New a posteriori error estimates for hp version of finite element methods of nonlinear parabolic optimal control problems

Abstract

In this paper, we investigate residual-based a posteriori error estimates for the hp version of the finite element approximation of nonlinear parabolic optimal control problems. By using the hp finite element approximation for both the state and the co-state variables and the hp discontinuous Galerkin finite element approximation for the control variable, we derive hp residual-based a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, can be used to construct a reliable hp adaptive finite element approximation for the nonlinear parabolic optimal control problems.

1 Introduction

Recently, optimal control problems have attracted substantial interest due to their applications in atmospheric pollution problems, exploration and extraction of oil and gas resources, and engineering. They must be solved with efficient numerical methods. The hp version of the finite element method is an important numerical method for the optimal control problems governed by partial differential equations. There have been extensive studies of the convergence of the finite element approximation for optimal control problems. Let us mention two early papers devoted to linear optimal control problems by Falk [1] and Geveci [2]. A systematic introduction of the finite element method for optimal control problems can be found in [3–5], but there are much less published results on this topic for hp-finite element methods for optimal control problems. The adaptive finite element method has been investigated extensively and became one of the most popular methods in the scientific computation and numerical modeling. In [6], the authors studied a posteriori error estimates for adaptive finite element discretizations of boundary control problems. A posteriori error estimates and adaptive finite element approximation for parameter estimation problems have been obtained in [7, 8]. The adaptive finite element approximation is among the most important means to boost the accuracy and efficiency of the finite element discretization. There are three main versions in adaptive finite element approximation, i.e., the p-version, h-version, and hp-version. The p-version of finite element methods uses a fixed mesh and improves the approximation of the solution by increasing the degrees of piecewise polynomials. The h-version is based on mesh refinement and piecewise polynomials of low and fixed degrees. In the hp-version adaptation, one has the option to split an element (h-refinement) or to increase its approximation order (p-refinement). Generally speaking, a local p-refinement is the more efficient method on regions where the solution is smooth, while a local h-refinement is the strategy suitable on elements where the solution is not smooth in [9]. There have been many theoretical studies as regards the hp finite element method in [9–16].

To the best of our knowledge, there are many h-versions of adaptive finite element methods for optimal control problems in [17, 18]. In fact, comparable literature for high order elements such as the hp-version of the finite element method for optimal control problems is rather limited. For the constrained optimal control problem governed by elliptic equations, the authors have derived a posteriori error estimates for the hp finite element approximation in [19]. The purpose of this work is to derive hp a posteriori error estimates for optimal control problems governed by nonlinear parabolic equations.

In this paper, we adopt the standard notation $W^{m,p}(\Omega)$ for Sobolev spaces on Ω with a norm $\Vert \cdot \Vert _{m,p}$ given by $\Vert v \Vert _{m,p}^{p}=\sum_{\vert \alpha \vert \leq m}\Vert D^{\alpha}v \Vert _{L^{p}(\Omega)}^{p}$, a semi-norm $\vert \cdot \vert _{m,p}$ given by $\vert v\vert _{m,p}^{p}=\sum_{\vert \alpha \vert = m}\Vert D^{\alpha}v \Vert _{L^{p}(\Omega)}^{p}$. 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 $\Vert \cdot \Vert _{m}=\Vert \cdot \Vert _{m,2}$, $\Vert \cdot \Vert =\Vert \cdot \Vert _{0,2}$. 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 $\Vert v\Vert _{L^{s}(J;W^{m,p}(\Omega))}= (\int_{0}^{T}\Vert v\Vert _{W^{m,p}(\Omega )}^{s}\,dt )^{\frac{1}{s}}$ for $s\in[1,\infty)$, and the standard modification for $s=\infty$. Similarly, one can define the spaces $H^{1}(J;W^{m,p}(\Omega))$ and $C^{k}(J;W^{m,p}(\Omega))$. The details can be found in [20].

The paper is organized as follows. In Section 2, we shall construct the hp finite element approximation for nonlinear parabolic optimal control problems. In Section 3, we derive hp a posteriori error estimates for the optimal control problems. In the last section, we briefly give conclusions and some possible future work.

2 The hp finite element of nonlinear parabolic optimal control

In this section, we study the hp finite element method and the backward Euler discretization approximation of convex optimal control problems governed by nonlinear parabolic equations. We shall take the state space $W=L^{2}(0,T;Y)$ with $Y=H^{1}_{0}(\Omega)$, the control space $X=L^{2}(0,T;U)$ with $U=L^{2}(\Omega_{U})$ and $H=L^{2}(\Omega)$ to fix the idea. Let B be a linear continuous operator from X to $L^{2}(0,T;Y)$. We are interested in the following nonlinear parabolic optimal control problems:

$$\begin{aligned}& \min_{u\in K} \biggl\{ \frac{1}{2} \int_{0}^{T}\bigl(\Vert y-y_{d}\Vert ^{2}_{L^{2}(\Omega)}+\Vert u\Vert ^{2}_{L^{2}(\Omega_{U})} \bigr)\,dt \biggr\} , \end{aligned}$$

(2.1)

$$\begin{aligned}& y_{t}-\operatorname {div}(A\nabla y)+ \phi(y)=f+Bu,\quad x\in\Omega, t \in(0,T], \end{aligned}$$

(2.2)

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

(2.3)

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

(2.4)

where Ω and $\Omega_{U}$ are bounded open sets in $\mathbb{R}^{2}$ with a Lipschitz boundary ∂Ω and $\partial\Omega_{U}$, K is a set defined by $K= \{v\in X:\int_{0}^{T}\int_{\Omega_{U}}v \,dx \,dt\geq0 \}$, and $f, y_{d}\in L^{2}(0,T;H)$, $y_{0}(x)\in V=H_{0}^{1}(\Omega)$, and $A(\cdot)=(a_{i,j}(\cdot))_{2\times 2}\in(C^{\infty}(\overline{\Omega}))^{2\times2}$, such that there is a constant $c>0$ satisfying $\xi^{t}A\xi\geq c\Vert \xi \Vert ^{2}$, $\xi\in\mathbb{R}^{2}$. The function $\phi(\cdot)\in W^{2,\infty}(-R,R)$ for any $R>0$, $\phi^{\prime}(y)\in L^{2}(\Omega)$ for any $y\in H^{1}(\Omega)$, and $\phi^{\prime}(y)\geq0$.

Let $a(v,w)=\int_{\Omega }(A\nabla v)\cdot\nabla w \,dx$, $\forall v,w\in V$, $(f_{1},f_{2})=\int_{\Omega}f_{1}f_{2}\,dx$, $\forall f_{1},f_{2}\in H$, $(v,w)_{U}= \int_{\Omega_{U}}vw \,dx$, $\forall v,w\in U$. It follows from the assumptions on A that there are constants c and $C>0$ such that

$$\begin{aligned} a(v,v)\geq c\Vert v\Vert _{H^{1}(\Omega)}^{2}, \qquad \bigl\vert a(v,w) \bigr\vert \leq C\vert v\vert _{H^{1}(\Omega)}^{2}\vert w\vert _{H^{1}(\Omega)}^{2}, \quad \forall v,w\in Y. \end{aligned}$$

Then a weak formula of the convex nonlinear parabolic optimal control problems (2.1)-(2.4) reads

$$\begin{aligned} &\min_{u(t)\in K} \biggl\{ \frac{1}{2} \int_{0}^{T} \bigl(\Vert y-y_{d} \Vert ^{2}_{L^{2}(\Omega )}+\Vert u\Vert ^{2}_{L^{2}(\Omega_{U})} \bigr)\,dt \biggr\} , \end{aligned}$$

(2.5)

where $y\in W$, $u\in X$, $u(t)\in K$, subject to

$$\begin{aligned}& (y_{t},w)+a(y,w)+\bigl(\phi(y),w\bigr)=(f+Bu,w),\quad \forall w\in Y, t \in(0,T], \end{aligned}$$

(2.6)

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

(2.7)

It is well known (see, e.g., [21]) that the optimal control problem (2.5)-(2.7) has at least a solution $(y,u)$, and if that a pair $(y,u)$ is the solution of (2.5)-(2.7), then there is a co-state $p\in W$ such that the triplet $(y,p,u)$ satisfies the following optimality conditions:

$$\begin{aligned}& (y_{t},w)+a(y,w)+\bigl(\phi(y),w\bigr)=(f+Bu,w),\quad \forall w\in Y, y(0)=y_{0}(x), \end{aligned}$$

(2.8)

$$\begin{aligned}& -(p_{t}, w)+a(q,p)+\bigl(\phi^{\prime}(y)p,q \bigr)=(y-y_{d},q),\quad \forall q\in Y, p(T)=0, \end{aligned}$$

(2.9)

$$\begin{aligned}& \int_{0}^{T}\bigl(u+B^{*}p,v-u \bigr)_{U}\,dt\geq0,\quad \forall v\in K, \end{aligned}$$

(2.10)

where $B^{*}$ is the adjoint operators of B, and $(\cdot,\cdot)_{U}$ is the inner product of U, which will be simply written as $(\cdot,\cdot)$ in the rest of the paper when no confusion is caused.

