Adaptive variational discretization approximation method for parabolic optimal control problems

Abstract

In this paper, we study variational discretization method for parabolic optimization problems. Firstly, we obtain some convergence and superconvergence analysis results of the approximation scheme. Secondly, we derive a posteriori error estimates of the approximation solutions. Finally, we present variational discretization approximation algorithm and adaptive variational discretization approximation algorithm for parabolic optimization problems and do some numerical experiments to confirm our theoretical results.

1 Introduction

Optimal control problems (OCPs) play an important role in scientific and engineering numerical simulation, and nowadays are strongly utilized in biology, economics, and finance. Finite element approximation is one of the widely used numerical methods in computing OCPs. A systematic introduction of finite element method (FEM) for partial differential equations or OPCs can be found in [1, 2, 8, 20, 30, 33].

There have been abundant researches on FEM approximation for elliptic optimal control problems (EOCPs). The pioneering work of the late 1970s in the area of finite element approximation for EOCPs includes [10, 11], where a priori error estimates were established. Then a lot of superconvergence and a posteriori error estimate results of FEM solving different kinds of EOCPs were obtained. For instance, superconvergence of FEM for linear, bilinear, and semilinear EOCPs were derived in [27, 32], and [4], respectively, while residual- and recovery-based a posteriori error estimates of FEM for distributed OCPs were constructed in [22] and [18], respectively. Moreover, adaptive FEM for EOCPs were presented in [3, 13, 17]. It is worth mentioning that some similar results of mixed FEM for EOCPs can be found in [5, 7, 16].

In the last decade, numerical solution of parabolic optimal control problems (POCPs) became a hot research topic. A priori error estimates of FEM, space-time FEM, and Petrov–Galerkin Crank–Nicolson approximation for POCPs were given in [12, 25], and [26], respectively. In [24], the authors constructed an adaptive space-time FEM for POCPs. There are also some results on residual-type a posteriori error estimates of FEM or mixed FEM for POCPs which can be found in [6, 23, 31], where the authors do not give any adaptive FEM approximation for POCPs.

Hinze presented a variational discretization (VD) concept for control constrained optimization problems in [14]. It can not only save some computation cost but also improve the error of the control variable. Recently, VD approximation of convection dominated diffusion EOCP with control constraints and POCP with pointwise state constraints were investigated in [15] and [9], respectively. We have investigated VD approximation for a linear POCP in [28].

The purpose of this work is to investigate VD approximation for a POCP with control constraints. We first analyze the convergence and superconvergence of the VD approximation scheme and derive a posteriori error estimates, then construct an adaptive VD approximation algorithm for the POCP. Finally, two numerical examples are provided to verify theoretical results.

We are interested in the following POCP:

$$ \textstyle\begin{cases} \min\limits_{u(x,t)\in K}\frac{1}{2}\int _{0}^{T} ( \Vert y(x,t)-y_{d}(x,t) \Vert ^{2}+\nu \Vert u(x,t) \Vert ^{2} )\,dt, \\ y_{t}(x,t)-\operatorname{div}(A(x)\nabla y(x,t))=f(x,t)+u(x,t), \quad x\in \Omega , t\in J, \\ y(x,t)=0, \quad x\in \partial \Omega , t\in J, \\ y(x,0)=y_{0}(x), \quad x\in \Omega , \end{cases} $$

(1)

where $\nu >0$ represents the weight of the cost of the control, Ω is a bounded domain in $\mathbb{R}^{2}$ with a Lipschitz boundary ∂Ω, $0< T<+\infty$ and $J=[0,T]$. The coefficient $A(x)=(a_{ij}(x))_{2\times 2}\in (W^{1,\infty }(\bar{\Omega }))^{2 \times 2}$ is such that for any $\xi \in \mathbb{R}^{2}$, $(A(x)\xi )\cdot \xi \geq c| \xi | ^{2}$ with $c>0$. We assume that K is a nonempty closed convex set in $L^{2}(J;L^{2}(\Omega ))$, defined by

$$ K= \bigl\{ v(x,t)\in L^{\infty }\bigl(J;L^{2}(\Omega )\bigr): a\leq v(x,t)\leq b, \mbox{a.e. } (x,t)\in \Omega \times J \bigr\} , $$

where a and b are constants.

In this paper, we adopt the standard notation $W^{m,q}(\Omega )$ for Sobolev spaces on Ω with norm $\|\cdot \|_{W^{m,q}(\Omega )}$ and seminorm $|\cdot |_{W^{m,q}(\Omega )}$. We denote $W^{m,2}(\Omega )$ by $H^{m}(\Omega )$ and set $H_{0}^{1}(\Omega )\equiv \{ v \in H^{1}(\Omega ): v|_{\partial \Omega } =0 \}$. We denote by $L^{s}(J;W^{m,q}(\Omega ))$ the Banach space of $L^{s}$ integrable functions from J into $W^{m,q}(\Omega )$ with norm $\|v\|_{L^{s}(J;W^{m,q}(\Omega ))}= (\int _{0}^{T}\|v\|_{W^{m,q}( \Omega )}^{s}\,dt )^{\frac{1}{s}}$ for $s\in [1,\infty )$ and the standard modification for $s=\infty$. As in [21], we can define the space $H^{l}(J;W^{m,q}(\Omega ))$. In addition, c or C denotes a generic positive constant independent of h and k.

