \, dx\, dt, \end{aligned}$$
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 Then, using the ϵ-Cauchy inequality, we can find an estimate as follows: Note that then, using the assumption on a, we can obtain Integrating (3.39) in time, and since \(r_{1}(0)=0\), using Lemma 3.2 to get implies (3.35).
$$ (\alpha\epsilon_{1},\epsilon_{1})+(r_{1t},r_{1})=(u-U_{h},r_{1})-( \boldsymbol {\beta}r_{1}, \epsilon_{1})-(cr_{1},r_{1}). $$
(3.37)
$$ (a\epsilon_{1},\epsilon_{1})+(r_{1t},r_{1}) \leq C \bigl(\| r_{1}\|_{L^{2}(\Omega)}^{2}+ \|u-U_{h}\|_{L^{2}(\Omega)}^{2} \bigr)+\frac {1}{2}(a \epsilon_{1},\epsilon_{1}). $$
(3.38)
$$({r_{1t}},r_{1})=\frac{1}{2}\frac{\partial}{\partial t}\| r_{1}\|_{L^{2}(\Omega)}^{2}, $$
$$ \frac{1}{2}a_{0}\|\epsilon_{1} \|_{L^{2}(\Omega)}^{2}+\frac{1}{2}\frac {\partial}{\partial t}\| r_{1}\|_{L^{2}(\Omega)}^{2} \leq C \bigl(\| r_{1} \|_{L^{2}(\Omega)}^{2}+\| u-U_{h}\|_{L^{2}(\Omega)}^{2} \bigr). $$
(3.39)
$$ \|\epsilon_{1}\|_{L^{2}(J;L^{2}(\Omega))}^{2}+\| r_{1}\|_{L^{\infty}(J;L^{2}(\Omega))}^{2} \leq C\| u-U_{h} \|_{L^{2}(J;L^{2}(\Omega))}^{2}, $$
(3.40)
Similarly, we can obtain Using (3.41) and (3.35), we complete the proof of Theorem 3.4. □
$$ \|\epsilon_{2}\|_{L^{2}(J;L^{2}(\Omega))}^{2}+ \|r_{2}\|_{L^{\infty }(J;L^{2}(\Omega))}^{2} \leq C\bigl(\| \epsilon_{1}\|_{L^{2}(J;L^{2}(\Omega))}^{2}+\| r_{1} \|_{L^{2}(J;L^{2}(\Omega))}^{2}\bigr). $$
(3.41)
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
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.
$$ \| u-U_{h}\|_{L^{2}(J;L^{2}(\Omega))}^{2}+\| y-Y_{h}\| _{L^{2}(J;L^{2}(\Omega))}^{2}+\|z-Z_{h} \|_{L^{2}(J;L^{2}(\Omega))}^{2}\leq C\sum_{i=1}^{14} \eta_{i}^{2}, $$
(3.42)
Acknowledgements
The first author is supported by the scientific research program in Hunan University of Science and Engineering (2015). The second author is supported by the National Natural Science Foundation of China (11401201), the Foundation of Hunan Educational Committee (13C338), and the construct program of the key discipline in Hunan University of Science and Engineering.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The first author carried out the molecular genetic studies, participated in the sequence alignment, and drafted the manuscript. The second author conceived of the study, and participated in its design and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.