Due to the special structure of the control constraint set K, we can derive a relationship between the control variable and the co-state variable of (2.8)-(2.10) in the following lemma.

Lemma 2.1

Let $(y,p,u)$ be the solution of (2.8)-(2.10). Then we have

$$\begin{aligned} u=\max \bigl\{ 0,\overline{B^{*}p} \bigr\} -B^{*}p, \end{aligned}$$

where $\overline{B^{*}p}=\frac{\int_{0}^{T}\int_{\Omega_{U}}B^{*}p \,dx \,dt}{\int_{0}^{T}\int_{\Omega_{U}}1 \,dx \,dt}$ denotes the integral average on $\Omega_{U}\times[0,T]$ of the function $B^{*}p$.

Now, we consider the hp finite element approximation for the nonlinear parabolic optimal control problems. We assume that Ω and $\Omega_{U}$ are polygonal. We consider the triangulation $\mathcal{T}$ of the set Ω⊂ $\mathbb{R}^{2}$, which is a collection of elements $\tau\in \mathcal{T}$ associated with each element τ, and an affine element map $F_{\tau}:\widehat{\tau}\rightarrow\tau$, where the reference element τ̂ is the reference triangle

$$\bigl\{ (x,y)\in\mathbb{R}^{2} : 0< x< 1,0< y< \min(x,1-x) \bigr\} . $$

We consider the triangulation $\mathcal{T}$ which satisfies the standard conditions defined in [16] and write $h_{\tau}=\operatorname {diam}\tau$. Additionally we assume that triangulation $\mathcal{T}$ is γ-shape regular, i.e.,

$$\begin{aligned} h_{\tau}^{-1}\bigl\Vert F_{\tau}^{\prime} \bigr\Vert _{L^{\infty }(\widehat{\tau})}+h_{\tau}\bigl\Vert \bigl(F_{\tau}^{\prime} \bigr)^{-1}\bigr\Vert _{L^{\infty }(\widehat{\tau})}\leq \gamma. \end{aligned}$$

(2.11)

This implies that there exists a constant $C>0$ that depends solely on γ such that

$$\begin{aligned} C^{-1}h_{\tau }\leq h_{\tau^{\prime}}\leq Ch_{\tau}, \quad \tau,\tau^{\prime}\in\mathcal {T} \mbox{ with } \overline{\tau}\cap\overline{\tau}^{\prime}\neq\emptyset, \end{aligned}$$

(2.12)

and there exists a constant $M\in\mathbb{N}$ that depends solely on γ such that no more than M elements share a common vertex. We assume that the triangulation $\mathcal{T}_{U}$ of $\Omega_{U}$ which is a collection of elements $\tau_{U}\in\mathcal{T}_{U}$, is γ-shape regular which satisfies the standard conditions as $\mathcal{T}$. Associated with each element $\tau_{U}$ is an affine element map $F_{\tau_{U}}:\widehat{\tau}\rightarrow\tau_{U}$. We further assume the triangulation $\mathcal{T}$ satisfies the relation between the patch and the reference patch in [16].

For each element $\tau\in\mathcal{T}$, we denote $\mathcal {E}(\tau)$ the set of edges of τ and $\mathcal{N}(\tau)$ the set of vertices of τ, and choose a polynomial degree $p_{\tau}\in\mathbb{N}$ and collect these numbers in the polynomial degree vector $\mathbf{p}_{1}=(p_{\tau})_{\tau\in\mathcal{T}}$. Similarly, for each element $\tau_{U}\in\mathcal{T}_{U}$, we choose a polynomial degree vector $\mathbf{p}_{2}=(p_{\tau_{U}})_{\tau_{U}\in \mathcal {T}_{U}}$ ($p_{\tau_{U}}\in\mathbb{N}$). $\mathcal{N}(\mathcal {T})$ denotes the set of all vertices of $\mathcal{T}$, $\mathcal {E}(\mathcal{T})$ denotes the set of all edges. Additionally, we introduce the following notation ($V\in\mathcal{N}(\mathcal{T})$, $e\in\mathcal{E}(\mathcal{T})$):

$$\begin{aligned}& \mathcal {N}(e)=\bigl\{ V\in\mathcal {N}(\mathcal{T}):V\in\overline{e}\bigr\} , \\& w_{V}=\bigl\{ x\in\Omega:x\in\overline{\tau}\mbox{ and } \overline{\tau}\cap\{V\}\neq\emptyset\bigr\} ^{0}, \\& w^{1}_{e}=\bigcup_{V\in\mathcal{N}(e)}w_{V},\qquad w^{1}_{\tau}=\bigcup_{V\in\mathcal{N}(\tau)}w_{V}, \\& h_{\tau_{U}}=\operatorname {diam}\tau_{U}, \qquad p_{e}=\max\bigl\{ p_{\tau}: e\in\mathcal{E}(\tau)\bigr\} , \end{aligned}$$

where $\chi^{0}$ denotes the interior of the set χ. Finally, we denote by $h_{e}$ the length of the edge e.

Next, we define the hp-finite element space $S^{\mathbf{p}_{1}}(\mathcal{T})\subset H^{1}(\Omega)$ and the hp-discontinuous Galerkin finite element space $U^{ \mathbf{p}_{2}}(\mathcal{T}_{U})\subset L^{2}(\Omega_{U})$ by

$$\begin{aligned}& S^{ \mathbf{p}_{1}}(\mathcal{T})= \bigl\{ v\in C(\Omega):v|_{\tau}\circ F_{\tau}\in\Pi_{p_{\tau}}(\widehat{\tau}) \bigr\} , \\& U^{ \mathbf{p}_{2}}(\mathcal{T}_{U})= \bigl\{ v\in L^{2}( \Omega_{U}):v|_{\tau_{U}}\circ F_{\tau_{U}}\in \Pi_{p_{\tau_{U}}}(\widehat{\tau}) \bigr\} , \end{aligned}$$

where we set

$$\Pi_{k}(\widehat{\tau})= \textstyle\begin{cases} {P}_{k}=\operatorname {span}\{x^{i}y^{j}:0\leq i+j\leq k\},& \mbox{if } \widehat{\tau}=T,\\ {Q}_{k}=\operatorname {span}\{x^{i}y^{j}:0\leq i,j\leq k\}, & \mbox{if } \widehat{\tau}=S. \end{cases} $$

We also assume that the polynomial degree vector $\mathbf{p}_{1}$ satisfies

$$\begin{aligned} \gamma^{-1}p_{\tau}\leq p_{\tau^{\prime}}\leq \gamma p_{\tau}, \quad \tau,\tau^{\prime}\in\mathcal{T} \mbox{ with } \overline{\tau}\cap\overline{\tau}^{\prime}\neq\emptyset. \end{aligned}$$

(2.13)

Then we can introduce the finite dimensional spaces $K_{hp}=K\cap U^{ \mathbf{p}_{2}}(\mathcal{T}_{U})$, $V_{hp}=V\cap S^{\mathbf{p}_{1}}(\mathcal{T})$.