The outline of the paper is as follows. In Sect. 2, we give a VD approximation for the model problem. In Sect. 3, we derive some convergence and superconvergence analysis results for the control, state, and costate variables. In Sect. 4, we establish a posteriori error estimates for the approximation scheme. We present a VD approximation algorithm and adaptive VD approximation algorithm for the POCP and do some numerical experiments to confirm our theoretical results in the last section.

2 VD approximation of POCP

In this section, we present a VD approximation for the model problem (1). For ease of exposition, we denote $L^{p}(J;W^{m,q}(\Omega ))$ by $L^{p}(W^{m,q})$. Let $W=H_{0}^{1}(\Omega )$ and $U=L^{2}(\Omega )$. Moreover, we denote $\|\cdot \|_{H^{m}(\Omega )}$ and $\|\cdot \|_{L^{2}(\Omega )}$ by $\|\cdot \|_{m}$ and $\|\cdot \|$, respectively.

Let

$$ \begin{aligned} &a(v,w)= \int _{\Omega }(A\nabla v)\cdot \nabla w,\quad \forall v, w\in W, \\ &(f_{1},f_{2})= \int _{\Omega }f_{1}\cdot f_{2},\quad \forall f_{1}, f_{2}\in U. \end{aligned} $$

It follows from the assumptions on A that

$$ a(v,v)\geq c \Vert v \Vert _{1}^{2},\qquad \bigl\vert a(v,w) \bigr\vert \leq C \Vert v \Vert _{1} \Vert w \Vert _{1},\quad \forall v, w\in W. $$

Thus a possible weak formula for the model problem (1) reads:

$$ \textstyle\begin{cases} \min\limits_{u\in K}\frac{1}{2}\int _{0}^{T} ( \Vert y-y_{d} \Vert ^{2}+\nu \Vert u \Vert ^{2} )\,dt, \\ (y_{t},w)+a(y,w)=(f+u,w), \quad \forall w\in W, t\in J, \\ y(x,0)=y_{0}(x),\quad x\in \Omega . \end{cases} $$

(2)

It is well known (see, e.g., [20, 23]) that problem (2) has a unique solution $(y,u)$, and the pair $(y,u)\in (H^{1}(L^{2})\cap L^{2}(H^{1}) )\times K$ is the solution of (2) if and only if there is an adjoint state $p\in H^{1}(L^{2})\cap L^{2}(H^{1})$ such that the triplet $(y,p,u)$ satisfies the following optimality conditions:

$$\begin{aligned}& (y_{t},w)+a(y,w)=(f+u,w),\quad \forall w\in W, t\in J, \end{aligned}$$

(3)

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

(4)

$$\begin{aligned}& -(p_{t},q)+a(q,p)=(y_{d}-y,q),\quad \forall q\in W, t\in J, \end{aligned}$$

(5)

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

(6)

$$\begin{aligned}& (\nu u-p,v-u)\geq 0, \quad \forall v\in K,t\in J. \end{aligned}$$

(7)

We introduce the following pointwise projection operator:

$$ \Pi _{[a,b]} \bigl(g(x,t)\bigr)=\min \biggl(b,\max \biggl(a, \frac{1}{\nu }g(x,t) \biggr) \biggr),\quad \forall (x,t)\in \Omega \times J. $$

Similar to [15], the variational inequality (7) can be rewritten as

$$ u(x,t)=\Pi _{[a,b]}\bigl(p(x,t)\bigr),\quad \forall (x,t)\in \Omega \times J.$$

(8)

Let $\mathcal{T}^{h}$ be regular triangulations of Ω and $\bar{\Omega }=\bigcup_{\tau \in \mathcal{T}^{h}}\bar{\tau }$. Let $h=\max_{\tau \in \mathcal{T}^{h}}\{h_{\tau }\}$, where $h_{\tau }$ denotes the diameter of the element τ. Moreover, we set

$$ W^{h}=\bigl\{ v_{h}\in C( \bar{\Omega }) : v_{h}|_{\tau }\in \mathbb{P}_{1}, \forall \tau \in \mathcal{T}^{h}, v_{h}|_{\partial \Omega }=0 \bigr\} , $$

where $\mathbb{P}_{1}$ is the space of polynomials up to order 1.

We now consider the time discretization for problem (2). Let $k>0$, $N=T/k\in \mathbb{Z}^{+}$, $t_{n}=n k$, $n=0,1,\ldots ,N$. Set $\varphi ^{n}=\varphi (x,t_{n})$ and

$$ d_{t}\varphi ^{n}=\frac{\varphi ^{n}-\varphi ^{n-1}}{k},\quad n=1,2, \ldots ,N. $$

We define for $1\leq p<\infty$ the discrete time-dependent norms

$$ |\!|\!|\varphi |\!|\!|_{l^{p}(J;W^{m,q}(\Omega ))}:= \Biggl(k\sum _{n=1-l}^{N-l} \bigl\Vert \varphi ^{n} \bigr\Vert ^{p}_{W^{m,q}(\Omega )} \Biggr)^{ \frac{1}{p}}, $$

where $l=0$ for the control u and the state y and $l=1$ for the adjoint state p, with the standard modification for $p=\infty$. Just for simplicity, we denote $|\!|\!|\cdot |\!|\!|_{l^{p}(J;W^{m,q}(\Omega ))}$ by $|\!|\!|\cdot |\!|\!|_{l^{p}(W^{m,q})}$ and let

$$ l^{p}_{D}\bigl(J;W^{m,q}(\Omega )\bigr):= \bigl\{ \varphi :|\!|\!|\varphi |\!|\!|_{l^{p}(W^{m,q})}< \infty \bigr\} ,\quad 1\leq p\leq \infty. $$