The semidiscrete hp finite element approximation of (2.1)-(2.4) is as follows:

$$\begin{aligned}& \min_{u_{hp}\in K_{hp}} \biggl\{ \frac{1}{2} \int_{0}^{T}\bigl(\Vert y_{hp}-y_{d} \Vert ^{2}_{L^{2}(\Omega )}+\Vert u_{hp}\Vert ^{2}_{L^{2}(\Omega_{U})}\bigr)\,dt \biggr\} , \end{aligned}$$

(2.14)

$$\begin{aligned}& \biggl(\frac{\partial y_{hp}}{\partial t},w_{hp} \biggr)+a(y_{hp},w_{hp})+ \bigl(\phi(y_{hp}),w\bigr)=(f+Bu_{hp},w_{hp}),\quad \forall w_{hp}\in V_{hp}, \end{aligned}$$

(2.15)

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

(2.16)

where $y_{hp}\in H^{1}(0,T;V_{hp})$ and $y_{0}^{hp}\in V_{hp}$ is an finite element approximation of $y_{0}$.

It follows that the optimal control problems (2.14)-(2.16) has at least a solution $(y_{hp},u_{hp})$ and if that a pair $(y_{hp},u_{hp})$ is the solution of (2.14)-(2.16), then there is a co-state $p_{hp}\in V_{hp}$ such that the triplet $(y_{hp},p_{hp},u_{hp})$ satisfies the following optimality conditions:

$$\begin{aligned}& \biggl(\frac{\partial y_{hp}}{\partial t},w_{hp} \biggr)+a(y_{hp},w_{hp})+ \bigl(\phi(y_{hp}),w\bigr)=(f+Bu_{hp},w_{hp}),\quad \forall w_{hp}\in V_{hp}, \end{aligned}$$

(2.17)

$$\begin{aligned}& y_{hp}(x,0)=y_{0}^{hp}(x),\quad x\in\Omega, \\& - \biggl(\frac{\partial p_{hp}}{\partial t},q_{hp} \biggr)+a(q_{hp},p_{hp})+ \bigl(\phi^{\prime }(y_{hp})p_{hp},q_{hp} \bigr)=(y_{hp}-y_{d},q_{hp}),\quad \forall w_{hp}\in V_{hp}, \end{aligned}$$

(2.18)

$$\begin{aligned}& p_{hp}(x,T)=0, \quad x\in\Omega, \\& \bigl(u_{hp}+B^{*}p_{hp},v_{hp}-u_{hp} \bigr)_{U}\geq0,\quad \forall v_{hp}\in K_{hp}. \end{aligned}$$

(2.19)

Furthermore, we consider the fully discrete finite element approximation for the above semidiscrete problems by using the backward Euler scheme. Let $0=t_{0}< t_{1}<\cdots<t_{M-1}<t_{M}=T$, $k_{i}=t_{i}-t_{i-1}$, $i=1,2,\ldots,M$, $k=\max\limits_{1\leq i\leq M}\{k_{i}\}$.

For $i=1,2,\ldots,M$, construct the hp finite element approximation spaces $V_{hp}^{i}\subset H^{1}_{0}(\Omega)$ (similar to $V_{hp}$) on the ith time step. Similarly, we construct the hp finite element approximation spaces $K^{i}_{hp}\subset L^{2}(\Omega_{U})$ (similar to $K_{hp}$) on the ith time step. The fully discrete hp finite element approximation scheme (2.17)-(2.19) is to find $(y_{hp}^{i},u_{hp}^{i})\in V^{i}_{hp}\times K^{i}_{hp}$, $i=1,2,\ldots,M$, such that

$$\begin{aligned}& \min_{u_{hp}^{i}\in K^{i}_{hp}} \Biggl\{ \sum _{i=1}^{M} \biggl(\frac{1}{2}\bigl\Vert y_{hp}^{i}-y_{d}(x,t_{i})\bigr\Vert ^{2}_{L^{2}(\Omega)}+\frac{1}{2}\bigl\Vert u_{hp}^{i} \bigr\Vert ^{2}_{L^{2}(\Omega_{U})} \biggr) \Biggr\} , \end{aligned}$$

(2.20)

$$\begin{aligned}& \biggl(\frac{y_{hp}^{i}-y_{hp}^{i-1}}{k_{i}},w_{hp} \biggr)+a \bigl(y_{hp}^{i},w_{hp}\bigr)+\bigl(\phi \bigl(y_{hp}^{i}\bigr),w_{hp}\bigr) \\& \quad = \bigl(f(x,t_{i})+Bu_{hp}^{i},w_{hp} \bigr), \quad \forall w_{hp}\in V_{hp}^{i}, \end{aligned}$$

(2.21)

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

(2.22)

It follows that the optimal control problem (2.20)-(2.22) has at least a solution $(Y_{hp}^{i},U_{hp}^{i})$, and if a pair $(Y_{hp}^{i},U_{hp}^{i})\in V^{i}_{hp}\times K^{i}_{hp}$ is the solution of (2.20)-(2.22), then there is a co-state $P_{hp}^{i-1}\in V^{i}_{hp}$, such that the triplet $(Y_{hp}^{i},P_{hp}^{i-1},U_{hp}^{i})\in V^{i}_{hp}\times V^{i}_{hp}\times K^{i}_{hp}$, satisfies the following optimality conditions:

$$\begin{aligned}& \biggl(\frac{Y_{hp}^{i}-Y_{hp}^{i-1}}{k_{i}},w_{hp} \biggr)+a\bigl(Y_{hp}^{i},w_{hp} \bigr)+\bigl(\phi \bigl(Y_{hp}^{i}\bigr),w_{hp} \bigr)=\bigl(f(x,t_{i})+BU_{hp}^{i},w_{hp} \bigr), \end{aligned}$$

(2.23)

$$\begin{aligned}& \forall w_{hp}\in V_{hp}^{i}\subset V=H_{0}^{1}(\Omega),\quad i=1,2,\ldots,M,\qquad Y_{hp}^{0}(x)=y_{0}^{hp}(x),\quad x\in\Omega, \\& \biggl(\frac{P_{hp}^{i-1}-P_{hp}^{i}}{k_{i}},q_{hp} \biggr)+a\bigl(q_{hp},P_{hp}^{i-1} \bigr)+\bigl(\phi^{\prime }\bigl(Y_{hp}^{i-1} \bigr){P_{hp}^{i-1}},q_{hp}\bigr)= \bigl(Y_{hp}^{i}-y_{d}(x,t_{i}),q_{hp} \bigr), \end{aligned}$$

(2.24)

$$\begin{aligned}& \forall q_{hp}\in V_{hp}^{i}\subset V=H_{0}^{1}(\Omega), \quad i=M,\ldots,2,1,\qquad P_{hp}^{M}(x)=0,\quad x\in\Omega, \\& \bigl(U_{hp}^{i}+B^{*}P_{hp}^{i-1},v_{hp}-U_{hp}^{i} \bigr)_{U}\geq0, \quad \forall v_{hp}\in K_{hp}^{i}, i=1,2,\ldots,M. \end{aligned}$$

(2.25)

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

$$\begin{aligned}& Y_{hp}|_{(t_{i-1},t_{i}]}= \bigl((t_{i}-t)Y_{hp}^{i-1}+(t-t_{i-1})Y_{hp}^{i} \bigr)/k_{i}, \\& P_{hp}|_{(t_{i-1},t_{i}]}= \bigl((t_{i}-t)P_{hp}^{i-1}+(t-t_{i-1})P_{hp}^{i} \bigr)/k_{i}, \\& U_{hp}|_{(t_{i-1},t_{i}]}=U_{hp}^{i}. \end{aligned}$$

For any function $w\in C(0,T;L^{2}(\Omega))$, let $\hat{w}(x,t)|_{t\in(t_{i-1},t_{i}]}=w(x,t_{i})$, $\tilde{w}(x,t)|_{t\in(t_{i-1},t_{i}]}=w(x,t_{i-1})$. Then the optimality conditions (2.23)-(2.25) can be restated as

$$\begin{aligned}& \biggl(\frac{\partial Y_{hp}}{\partial t},w_{hp} \biggr)+a(\hat {Y}_{hp},w_{hp})+\bigl(\phi(\hat{Y}_{hp}),w_{hp} \bigr)=(\hat {f}+BU_{hp},w_{hp}), \end{aligned}$$

(2.26)

$$\begin{aligned}& \forall w_{hp}\in V_{hp}^{i}\subset V=H_{0}^{1}(\Omega), \quad t\in(t_{i-1},t_{i}], i=1,2,\ldots,M, \\& Y_{hp}(x,0)=y_{0}^{hp}(x),\quad x\in\Omega, \\& - \biggl(\frac{\partial P_{hp}}{\partial t},q_{hp} \biggr)+a(q_{hp}, \tilde{P}_{hp})+\bigl(\phi^{\prime}(\tilde{Y}_{hp}){ \tilde {P}_{hp}},q_{hp}\bigr)=(\hat{Y}_{hp}- \hat{y}_{d},q_{hp}), \end{aligned}$$

(2.27)

$$\begin{aligned}& \forall q_{hp}\in V_{hp}^{i}\subset V=H_{0}^{1}(\Omega),\quad t\in(t_{i-1},t_{i}], i=M,\ldots,2,1, \\& P_{hp}(x,T)=0, \quad x\in\Omega, \\& \bigl(U_{hp}+B^{*}\tilde{P}_{hp},v_{hp}-U_{hp} \bigr)_{U}\geq0, \\& \forall v_{hp}\in K_{hp}^{i},\quad t \in(t_{i-1},t_{i}], i=1,2,\ldots,M. \end{aligned}$$

(2.28)

The following lemmas [10, 16, 19] are important in deriving a posteriori error estimates of residual type.

Lemma 2.2

There exist a constant $C>0$ independent of v, $h_{\tau_{U}}$, and $p_{\tau_{U}}$ and a mapping $\pi_{p_{\tau_{U}}}^{h_{\tau_{U}}}:H^{1}(\tau_{U})\rightarrow \mathscr{P}_{p_{\tau_{U}}}(\tau_{U})$ such that $\forall v\in H^{1}(\tau_{U})$, $\tau_{U}\in\mathcal{T}_{U}$ the following inequality is valid:

$$\bigl\Vert v-\pi_{p_{\tau_{U}}}^{h_{\tau_{U}}}\bigr\Vert _{L^{2}(\tau _{U})}\leq C\frac{h_{\tau_{U}}}{p_{\tau_{U}}}\Vert v\Vert _{H^{1}(\tau_{U})}, $$

where we will write $v\in\mathscr{P}_{p_{\tau_{U}}}(\tau_{U})$ if the following satisfied: $v|_{\tau_{U}}\circ F_{\tau_{U}}\in P_{p_{\tau_{U}}}(\widehat{\tau})$ if $\tau_{U}$ is a triangle; $v|_{\tau_{U}}\circ F_{\tau_{U}}\in Q_{p_{\tau_{U}}}(\widehat{\tau})$ if $\tau_{U}$ is a parallelogram.

Lemma 2.3

Let $\mathbf{p}_{1}$ be an arbitrary polynomial degree distribution satisfies (2.13). Then there exists a linear operator $E_{1}:H^{1}_{0}(\Omega)\rightarrow S^{ \mathbf{p}_{1}}(\mathcal {T})\cap H^{1}_{0}(\Omega)$, and there exists a constant $C>0$ depending solely on γ such that for every $v\in H^{1}_{0}(\Omega)$ and all elements $\tau\in\mathcal{T}$ and all edges $e\in\mathcal{E}(\mathcal{T})$:

$$\begin{aligned}& \Vert v-E_{1}v\Vert _{L^{2}(\tau)}+\frac{h_{\tau}}{p_{\tau }}\bigl\Vert \nabla (v-E_{1}v)\bigr\Vert _{L^{2}(\tau)}\leq C \frac{h_{\tau}}{p_{\tau}} \Vert \nabla v\Vert _{L^{2}(w_{\tau }^{1})}, \\& \Vert v-E_{1}v\Vert _{L^{2}(e)}\leq C \biggl( \frac{h_{e}}{p_{e}} \biggr)^{\frac{1}{2}}\Vert \nabla v\Vert _{L^{2}(w_{e}^{1})}. \end{aligned}$$

Lemma 2.4

Let $\mathbf{p}_{1}$ be an arbitrary polynomial degree distribution satisfying (2.13) and $p_{\tau}\geq2$, $\forall \tau\in\mathcal{T}$. Then there exists a bounded linear operator $E_{2}:H^{1}_{0}(\Omega)\cap H^{2}(\Omega)\rightarrow S^{ \mathbf{p}_{1}}(\mathcal{T})\cap H^{1}_{0}(\Omega)$, and there exists a constant $C>0$ that depends solely on γ such that for every $v\in H^{1}_{0}(\Omega)\cap H^{2}(\Omega)$ and all elements $\tau\in\mathcal{T}$ and all edges $e\in\mathcal{E}(\mathcal {T})$:

$$\begin{aligned}& \Vert v-E_{2}v\Vert _{L^{2}(\tau)}+\frac{h_{\tau}}{p_{\tau }}\bigl\Vert \nabla (v-E_{2}v)\bigr\Vert _{L^{2}(\tau)}\leq C \biggl( \frac{h_{\tau}}{p_{\tau}} \biggr)^{2}\vert v\vert _{H^{2}(w_{\tau }^{1})}, \\& \Vert v-E_{2}v\Vert _{L^{2}(e)}\leq C \biggl( \frac{h_{e}}{p_{e}} \biggr)^{\frac {3}{2}}\vert v\vert _{H^{2}(w_{e}^{1})}. \end{aligned}$$

For $\varphi\in W_{h}$, we shall write

$$\begin{aligned} \phi(\varphi)-\phi(\rho)=-\tilde{\phi}^{\prime}(\varphi) ( \rho -\varphi )=-\phi^{\prime}(\rho) (\rho-\varphi) +\tilde{ \phi}^{\prime\prime}(\varphi) (\rho-\varphi)^{2}, \end{aligned}$$

(2.29)

where

$$\begin{aligned}& \tilde{\phi}^{\prime}(\varphi)= \int_{0}^{1}\phi^{\prime}\bigl(\varphi +s(\rho -\varphi)\bigr)\,ds, \qquad \tilde{\phi}^{\prime\prime}(\varphi)= \int_{0}^{1}(1-s)\phi^{\prime \prime }\bigl(\rho+s( \varphi-\rho)\bigr)\,ds \end{aligned}$$

are bounded functions in Ω̄ [22].

3 A posteriori error estimates

In this section, we shall derive some a posteriori error estimates for the hp finite element approximation of the optimal control problems governed by nonlinear parabolic equations.