Then a VD approximation of (2) is as follows:

$$ \textstyle\begin{cases} \min\limits_{u_{h}^{n}\in K} \frac{1}{2}\sum_{n=1}^{N}k ( \Vert y_{h}^{n}-y_{d}^{n} \Vert ^{2}+\nu \Vert u_{h}^{n} \Vert ^{2} ), \\ (d_{t}y_{h}^{n},w_{h} )+a (y_{h}^{n},w_{h} )= (f^{n}+u_{h}^{n},w_{h} ), \quad \forall w_{h}\in W^{h}, n=1,2,\ldots ,N, \\ y_{h}^{0}(x)=P_{h}y_{0}(x), \quad x\in \Omega , \end{cases} $$

(9)

where $P_{h}$ is an elliptic projection operator which will be specified later.

It follows (see, e.g., [15, 23]) that the control problem (9) has a unique solution $(y_{h}^{n},u_{h}^{n} )$, $n=1,2,\ldots ,N$, and $(y_{h}^{n},u_{h}^{n} )\in W^{h}\times K$, $n=1,2,\ldots ,N$, is the solution of (9) if and only if there is an adjoint state $p_{h}^{n-1}\in W^{h}$, $n=1,2,\ldots ,N$, such that the triplet $(y_{h}^{n},p_{h}^{n-1},u_{h}^{n} )\in W^{h}\times W^{h} \times K$, $n=1,2,\ldots ,N$, satisfies the following optimality conditions:

$$\begin{aligned}& \bigl(d_{t}y_{h}^{n},w_{h} \bigr)+a \bigl(y_{h}^{n},w_{h} \bigr)= \bigl(f^{n}+u_{h}^{n},w_{h} \bigr),\quad \forall w_{h}\in W^{h}, \end{aligned}$$

(10)

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

(11)

$$\begin{aligned}& - \bigl(d_{t}p_{h}^{n},q_{h} \bigr)+a \bigl(q_{h},p_{h}^{n-1} \bigr)= \bigl(y_{d}^{n}-y_{h}^{n},q_{h} \bigr),\quad \forall q_{h} \in W^{h}, \end{aligned}$$

(12)

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

(13)

$$\begin{aligned}& \bigl(\nu u_{h}^{n}-p_{h}^{n-1},v-u_{h}^{n} \bigr)\geq 0 ,\quad \forall v\in K. \end{aligned}$$

(14)

Similar to [15], the optimality condition (14) can be equivalently expressed as

$$ u_{h}^{n}=\Pi _{[a,b]} \bigl(p_{h}^{n-1} \bigr),\quad n=1,2,\ldots ,N.$$

(15)

3 Convergence and superconvergence analysis

For the approximation scheme (10)–(14), some convergence and superconvergence results will be derived in this section. We define the following intermediate variables. Let $(y_{h}^{n}(u),p_{h}^{n-1}(u) )\in W^{h}\times W^{h}$, $n=1,2, \ldots ,N$, satisfy the following system:

$$\begin{aligned}& \bigl(d_{t}y_{h}^{n}(u),w_{h} \bigr)+a \bigl(y_{h}^{n}(u),w_{h} \bigr)= \bigl(f^{n}+u^{n},w_{h} \bigr),\quad \forall w_{h}\in W^{h}, \end{aligned}$$

(16)

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

(17)

$$\begin{aligned}& - \bigl(d_{t}p_{h}^{n}(u),q_{h} \bigr)+a \bigl(q_{h},p_{h}^{n-1}(u) \bigr)= \bigl(y_{d}^{n}-y_{h}^{n}(u),q_{h} \bigr),\quad \forall q_{h}\in W^{h}, \end{aligned}$$

(18)

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

(19)

and introduce elliptic projection operator $P_{h}: W\rightarrow W^{h}$, which satisfies, for any $\phi \in W$,

$$ a(\phi -P_{h}\phi ,w_{h})=0,\quad \forall w_{h}\in W^{h}.$$

(20)

It has the following approximation properties (see, e.g., [4]):

$$ \Vert \phi -P_{h}\phi \Vert \leq Ch^{2} \Vert \phi \Vert _{2}, \quad \forall \phi \in H^{2}(\Omega ).$$

(21)

Lemma 3.1

Let $(y,p,u)$ and $(y_{h}(u),p_{h}(u) )$ be the solutions of (3)–(7) and (16)–(18), respectively. Suppose that $y,p\in l^{2}_{D}(H^{2})\cap H^{1}(H^{2})\cap H^{2}(L^{2})$. Then

$$\begin{aligned} \bigl|\!\bigl|\!\bigl|y_{h}(u)-y\bigr|\!\bigr|\!\bigr|_{l^{2}(L^{2})}+\bigl|\!\bigl|\!\bigl|p_{h}(u)-p\bigr|\!\bigr|\!\bigr|_{l^{2}(L^{2})}\leq C \bigl(h^{2}+k\bigr). \end{aligned}$$

(22)