Let

$$\begin{aligned}& S(u)=\frac{1}{2} \bigl(\Vert y-y_{d}\Vert ^{2}_{L^{2}(\Omega)}+\Vert u\Vert ^{2}_{L^{2}(\Omega_{U})} \bigr), \end{aligned}$$

(3.1)

$$\begin{aligned}& S_{hp}(U_{hp})=\frac{1}{2} \bigl( \Vert Y_{hp}-y_{d}\Vert ^{2}_{L^{2}(\Omega)}+ \Vert U_{hp}\Vert ^{2}_{L^{2}(\Omega_{U})} \bigr). \end{aligned}$$

(3.2)

It can be shown [19] that

$$\begin{aligned}& \bigl(S^{\prime}(u),v\bigr)=\bigl(u+B^{*}p, v \bigr), \end{aligned}$$

(3.3)

$$\begin{aligned}& \bigl(S^{\prime}(U_{hp}),v\bigr)= \bigl(U_{hp}+B^{*}p(U_{hp}), v\bigr), \end{aligned}$$

(3.4)

$$\begin{aligned}& \bigl(S_{hp}^{\prime}(U_{hp}),v\bigr)= \bigl(U_{hp}+B^{*}\tilde{P}_{hp}, v\bigr). \end{aligned}$$

(3.5)

It is clear that S and $S_{hp}$ are well defined and continuous on K and $K_{hp}$. Also the functional $S_{hp}$ can be naturally extended on K. Then (2.5) and (2.14) can be represented as

$$ \min_{u\in K} \biggl\{ \int_{0}^{T}S(u)\,dt \biggr\} $$

(3.6)

and

$$ \min_{U_{hp}\in K_{hp}} \biggl\{ \int_{0}^{T} S_{hp}(U_{hp})\,dt \biggr\} . $$

(3.7)

In many application, $S(\cdot)$ is uniform convex near the solution u. The convexity of $S(\cdot)$ is closely related to the second order sufficient conditions of the optimal control problems, which are assumed in many studies on numerical methods of the problem. For instance, in many applications, there is a $c>0$, independent of h, such that

$$ \int_{0}^{T}\bigl(S^{\prime}(u)-S^{\prime}(U_{hp}),u-U_{hp} \bigr)_{U}\,dt\geq c\Vert u-U_{hp}\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}. $$

(3.8)

The following theorem is the first step to derive a posteriori error estimates.

Theorem 3.1

Let $(y,u,p)$ and $(Y_{hp},P_{hp},U_{hp})$ be the solutions of (2.8)-(2.10) and (2.23)-(2.25), respectively. Assume that $(S_{hp}^{\prime}(U_{hp}))|_{\tau}\in H^{s}(\tau)$, $\forall\tau\in \mathcal {T}_{h}$ ($s=0,1$), and there is a $v_{hp}\in K_{hp}$ such that

$$ \bigl\vert \bigl(S_{hp}^{\prime}(U_{hp}),v_{hp}-u \bigr)\bigr\vert \leq C \sum_{\tau\in \mathcal {T}_{h}}h_{\tau} \bigl\Vert S_{hp}^{\prime}(U_{hp})\bigr\Vert _{H^{s}(\tau)}\Vert u-U_{hp}\Vert _{L^{2}(\tau)}^{s}. $$

(3.9)

Then we have

$$ \Vert u-U_{hp}\Vert _{L^{2}(J;L^{2}(\Omega))}^{2} \leq C\eta_{1}^{2}+C\bigl\Vert p(U_{hp})- \tilde{P}_{hp}\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, $$

(3.10)

where

$$\eta_{1}^{2}= \int_{0}^{T}\sum_{\tau\in \mathcal {T}_{h}}h_{\tau}^{1+s} \bigl\Vert U_{hp}+B^{*}\tilde {P}_{hp}\bigr\Vert _{H^{1}(\tau)}^{1+s}\,dt, $$

and $p(U_{hp})$ is defined by the following equations:

$$\begin{aligned}& \biggl(\frac{\partial}{\partial t}y(U_{hp}),w \biggr)+a\bigl(y(U_{hp}),w \bigr)+\bigl(\phi \bigl(y(U_{hp})\bigr),w\bigr)=(f+BU_{hp},w),\quad \forall w\in V, \end{aligned}$$

(3.11)

$$\begin{aligned}& y(U_{hp}) (x,0)=y_{0}(x), \quad x\in\Omega, \\& - \biggl(\frac{\partial}{\partial t}p(U_{hp}),q \biggr)+a\bigl(q,p(U_{hp}) \bigr)+\bigl(\phi^{\prime }\bigl(y(U_{hp})\bigr){p(U_{hp})},q \bigr) \\& \quad =\bigl(y(U_{hp})-y_{d},q\bigr), \quad \forall q\in V, \\& p(U_{hp}) (x,T)=0,\quad x\in\Omega. \end{aligned}$$

(3.12)

Proof

It follows from (3.6) and (3.7) that $\int_{0}^{T}(S^{\prime}(u),u-v)\,dt\leq0$, $\forall v\in K$, and $\int_{0}^{T}(S_{hp}^{\prime}(U_{hp}), U_{hp}-v_{hp})\,dt\leq0$, $\forall v_{hp}\in K_{hp}\subset K$. Then it follows from the assumptions (3.8), (3.9), and the Schwartz inequality that

$$\begin{aligned} c\Vert u-U_{hp}\Vert _{L^{2}(J;L^{2}(\Omega))}^{2} \leq{}& \int_{0}^{T} \bigl(S^{\prime}(u)-S^{\prime}(U_{hp}),u-U_{hp} \bigr)\,dt \\ \leq{}& \int_{0}^{T} \bigl\{ \bigl(S_{hp}^{\prime }(U_{hp}),v_{hp}-u \bigr)+\bigl(S_{hp}^{\prime}(U_{hp})-S^{\prime }(U_{hp}),u-U_{hp} \bigr) \bigr\} \,dt \\ \leq{}& C \int_{0}^{T} \biggl\{ \sum _{\tau\in \mathcal {T}_{h}}h_{\tau }^{1+s}\bigl\Vert S_{hp}^{\prime}(U_{hp})\bigr\Vert _{H^{s}(\tau)}^{1+s}+\bigl\Vert S_{hp}^{\prime}(U_{hp})-S^{\prime}(U_{hp}) \bigr\Vert _{L^{2}(\Omega)}^{2} \biggr\} \,dt \\ &{} +\frac{c}{2}\Vert u-U_{hp}\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}. \end{aligned}$$

(3.13)

It is not difficult to show

$$ S_{hp}^{\prime}(U_{hp})=U_{hp}+B^{*} \tilde {P}_{hp},\qquad S^{\prime}(U_{hp})=U_{hp}+B^{*}p(U_{hp}), $$

(3.14)

where $p(U_{hp})$ is defined in (3.11)-(3.12). Thanks to (3.14), it is easy to derive

$$\begin{aligned} \bigl\Vert S_{hp}^{\prime}(U_{hp})-S^{\prime }(U_{hp}) \bigr\Vert _{L^{2}(\Omega)}=\bigl\Vert p(U_{hp})- \tilde{P}_{hp}\bigr\Vert _{L^{2}(\Omega)}. \end{aligned}$$

(3.15)

Then by using the estimates (3.13) and (3.15) we can prove the requested result (3.10). □

Next, we are in the position to estimate the errors $\Vert Y_{hp}-y(U_{hp})\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}$ and ${\Vert P_{hp}-p(U_{hp})\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}}$.

Theorem 3.2

Let $(Y_{hp},P_{hp},U_{hp})$ be the solutions of (2.26)-(2.28), let $(y(U_{hp}),p(U_{hp}))$ be defined by (3.11)-(3.12). Then we have

$$\begin{aligned} \bigl\Vert Y_{hp}-y(U_{hp})\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +\bigl\Vert P_{hp}-p(U_{hp}) \bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}\leq C\sum _{i=2}^{6}\eta_{i}^{2}, \end{aligned}$$

(3.16)

where

$$\begin{aligned}& \eta_{2}^{2}= \int_{0}^{T}\sum_{\tau} \frac{h_{\tau }^{2}}{p_{\tau}^{2}} \int_{\tau} \biggl(\hat{f}+BU_{hp}+\operatorname {div}(A\nabla \hat{Y}_{hp})-\phi(\hat {Y}_{hp})-\frac {\partial Y_{hp}}{\partial t} \biggr)^{2}\,dx\,dt, \\& \eta_{3}^{2}= \int_{0}^{T}\sum_{\tau} \frac{h_{\tau }^{2}}{p_{\tau }^{2}} \int_{\tau} \biggl(\hat{Y}_{hp}-\hat{y}_{d}+ \operatorname {div}\bigl(A^{*}\nabla\tilde{P}_{hp}\bigr)-\phi ^{\prime}(\tilde{Y}_{hp}){\tilde{P}_{hp}}+ \frac{\partial P_{hp}}{\partial t} \biggr)^{2}\,dx\,dt, \\& \eta_{4}^{2}= \int_{0}^{T}\sum_{e} \int_{e}\frac {h_{e}}{p_{e}}\bigl[(A\nabla \hat{Y}_{hp}) \cdot n\bigr]^{2}\,de\,dt+ \int_{0}^{T}\sum_{e} \int_{e}\frac{h_{e}}{p_{e}}\bigl[\bigl(A^{*}\nabla \tilde{P}_{hp}\bigr)\cdot n\bigr]^{2}\,de\,dt, \\& \eta_{5}^{2}= \int_{0}^{T} \int_{\Omega}\bigl\vert A\nabla(\hat{Y}_{hp}-Y_{hp}) \bigr\vert ^{2}\,dx\,dt+ \int_{0}^{T} \int_{\Omega}\bigl\vert A^{*}\nabla( \tilde{P}_{hp}-P_{hp})\bigr\vert ^{2}\,dx\,dt , \\& \eta_{6}^{2}= \Vert Y_{hp}- \tilde{Y}_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}+\Vert Y_{hp}-\hat{Y}_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))}+ \Vert y_{d}-\hat {y}_{d}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))} \\& \hphantom{\eta_{6}^{2}=}{} +\Vert f-\hat{f}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))}+\bigl\Vert y_{0}(x)-Y_{hp}(x,0)\bigr\Vert ^{2}_{L^{2}(\Omega)}, \end{aligned}$$

where $e=\bar{\tau}_{e}^{1}\cap\bar{\tau}_{e}^{2}$, $\tau_{e}^{1}$, $\tau_{e}^{2}$ are two neighboring elements in $\mathcal{T}$, $[A\nabla\hat{Y}_{hp}\cdot n]_{e}$ and $[A^{*}\nabla\tilde{P}_{hp}\cdot n]_{e}$ are the A-normal and $A^{*}$-normal derivative jumps over the interior edge e, respectively, defined by

$$\begin{aligned}& \bigl[(A\nabla\hat{Y}_{hp})\cdot n\bigr]_{e}=(A\nabla \hat{Y}_{hp}|_{\tau_{e}^{1}}-A\nabla\hat {Y}_{hp}|_{\tau _{e}^{2}}) \cdot n, \\& \bigl[\bigl(A^{*}\nabla\tilde{P}_{hp}\bigr)\cdot n \bigr]_{e}=\bigl(A^{*}\nabla\tilde{P}_{hp}|_{\tau_{e}^{1}}-A^{*} \nabla\tilde {P}_{hp}|_{\tau_{e}^{2}}\bigr)\cdot n, \end{aligned}$$

where n is the unit normal vector on $e=\bar{\tau}_{e}^{1}\cap\bar{\tau}_{e}^{2}$ outwards $\tau_{e}^{1}$. For later convenience, we define $[A\nabla\hat{Y}_{hp}\cdot n]_{e}=0$ and $[A^{*}\nabla\tilde{P}_{hp}\cdot n]_{e}=0$ when $e\subset\partial\Omega$.

Proof

Let $r_{hp}=p(U_{hp})-P_{hp}$ and $E_{1}$ be the linear operator defined in Lemma 2.3. Note that $(p(U_{hp})-P_{hp})(x,T)=0$, hence

$$\begin{aligned} \int_{0}^{T}- \biggl(\frac{\partial (p(U_{hp})-P_{hp})}{\partial t},r_{hp} \biggr)\,dt\geq0. \end{aligned}$$

(3.17)

By using the assumption of ϕ, (2.29), and (3.17), we obtain

$$\begin{aligned} &c\Vert r_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad \leq \int_{0}^{T}a\bigl(r_{hp},p(U_{hp})-P_{hp} \bigr)\,dt+ \int_{0}^{T}\bigl(\phi ^{\prime } \bigl(y(U_{hp})\bigr) \bigl(p(U_{hp})-P_{hp} \bigr),p(U_{hp})-P_{hp}\bigr)\,dt \\ &\quad \leq \int_{0}^{T}\bigl(\nabla r_{hp},A^{*} \nabla\bigl(p(U_{hp})-P_{hp}\bigr)\bigr)\,dt- \int_{0}^{T} \biggl(\frac{\partial(p(U_{hp})-P_{hp})}{\partial t},r_{hp} \biggr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr) \bigl(p(U_{hp})-\tilde {P}_{hp}\bigr),p(U_{hp})-P_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr) (\tilde {P}_{hp}-{P}_{hp}),p(U_{hp})-P_{hp} \bigr)\,dt \\ &\quad = \int_{0}^{T}\bigl(\nabla r_{hp},A^{*} \nabla\bigl(p(U_{hp})-\tilde{P}_{hp}\bigr)\bigr)\,dt- \int_{0}^{T} \biggl(\frac{\partial(p(U_{hp})-P_{hp})}{\partial t},r_{hp} \biggr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr)p(U_{hp})-\phi^{\prime }(\tilde {Y}_{hp})\tilde{P}_{hp},p(U_{hp})-P_{hp} \bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\tilde{\phi}^{\prime\prime}( \tilde{Y}_{hp}) \bigl(\tilde {Y}_{hp}-y(U_{hp}) \bigr)\tilde{P}_{hp},p(U_{hp})-P_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr) (\tilde {P}_{hp}-{P}_{hp}),p(U_{hp})-P_{hp} \bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\nabla r_{hp},A^{*} \nabla(\tilde{P}_{hp}-P_{hp})\bigr)\,dt. \end{aligned}$$

(3.18)

Connecting (2.27), (3.12), and (3.18), we have

$$\begin{aligned} &c\Vert r_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad \leq \int_{0}^{T}\bigl(\nabla (r_{hp}-E_{1}r_{hp}),A^{*} \nabla\bigl(p(U_{hp})-\tilde{P}_{hp}\bigr)\bigr)\,dt \\ &\qquad {}- \int_{0}^{T} \biggl(\frac{\partial(p(U_{hp})-P_{hp})}{\partial t},r_{hp}-E_{1}r_{hp} \biggr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr)p(U_{hp})-\phi ^{\prime }( \tilde{Y}_{hp})\tilde{P}_{hp},r_{hp}-E_{1}r_{hp} \bigr)\,dt \\ & \qquad {}- \int_{0}^{T} \biggl(\frac{\partial(p(U_{hp})-P_{hp})}{\partial t},E_{1}r_{hp} \biggr)\,dt+ \int_{0}^{T}\bigl(\nabla E_{1}r_{hp},A^{*} \nabla\bigl(p(U_{hp})-\tilde{P}_{hp}\bigr)\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr)p(U_{hp})-\phi^{\prime }(\tilde {Y}_{hp})\tilde{P}_{hp},E_{1}r_{hp} \bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\tilde{\phi}^{\prime\prime}( \tilde {Y}_{hp}) \bigl(\tilde {Y}_{hp}-y(U_{hp}) \bigr)\tilde{P}_{hp},p(U_{hp})-P_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr) (\tilde {P}_{hp}-{P}_{hp}),p(U_{hp})-P_{hp} \bigr)\,dt+ \int_{0}^{T}\bigl(\nabla r_{hp},A^{*} \nabla(\tilde{P}_{hp}-P_{hp})\bigr)\,dt \\ &\quad = \int_{0}^{T} \biggl(y(U_{hp})-y_{d}+ \operatorname {div}\bigl(A^{*}\nabla\tilde {P}_{hp}\bigr)-\phi ^{\prime}(\tilde{Y}_{hp})\tilde{P}_{hp}+ \frac{\partial P_{hp}}{\partial t},r_{hp}-E_{1}r_{hp} \biggr)\,dt \\ &\qquad {}+ \int_{0}^{T}\sum_{e} \int_{e}\bigl[\bigl(A^{*}\nabla \tilde{P}_{hp}\bigr)\cdot n\bigr](r_{hp}-E_{1}r_{hp}) \,de\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\tilde {\phi}^{\prime\prime}( \tilde{Y}_{hp}) \bigl(\tilde {Y}_{hp}-y(U_{hp}) \bigr)\tilde {P}_{hp},p(U_{hp})-P_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(y(U_{hp})- \hat{Y}_{hp},E_{1}r_{hp}\bigr)\,dt+ \int_{0}^{T}(\hat{y}_{d}-y_{d},E_{1}r_{hp}) \,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr) (\tilde {P}_{hp}-{P}_{hp}),p(U_{hp})-P_{hp} \bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\nabla r_{hp},A^{*} \nabla(\tilde{P}_{hp}-P_{hp})\bigr)\,dt. \end{aligned}$$