Proof

From the definition of $P_{h}$, (3) and (16), for any $w_{h}\in W^{h}$, $n=1,2,\ldots ,N$, we have

$$ \begin{aligned} &\bigl(d_{t}y_{h}^{n}(u)-d_{t}P_{h}y^{n},w_{h} \bigr)+a \bigl(y_{h}^{n}(u)-P_{h}y^{n},w_{h} \bigr) \\ &\quad =-\bigl(d_{t}P_{h}y^{n},w_{h} \bigr)-a\bigl(y^{n},w_{h}\bigr)+\bigl(f^{n}+u^{n},w_{h} \bigr) \\ &\quad =-\bigl(d_{t}P_{h}y^{n}-d_{t}y^{n},w_{h} \bigr)-\bigl(d_{t}y^{n}-y_{t}^{n},w_{h} \bigr). \end{aligned} $$

(23)

By selecting $w_{h}=y_{h}^{n}(u)-P_{h}y^{n}$, $n=1,2,\ldots ,N$, we obtain

$$ \begin{aligned} &\bigl(d_{t}y_{h}^{n}(u)-d_{t}y^{n},y_{h}^{n}(u)-P_{h}y^{n} \bigr)+a \bigl(y_{h}^{n}(u)-P_{h}y^{n},y_{h}^{n}(u)-P_{h}y^{n} \bigr) \\ &\quad =-\bigl(d_{t}P_{h}y^{n}-d_{t}y^{n},y_{h}^{n}(u)-P_{h}y^{n} \bigr)-\bigl(d_{t}y^{n}-y_{t}^{n},y_{h}^{n}(u)-P_{h}y^{n} \bigr). \end{aligned} $$

(24)

Note that $a (y_{h}^{n}(u)-P_{h}y^{n},y_{h}^{n}(u)-P_{h}y^{n} )\geq 0$, and

$$ \begin{aligned} &\bigl(d_{t}y_{h}^{n}(u)-d_{t}P_{h}y^{n},y_{h}^{n}(u)-P_{h}y^{n} \bigr) \\ &\quad \geq \frac{1}{k} \bigl( \bigl\Vert y_{h}^{n}(u)-P_{h}y^{n} \bigr\Vert ^{2}- \bigl\Vert y_{h}^{n}(u)-P_{h}y^{n} \bigr\Vert \bigl\Vert y_{h}^{n-1}(u)-P_{h}y^{n-1} \bigr\Vert \bigr). \end{aligned} $$

(25)

It follows from (24)–(25) and Hölder’s inequality that

$$ \begin{aligned} & \bigl\Vert y_{h}^{n}(u)-P_{h}y^{n} \bigr\Vert \\ &\quad \leq \bigl\Vert y_{h}^{n-1}(u)-P_{h}y^{n-1} \bigr\Vert + \bigl\Vert (P_{h}-I) \bigl(y^{n}-y^{n-1} \bigr) \bigr\Vert + \bigl\Vert y^{n}-y^{n-1}-k y_{t}^{n} \bigr\Vert .\end{aligned} $$

(26)

Summing over n from 1 to $N^{*}$ ($1\leq N^{*}\leq N$), we get

$$ \begin{aligned} \bigl\Vert y_{h}^{N^{*}}(u)-P_{h}y^{N^{*}} \bigr\Vert &\leq \sum_{n=1}^{N^{*}} \bigl\Vert (P_{h}-I) \bigl(y^{n}-y^{n-1}\bigr) \bigr\Vert +\sum_{n=1}^{N^{*}} \bigl\Vert y^{n}-y^{n-1}-k y_{t}^{n} \bigr\Vert \\ &\leq \sum_{n=1}^{N^{*}}Ch^{2} \bigl\Vert y^{n}-y^{n-1} \bigr\Vert _{2}+ \sum_{n=1}^{N^{*}} \int _{t_{n-1}}^{t_{n}} \bigl\Vert (t_{n-1}-t)y_{tt} \bigr\Vert \,dt \\ &\leq Ch^{2}\sum_{n=1}^{N^{*}} \int _{t_{n-1}}^{t_{n}} \Vert y_{t} \Vert _{2}\,dt+k\sum_{n=1}^{N^{*}} \int _{t_{n-1}}^{t_{n}} \Vert y_{tt} \Vert \,dt \\ &\leq Ch^{2} \int _{0}^{t_{N^{*}}} \Vert y_{t} \Vert _{2}\,dt+k \int _{0}^{t_{N^{*}}} \Vert y_{tt} \Vert \,dt \\ &\leq C \bigl(h^{2} \Vert y_{t} \Vert _{L^{2}(H^{2})}+k \Vert y_{tt} \Vert _{L^{2}(L^{2})} \bigr).\end{aligned} $$

(27)

Thus, we have

$$ \bigl|\!\bigl|\!\bigl|y_{h}(u)-P_{h}y \bigr|\!\bigr|\!\bigr|_{l^{\infty }(L^{2})}\leq C \bigl(h^{2}+k \bigr).$$

(28)

From (21), we derive

$$\begin{aligned} |\!|\!|P_{h}y-y|\!|\!|^{2}_{l^{2}(L^{2})}=\sum _{n=1}^{N}k \bigl\Vert P_{h}y^{n}-y^{n} \bigr\Vert ^{2}\leq Ch^{4}\sum _{n=1}^{N}k \bigl\Vert y^{n} \bigr\Vert _{2}^{2}=Ch^{4}|\!|\!|y|\!|\!|_{l^{2}(H^{2})}^{2}. \end{aligned}$$

(29)

According to embedding theorem and (28)–(29), we have

$$ \bigl|\!\bigl|\!\bigl|y_{h}(u)-y\bigr|\!\bigr|\!\bigr|_{l^{2}(L^{2})}\leq C \bigl(h^{2}+k \bigr). $$

(30)

Similarly, we can prove that

$$ \bigl|\!\bigl|\!\bigl|p_{h}(u)-p\bigr|\!\bigr|\!\bigr|_{l^{2}(L^{2})}\leq C \bigl(h^{2}+k \bigr).$$

(31)

Then (22) follows from (30) and (31). □

For the control variable, we derive the following convergence result.