(3.19)

Then we have

$$\begin{aligned} &c\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad \leq \int_{0}^{T} \biggl(\hat{Y}_{hp}- \hat{y}_{d}+\operatorname {div}\bigl(A^{*}\nabla \tilde {P}_{hp} \bigr)-\phi^{\prime}(\tilde{Y}_{hp})\tilde{P}_{hp}+ \frac {\partial P_{hp}}{\partial t},r_{hp}-E_{1}r_{hp} \biggr)\,dt \\ &\qquad {}+ \int_{0}^{T}\sum_{e} \int_{e}\bigl[\bigl(A^{*}\nabla \tilde{P}_{hp}\bigr)\cdot n\bigr](r_{hp}-E_{1}r_{hp}) \,de\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(y(U_{hp})- \hat{Y}_{hp},r_{hp}\bigr)\,dt+ \int_{0}^{T}(\hat{y}_{d}-y_{d},r_{hp}) \,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\nabla r_{hp},A^{*} \nabla(\tilde{P}_{hp}-P_{hp})\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\tilde{\phi}^{\prime\prime}( \tilde{Y}_{hp}) \bigl(\tilde {Y}_{hp}-y(U_{hp}) \bigr)\tilde{P}_{hp},p(U_{hp})-P_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr) (\tilde {P}_{hp}-{P}_{hp}),p(U_{hp})-P_{hp} \bigr)\,dt \\ &\quad \equiv\sum_{i=1}^{7}I_{i}. \end{aligned}$$

(3.20)

It follows from Lemma 2.3 that

$$\begin{aligned} I_{1}={}& \int_{0}^{T} \biggl(\hat{Y}_{hp}- \hat{y}_{d}+\operatorname {div}\bigl(A^{*}\nabla \tilde {P}_{hp} \bigr)-\phi^{\prime}(\tilde{Y}_{hp})\tilde{P}_{hp}+ \frac {\partial P_{hp}}{\partial t},r_{hp}-E_{1}r_{hp} \biggr)\,dt \\ \leq{}& C(\delta) \int_{0}^{T}\sum_{\tau} \frac{h_{\tau}^{2}}{p_{\tau }^{2}} \int _{\tau} \biggl(\hat{Y}_{hp}-\hat{y}_{d}+ \operatorname {div}\bigl(A^{*}\nabla\tilde {P}_{hp}\bigr)-\phi ^{\prime}(\tilde{Y}_{hp})\tilde{P}_{hp}+ \frac{\partial P_{hp}}{\partial t} \biggr)^{2}\,dx\,dt \\ &{}+\delta \int_{0}^{T}\Vert r_{hp}\Vert _{H^{1}(\Omega)}^{2}\,dt \\ ={}&C(\delta)\eta^{2}_{3}+\delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}. \end{aligned}$$

(3.21)

Similarly,

$$\begin{aligned} I_{2}&= \int_{0}^{T}\sum_{e} \int_{e}\bigl[\bigl(A^{*}\nabla \tilde{P}_{hp}\bigr)\cdot n\bigr](r_{hp}-E_{1}r_{hp}) \,de\,dt \\ &\leq C(\delta) \int_{0}^{T}\sum_{e} \int_{e}\frac {h_{e}}{p_{e}}\bigl[\bigl(A^{*}\nabla \tilde{P}_{hp}\bigr)\cdot n\bigr]^{2}\,de\,dt+\delta \int_{0}^{T}\Vert r_{hp}\Vert _{H^{1}(\Omega)}^{2}\,dt \\ &\leq C(\delta)\eta_{4}^{2}+\delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}. \end{aligned}$$

(3.22)

For $I_{3}$-$I_{5}$, we have

$$\begin{aligned}& \begin{aligned}[b] I_{3}={}& \int_{0}^{T}\bigl(y(U_{hp})- \hat{Y}_{hp},r_{hp}\bigr)\,dt \\ \leq{}& C(\delta) \int_{0}^{T} \int_{\Omega}\bigl\vert y(U_{hp})-\hat{Y}_{hp} \bigr\vert ^{2}\,dx\,dt +\delta\bigl\Vert p(U_{hp})-P_{hp} \bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ \leq{}&C(\delta)\bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +C(\delta)\Vert Y_{hp}- \hat{Y}_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))} \\ &{}+ \delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ \leq{}& C(\delta)\eta_{6}^{2}+C(\delta)\bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}+ \delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}, \end{aligned} \end{aligned}$$

(3.23)

$$\begin{aligned}& \begin{aligned}[b] I_{4}&= \int_{0}^{T}(\hat{y}_{d}-y_{d},r_{hp}) \,dt \\ &\leq C(\delta)\Vert y_{d}-\hat{y}_{d}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))}+\delta\bigl\Vert p(U_{hp})-P_{hp} \bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\leq C(\delta)\eta_{6}^{2}+\delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}, \end{aligned} \end{aligned}$$

(3.24)

and

$$\begin{aligned} I_{5}&= \int_{0}^{T}\bigl(\nabla r_{hp},A^{*} \nabla(\tilde{P}_{hp}-P_{hp})\bigr)\,dt \\ &\leq C(\delta) \int_{0}^{T} \int_{\Omega}\bigl\vert A^{*}\nabla( \tilde{P}_{hp}-P_{hp})\bigr\vert ^{2}\,dx\,dt+ \delta \int_{0}^{T} \int_{\Omega} \vert \nabla r_{hp}\vert ^{2} \,dx\,dt \\ &\leq C(\delta)\eta_{5}^{2}+\delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}. \end{aligned}$$

(3.25)

Similarly, we can obtain

$$\begin{aligned} I_{6}={}& \int_{0}^{T}\bigl(\tilde{\phi}^{\prime\prime }( \tilde {Y}_{hp}) \bigl(\tilde{Y}_{hp}-y(U_{hp}) \bigr)\tilde{P}_{hp},p(U_{hp})-P_{hp}\bigr)\,dt \\ \leq{}& C(\delta) \int_{0}^{T} \int_{\Omega}\bigl\vert y(U_{hp})-\tilde{Y}_{hp} \bigr\vert ^{2}\,dx\,dt +\delta\bigl\Vert p(U_{hp})-P_{hp} \bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ \leq{}&C(\delta)\bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +C(\delta)\Vert Y_{hp}- \tilde{Y}_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))} \\ &{}+ \delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ \leq{}& C(\delta)\eta_{6}^{2}+C(\delta)\bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}+ \delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \end{aligned}$$

(3.26)

and

$$\begin{aligned} I_{7}&= \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr) (\tilde {P}_{hp}-{P}_{hp}),p(U_{hp})-P_{hp} \bigr)\,dt \\ &\leq C(\delta)\Vert \tilde{P}_{hp}-{P}_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}+\delta\bigl\Vert p(U_{hp})-P_{hp} \bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega ))} \\ &\leq C(\delta) \int_{0}^{T} \int_{\Omega}\bigl\vert A^{*}\nabla(\tilde {P}_{hp}-P_{hp})\bigr\vert ^{2}\,dx\,dt+\delta \bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\leq C(\delta)\eta_{5}^{2}+\delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}. \end{aligned}$$

(3.27)

Then let δ be small enough, from (3.20)-(3.27), we derive

$$\begin{aligned} &\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega ))}\leq C(\delta)\sum_{i=2}^{6} \eta_{i}^{2}+C(\delta)\bigl\Vert y(U_{hp})-Y_{hp} \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))}. \end{aligned}$$