Theorem 3.1

Let $(y,p,u)$ and $(y_{h},p_{h},u_{h} )$ be the solutions of (3)–(7) and (10)–(14), respectively. Assume that all the conditions in Lemma 3.1are satisfied. Then, we have

$$ |\!|\!|u-u_{h}|\!|\!|_{l^{2}(L^{2})}\leq C \bigl(h^{2}+k \bigr).$$

(32)

Proof

From (7) and (14), we obtain

$$ \begin{aligned} &\nu |\!|\!|u-u_{h}|\!|\!|^{2}_{l^{2}(L^{2})} \\ &\quad =\sum_{n=1}^{N}k \bigl(\nu \bigl(u^{n}-u_{h}^{n}\bigr),u^{n}-u_{h}^{n} \bigr) \\ &\quad \leq \sum_{n=1}^{N}k \bigl(\nu u_{h}^{n}-p_{h}^{n-1}(u),u_{h}^{n}-u^{n} \bigr) +\sum_{n=1}^{N}k \bigl(p^{n}-p_{h}^{n-1}(u),u^{n}-u_{h}^{n} \bigr) \\ &\quad \leq \sum_{n=1}^{N}k \bigl(p_{h}^{n-1}-p_{h}^{n-1}(u),u_{h}^{n}-u^{n} \bigr) +\sum_{n=1}^{N}k \bigl(p^{n}-p_{h}^{n-1}(u),u^{n}-u_{h}^{n} \bigr).\end{aligned} $$

(33)

It follows from (10)–(12) and (16)–(18) that

$$ \sum_{n=1}^{N}k \bigl(p_{h}^{n-1}-p_{h}^{n-1}(u),u_{h}^{n}-u^{n} \bigr)=-\bigl|\!\bigl|\!\bigl|y_{h}-y_{h}(u)\bigr|\!\bigr|\!\bigr|_{l^{2}(L^{2})}^{2}\leq 0.$$

(34)

By using Hölder and Young inequalities, we get

$$ \begin{aligned} &\sum _{n=1}^{N}k \bigl(p^{n}-p_{h}^{n-1}(u),u^{n}-u_{h}^{n} \bigr) \\ &\quad =\sum_{n=1}^{N}k \bigl(p^{n-1}-p_{h}^{n-1}(u),u^{n}-u_{h}^{n} \bigr)+\sum_{n=1}^{N}k \bigl(p^{n}-p^{n-1},u^{n}-u_{h}^{n} \bigr) \\ &\quad \leq C(\delta )\sum_{n=1}^{N}k \bigl\Vert p^{n-1}-p_{h}^{n-1}(u) \bigr\Vert ^{2}+C(\delta )\sum_{n=1}^{N}k \bigl\Vert p^{n}-p^{n-1} \bigr\Vert ^{2}+ \delta \sum_{n=1}^{N}k \bigl\Vert u^{n}-u_{h}^{n} \bigr\Vert ^{2} \\ &\quad \leq C(\delta ) \bigl(\bigl|\!\bigl|\!\bigl|p-p_{h}(u)\bigr|\!\bigr|\!\bigr|_{l^{2}(L^{2})}^{2}+(k)^{2} \Vert p_{t} \Vert ^{2}_{L^{2}(J;L^{2}(\Omega ))} \bigr)+ \delta |\!|\!|u-u_{h}|\!|\!|_{l^{2}(L^{2})}^{2}.\end{aligned} $$

(35)

From (22) and (33)–(35), we obtain (32). □

For the control, state and costate variables, we have the following results.

Theorem 3.2

Let $(y,p,u )$ and $(y_{h},p_{h},u_{h} )$ be the solutions (3)–(7) and (10)–(14), respectively. Assume that all the conditions in Theorem 3.1are valid. Then

$$ |\!|\!|P_{h}y-y_{h} |\!|\!|_{l^{2}(H^{1})}+|\!|\!|P_{h}p-p_{h}|\!|\!|_{l^{2}(H^{1})} \leq C \bigl(h^{2}+k \bigr). $$

(36)

Proof

From (3) and (10), for any $w_{h}\in W^{h}$, $n=1,2,\ldots ,N$, we obtain the following error equation:

$$ \bigl(y_{t}^{n}-d_{t}y_{h}^{n},w_{h} \bigr)+a \bigl(y^{n}-y_{h}^{n},w_{h} \bigr)= \bigl(u^{n}-u_{h}^{n},w_{h} \bigr). $$

(37)

By choosing $w_{h}=P_{h}y^{n}-y_{h}^{n}$ and using the definition of $P_{h}$, we get

$$\begin{aligned} \begin{aligned} &\bigl(d_{t}P_{h}y^{n}-d_{t}y_{h}^{n},P_{h}y^{n}-y_{h}^{n} \bigr)+a \bigl(P_{h}y^{n}-y_{h}^{n},P_{h}y^{n}-y_{h}^{n} \bigr) \\ &\quad = \bigl(d_{t}P_{h}y^{n}-d_{t}y^{n}+d_{t}y^{n}-y_{t}^{n}+u^{n}-u_{h}^{n},P_{h}y^{n}-y_{h}^{n} \bigr). \end{aligned} \end{aligned}$$

(38)

Note that

$$ \bigl(d_{t}P_{h}y^{n}-d_{t}y_{h}^{n},P_{h}y^{n}-y_{h}^{n} \bigr) \geq \frac{1}{2k} \bigl( \bigl\Vert P_{h}y^{n}-y_{h}^{n} \bigr\Vert ^{2}- \bigl\Vert P_{h}y^{n-1}-y_{h}^{n-1} \bigr\Vert ^{2} \bigr)$$

(39)

and

$$ \begin{aligned} \bigl(d_{t}P_{h}y^{n}-d_{t}y^{n},P_{h}y^{n}-y^{n}_{h} \bigr) &\leq \bigl\Vert d_{t}P_{h}y^{n}-d_{t}y^{n} \bigr\Vert \bigl\Vert P_{h}y^{n}-y^{n}_{h} \bigr\Vert \\ &\leq Ch^{2} \bigl\Vert d_{t}y^{n} \bigr\Vert _{2} \bigl\Vert P_{h}y^{n}-y^{n}_{h} \bigr\Vert \\ &\leq Ch^{2}k^{-1} \int _{t_{n-1}}^{t_{n}} \Vert y_{t} \Vert _{2}\,dt \bigl\Vert P_{h}y^{n}-y^{n}_{h} \bigr\Vert \\ &\leq Ch^{2}k^{-\frac{1}{2}}\|y_{t} \|_{L^{2}(t_{n-1},t_{n};H^{2}( \Omega ))} \bigl\Vert P_{h}y^{n}-y^{n}_{h} \bigr\Vert .\end{aligned} $$

(40)

Additionally,

$$ \begin{aligned} \bigl(d_{t}y^{n}-y_{t}^{n},P_{h}y^{n}-y^{n}_{h} \bigr) &=k^{-1} \bigl(y^{n}-y^{n-1}-k y_{t}^{n},P_{h}y^{n}-y^{n}_{h} \bigr) \\ &\leq k^{-1} \bigl\Vert y^{n}-y^{n-1}k y_{t}^{n} \bigr\Vert \bigl\Vert P_{h}y^{n}-y^{n}_{h} \bigr\Vert \\ &=k^{-1} \biggl\Vert \int _{t_{n-1}}^{t_{n}}(t_{n-1}-t)y_{tt}\,dt \biggr\Vert \bigl\Vert P_{h}y^{n}-y^{n}_{h} \bigr\Vert \\ &\leq Ck^{\frac{1}{2}}\|y_{tt}\|_{L^{2}(t_{n-1},t_{n};L^{2}(\Omega ))} \bigl\Vert P_{h}y^{n}-y^{n}_{h} \bigr\Vert .\end{aligned} $$

(41)

Multiplying both sides of (38) by 2k, summing over n from 1 to N, and by using Hölder and Young inequalities, we have

$$ \begin{aligned} & \bigl\Vert P_{h}y^{N}-y^{N}_{h} \bigr\Vert ^{2}+c\sum_{n=1}^{N}k \bigl\Vert P_{h}y^{n}-y^{n}_{h} \bigr\Vert ^{2}_{1} \\ &\quad \leq C(\delta ) \bigl(h^{4} \Vert y_{t} \Vert _{L^{2}(H^{2})}^{2}+k^{2} \Vert y_{tt} \Vert _{L^{2}(L^{2})}^{2}+ |\!|\!|u-u_{h}|\!|\!|_{l^{2}(L^{2})}^{2} \bigr) +\delta \sum_{n=1}^{N}k \bigl\Vert P_{h}y^{n}-y^{n}_{h} \bigr\Vert ^{2}. \end{aligned}$$

(42)

From (32) and (42), we get

$$ |\!|\!|P_{h}y-y_{h}|\!|\!|_{l^{2}(H^{1})} \leq C \bigl(h^{2}+k \bigr).$$

(43)

Similarly, we can prove that

$$ |\!|\!|P_{h}p-p_{h}|\!|\!|_{l^{2}(H^{1})} \leq C \bigl(h^{2}+k \bigr).$$

(44)

Then (36) follows from (43)–(44). □

Theorem 3.3

Assume that $u\in l^{2}(H^{1})$ and all the conditions in Theorem 3.2are valid. Then

$$ |\!|\!|P_{h}u-u_{h} |\!|\!|_{l^{2}(H^{1})}\leq C \bigl(h^{2}+k \bigr).$$

(45)

Proof

Notice that $\Pi _{[a,b]}$ is Lipschitz continuous with constant 1. From (8) and (15), we have

$$ \begin{aligned} |\!|\!|P_{h}u-u_{h}|\!|\!|_{l^{2}(H^{1})}&\leq C\bigl|\!\bigl|\!\bigl|\Pi _{[a,b]}(P_{h}p-p_{h})\bigr|\!\bigr|\!\bigr|_{l^{2}(H^{1})} \\ &\leq C|\!|\!|P_{h}p-p_{h}|\!|\!|_{l^{2}(H^{1})}.\end{aligned} $$

(46)

From (36) and (46), we obtain (45). □

4 A posteriori error estimates

In this section, a posteriori error estimates of recovery type will be established by utilizing the superconvergence results derived in Sect. 3. Similar to the Z–Z patch recovery in [34, 35], we introduce recovery operators $R_{h}$ and $G_{h}$ for the state and the adjoint state. Let $R_{h}v$ be a continuous piecewise linear function (without zero boundary constraint) and let the values of $R_{h}v$ on the nodes be defined by a least-squares argument on element patches surrounding the nodes. The gradient recovery operator $G_{h}v= (R_{h}v_{x_{1}},R_{h}v_{x_{2}} )$. The details also can be found in [18].

Theorem 4.1

Let $(y,p,u)$ and $(y_{h},p_{h},u_{h})$ be the solutions of (3)–(7) and (10)–(14), respectively. Suppose that all the conditions in Theorem 3.2are valid and $y,p\in L^{2}_{D}(J;H^{3}(\Omega ))$. Then

$$ |\!|\!|G_{h}y_{h}- \nabla y|\!|\!|_{l^{2}(L^{2})}+|\!|\!|G_{h}p_{h}- \nabla p|\!|\!|_{l^{2}(L^{2})} \leq C \bigl(h^{2}+k \bigr).$$

(47)

Proof

Let $y_{I}$ be the piecewise linear Lagrange interpolation of y. From Theorem 2.1.1 in [19], we have

$$ \Vert G_{h}y-y_{I} \Vert _{1}\leq Ch^{2} \Vert y \Vert _{3}.$$

(48)

According to the standard interpolation error estimate technique (see, e.g., [8]), we have

$$ \Vert G_{h}y_{I}- \nabla y \Vert \leq Ch^{2} \vert y \vert _{3}.$$

(49)

By using (47)–(48), we get

$$ \begin{aligned} \bigl\Vert G_{h}y_{h}^{n}-\nabla y^{n} \bigr\Vert &= \bigl\Vert G_{h}y_{h}^{n}-G_{h}P_{h}y^{n} \bigr\Vert + \bigl\Vert G_{h}P_{h}y^{n}-G_{h}y_{I}^{n} \bigr\Vert + \bigl\Vert G_{h}y_{I}^{n}- \nabla y^{n} \bigr\Vert \\ &\leq C \bigl\Vert y_{h}^{n}-P_{h}y^{n} \bigr\Vert _{1}+C \bigl\Vert P_{h}y^{n}-y_{I}^{n} \bigr\Vert _{1}+ \bigl\Vert G_{h}y_{I}^{n}- \nabla y^{n} \bigr\Vert \\ &\leq C \bigl\Vert y_{h}^{n}-P_{h}y^{n} \bigr\Vert _{1}+Ch^{2} \bigl\Vert y^{n} \bigr\Vert _{3}.\end{aligned} $$

(50)

Therefore,

$$ \sum_{n=1}^{N}k \bigl\Vert G_{h}y_{h}^{n}-\nabla y^{n} \bigr\Vert ^{2} \leq C\sum _{n=1}^{N}k \bigl\Vert y_{h}^{n}-P_{h} y^{n} \bigr\Vert _{1}^{2}+Ch^{4} \sum_{n=1}^{N}k \bigl\Vert y^{n} \bigr\Vert _{3}^{2}.$$

(51)

From Theorem 3.2 and (51), we derive

$$ |\!|\!|G_{h}y_{h}- \nabla y|\!|\!|_{l^{2}(L^{2})}\leq C \bigl(h^{2}+k \bigr).$$

(52)

Similarly, we can prove that

$$ |\!|\!|G_{h}p_{h}- \nabla p|\!|\!|_{l^{2}(L^{2})}\leq C \bigl(h^{2}+k \bigr).$$

(53)

Then (47) follows from (52)–(53). □

By using the above superconvergence properties, it is easy to prove the following a posteriori error estimate results.

Theorem 4.2

Assume that all the conditions in Theorem 4.1are valid. Then

$$\begin{aligned}& \eta _{1}:=|\!|\!|G_{h}y_{h}-\nabla y_{h}|\!|\!|_{l^{2}(L^{2})}=\bigl|\!\bigl|\!\bigl|\nabla (y- y_{h} )\bigr|\!\bigr|\!\bigr|_{l^{2}(L^{2})}+ \mathcal{O} \bigl(h^{2}+k \bigr), \end{aligned}$$

(54)

$$\begin{aligned}& \eta _{2}:=|\!|\!|G_{h}p_{h}- \nabla p_{h}|\!|\!|_{l^{2}(L^{2})}=\bigl|\!\bigl|\!\bigl|\nabla ( p-p_{h} )\bigr|\!\bigr|\!\bigr|_{l^{2}(L^{2})}+\mathcal{O} \bigl(h^{2}+k \bigr). \end{aligned}$$

(55)

5 Numerical experiments

We do some numerical experiments to demonstrate our theoretical results. For an acceptable error Tol, we present a VD approximation algorithm (see Algorithm 5.1) for the constrained POCP (1). For ease of exposition, we have omitted the subscript h.

Algorithm 5.1

VD approximation algorithm

Similar to [18], by selecting the same meshes for the state and the adjoint state and using $\eta _{1}$ and $\eta _{2}$ as mesh refinement indicators for the state and the adjoint state, for an acceptable error $\mathit{Tol}^{\prime }$, we construct adaptive VD approximation algorithm (see Algorithm 5.2).

Algorithm 5.2

Adaptive VD approximation algorithm

The following numerical examples were solved with AFEPack which is freely available. Just for simplicity, we let $\Omega =[0,1]\times [0,1]$, $T=1$, $A(x)=E$ be the $2\times 2$ identity matrix and denote $|\!|\!|\cdot |\!|\!|_{l^{2}(L^{2})}$ and $|\!|\!|\cdot |\!|\!|_{l^{2}(H^{1})}$ by $|\!|\!|\cdot |\!|\!|$ and $|\!|\!|\cdot |\!|\!|_{1}$, respectively. The convergence order is computed by the following formula: $\mathit{Rate}=\frac{\log (e_{i+1})-\log (e_{i})}{\log (h_{i+1})-\log (h_{i})}$, where $e_{i} (e_{i+1})$ denotes the error when the spatial partition size is $h_{i} (h_{i+1})$. We solve the following POCP:

$$ \textstyle\begin{cases} \min_{u\in K}\frac{1}{2}\int _{0}^{T} ( \Vert y(x,t)-y_{d}(x,t) \Vert ^{2}+\nu \Vert u(x,t) \Vert ^{2} )\,dt, \\ y_{t}(x,t)-\operatorname{div}(A(x)\nabla y(x,t))=f(x,t)+u(x,t),\quad x\in \Omega , t\in J, \\ y(x,t)=0, \quad x\in \partial \Omega , t\in J, \\ y(x,0)=y_{0}(x), \quad x\in \Omega . \end{cases} $$

Moreover, we assume that

$$ K= \bigl\{ v(x,t)\in L^{\infty }\bigl(L^{2}\bigr): a\leq v(x,t)\leq b, (x,t) \in \Omega \times J \bigr\} . $$

Example 1

The data are as follows:

$$\begin{aligned}& \nu =1,\qquad a=-0.25,\qquad b=0.25, \\& p(x,t)=\sin(2\pi x_{1})\sin(2\pi x_{2}) (1-t), \\& y(x,t)=\sin(2\pi x_{1})\sin(2\pi x_{2})t, \\& u(x,t)=\max\bigl(-0.25,\min\bigl(0.25,-p(x,t)\bigr)\bigr), \\& f(x,t)=y_{t}(x,t)-\operatorname{div}\bigl(A(x) \nabla y(x,t) \bigr)-u(x,t), \\& y_{d}(x,t)=y(x,t)+p_{t}(x,t)+\operatorname{div} \bigl(A^{*}(x) \nabla p(x,t)\bigr). \end{aligned}$$

We use the Algorithm 5.1 to solve the first example. In Fig. 1, we plot the profile of the numerical solution $u_{h}$ at $t=0.5$ when $h=\frac{1}{80}$ and $k=\frac{1}{640}$. In Table 1, the errors $|\!|\!|u-u_{h}|\!|\!|$, $|\!|\!|P_{h}y-y_{h}|\!|\!|_{1}$, and $|\!|\!|P_{h}p-p_{h}|\!|\!|_{1}$ on a sequence of uniformly refined meshes are shown. It is easy to see $|\!|\!|u-u_{h}|\!|\!|=\mathcal{O} (h^{2}+k )$, $|\!|\!|P_{h}y-y_{h}|\!|\!|_{1}=\mathcal{O} (h^{2}+k )$, and $|\!|\!|P_{h}p-p_{h}|\!|\!|_{1}=\mathcal{O} (h^{2}+k )$, which confirm our theoretical results.

Figure 1

The numerical solution $u_{h}$ at $t=0.5$ when $h=\frac{1}{80}$ and $k=\frac{1}{640}$

Table 1 Numerical results, Example 1

Example 2

The data are as follows:

$$\begin{aligned}& \nu =1,\qquad a=0,\qquad b=10, \\& p(x,t)= \frac{(t-1)\sin (\pi x_{1})\sin (\pi x_{2})}{(x_{1}-0.5)^{2}+(x_{2}-0.5)^{2}+0.05}, \\& y(x,t)= \frac{t \sin (\pi x_{1})\sin (\pi x_{2})}{(x_{1}-0.5)^{2}+(x_{2}-0.5)^{2}+0.05}, \\& u(x,t)=\min\bigl(10,\max\bigl(0,-p(x,t)\bigr)\bigr), \\& f(x,t)=y_{t}(x,t)-\operatorname{div}\bigl(A(x) \nabla y(x,t) \bigr)-u(x,t), \\& y_{d}(x,t)=y(x,t)+p_{t}(x,t)+\operatorname{div} \bigl(A^{*}(x) \nabla p(x,t)\bigr). \end{aligned}$$

We take a small time size $k=\frac{1}{100}$ and solve the second example by using the Algorithms 5.1 and 5.2, respectively. Numerical results based on a sequence of uniformly refined meshes and adaptive meshes are listed in Tables 2 and 3, respectively. It is clear that the adaptive meshes generated via the error estimators $\eta _{1}$ and $\eta _{2}$ are able to save substantial computational work, in comparison with the uniform meshes.

Table 2 Numerical results on uniform meshes, Example 2

Table 3 Numerical results on adaptive meshes, Example 2

6 Conclusions

Although there has been extensive research on FEMs for various POCPs, they mostly focused on convergence and superconvergence (see, e.g., [12, 23–26, 28, 29]), and the results on convergence and superconvergence were $\mathcal{O}(h+k)$ and $\mathcal{O}(h^{\frac{3}{2}}+k)$, respectively. Recently, VD were used to deal with different OCPs in [9, 14, 15, 28], while there was little work on POCPs. Hence, our results on adaptive VD for POCPs are new.