(3.28)

Furthermore, we estimate the error $\Vert y(U_{hp})-Y_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))}$. Let $e_{hp}=y(U_{hp})-Y_{hp}$ and $E_{1}$ be the linear operator defined in Lemma 2.3. Note that

$$\begin{aligned} & \int_{0}^{T} \biggl(\frac{\partial (y(U_{hp})-Y_{hp})}{\partial t},e_{hp} \biggr)\,dt\\ &\quad = \int_{0}^{T} \int_{\Omega}\frac{\partial (y(U_{hp})-Y_{hp})}{\partial t}e_{hp}\,dx\,dt \\ &\quad = \int_{\Omega}\bigl(\bigl(y(U_{hp})-Y_{hp}\bigr) (x,T)\bigr)^{2}\,dx- \int_{\Omega}\bigl(\bigl(y(U_{hp})-Y_{hp}\bigr) (x,0)\bigr)^{2}\,dx \\ &\qquad {}- \int_{0}^{T} \biggl(\frac{\partial (y(U_{hp})-Y_{hp})}{\partial t},e_{hp} \biggr)\,dt. \end{aligned}$$

Thus

$$\begin{aligned} \int_{0}^{T} \biggl(\frac{\partial (y(U_{hp})-Y_{hp})}{\partial t},e_{hp} \biggr)\,dt+\frac{1}{2}\bigl\Vert y_{0}(x)-Y_{hp}(x,0) \bigr\Vert ^{2}_{L^{2}(\Omega)}\geq 0. \end{aligned}$$

(3.29)

From (2.29), it is easy to see that

$$\begin{aligned} \bigl({\phi}\bigl(y(U_{hp})\bigr)-\phi(Y_{hp}),e_{hp} \bigr)=\bigl(\tilde{\phi}^{\prime }\bigl(y(U_{hp})\bigr) \bigl(y(U_{hp})-Y_{hp}\bigr),e_{hp}\bigr) \geq0. \end{aligned}$$

(3.30)

By using (3.29) and (3.30), we have

$$\begin{aligned} &c\Vert e_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad \leq \int_{0}^{T}a\bigl(y(U_{hp})-Y_{hp},e_{hp} \bigr)\,dt+ \int_{0}^{T}\bigl({\phi }\bigl(y(U_{hp}) \bigr)-\phi(Y_{hp}),e_{hp}\bigr)\,dt \\ &\quad \leq \int_{0}^{T}a\bigl(y(U_{hp})-Y_{hp},e_{hp} \bigr)\,dt+ \int_{0}^{T}\bigl({\phi}\bigl(y(U_{hp}) \bigr)-{\phi}(\hat{Y}_{hp}),e_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T} \biggl(\frac{\partial (y(U_{hp})-Y_{hp})}{\partial t},e_{hp} \biggr)\,dt+ \int_{0}^{T}\bigl({\phi}(\hat{Y}_{hp})- \phi (Y_{hp}),e_{hp}\bigr)\,dt \\ &\qquad {}+\frac{1}{2}\bigl\Vert y_{0}(x)-Y_{hp}(x,0)\bigr\Vert ^{2}_{L^{2}(\Omega)} \\ &\quad = \int_{0}^{T}\bigl(A\nabla\bigl(y(U_{hp})- \hat{Y}_{hp}\bigr),\nabla e_{hp}\bigr)\,dt+ \int_{0}^{T}\bigl({\phi}\bigl(y(U_{hp}) \bigr)-\phi(\hat{Y}_{hp}),e_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T} \biggl(\frac{\partial(y(U_{hp})-Y_{hp})}{\partial t},e_{hp} \biggr)\,dt+ \int_{0}^{T}\bigl(A\nabla(\hat{Y}_{hp}-Y_{hp}), \nabla e_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl({\phi}(\hat{Y}_{hp})- \phi(Y_{hp}),e_{hp}\bigr)\,dt \\ &\qquad {}+\frac{1}{2}\bigl\Vert y_{0}(x)-Y_{hp}(x,0)\bigr\Vert ^{2}_{L^{2}(\Omega)}. \end{aligned}$$

(3.31)

Connecting (2.26), (2.29), (3.11), and (3.31), we obtain

$$\begin{aligned} &c\Vert e_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad \leq \int_{0}^{T} \biggl(\hat{f}+BU_{hp}+\operatorname {div}(A \nabla\hat {Y}_{hp})-\phi (\hat{Y}_{hp})-\frac{\partial Y_{hp}}{\partial t},e_{hp}-E_{1}e_{hp} \biggr)\,dt \\ &\qquad {}+ \int_{0}^{T}\sum_{e} \int_{e}\bigl[(A\nabla\hat{Y}_{hp})\cdot n \bigr](e_{hp}-E_{1}e_{hp})\,de\,dt+ \int_{0}^{T}(f-\hat{f},e_{hp})\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(A\nabla(\hat{Y}_{hp}-Y_{hp}), \nabla e_{hp}\bigr)\,dt+\frac{1}{2}\bigl\Vert y_{0}(x)-Y_{hp}(x,0)\bigr\Vert ^{2}_{L^{2}(\Omega )} \\ &\qquad {}+ \int_{0}^{T}\bigl(\tilde{\phi}^{\prime}( \hat{Y}_{hp}) (\hat {Y}_{hp}-Y_{hp}),e_{hp} \bigr)\,dt \\ &\quad \leq C(\delta) \int_{0}^{T}\sum_{\tau} \frac{h_{\tau}^{2}}{p_{\tau }^{2}} \int_{\tau} \biggl(\hat{f}+BU_{hp}+\operatorname {div}(A\nabla \hat{Y}_{hp})-\phi(\hat {Y}_{hp})-\frac {\partial Y_{hp}}{\partial t} \biggr)^{2}\,dx\,dt \\ &\qquad {}+C(\delta) \int_{0}^{T}\sum_{e} \int_{e}\frac {h_{e}}{p_{e}}\bigl[(A\nabla \hat{Y}_{hp}) \cdot n\bigr]^{2}\,de\,dt +C(\delta)\Vert f-\hat{f}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))} \\ &\qquad {}+C(\delta ) \int_{0}^{T} \int_{\Omega}\bigl\vert A\nabla(\hat{Y}_{hp}-Y_{hp}) \bigr\vert ^{2}\,dx\,dt+C(\delta)\Vert \hat{Y}_{hp}-Y_{hp} \Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\qquad {}+\frac{1}{2}\bigl\Vert y_{0}(x)-Y_{hp}(x,0) \bigr\Vert ^{2}_{L^{2}(\Omega )}+\delta\bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad \leq C(\delta)\sum_{i=2}^{6} \eta_{i}^{2}+\delta\bigl\Vert y(U_{hp})-Y_{hp} \bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}. \end{aligned}$$

(3.32)

Hence, let δ be small enough, we have

$$\begin{aligned} \bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega ))}\leq C(\delta)\sum_{i=1}^{6} \eta_{i}^{2}. \end{aligned}$$

(3.33)

Then (3.16) follows from (3.28) and (3.33). □

Finally, collecting Theorems 3.1-3.2, we derive the following residual-based a posteriori error estimates.

Theorem 3.3

Let $(y,p,u)$ and $(Y_{hp},P_{hp},U_{hp})$ are the solutions of (2.8)-(2.10) and (2.26)-(2.28) respectively. Assume that all the conditions in Theorem 3.1 are valid. Then

$$\begin{aligned} &\Vert y-Y_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}+\Vert p-P_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +\Vert u-U_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega_{U}))} \\ &\quad \leq C\sum _{i=1}^{6}\eta_{i}^{2}, \end{aligned}$$

(3.34)

where $\eta_{i}^{2}$, $i=1,\ldots,6$ are defined in Theorem 3.1 and Theorem 3.2.

Proof

It follows from Theorem 3.1 and Theorem 3.2 that

$$\begin{aligned} \Vert u-U_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega_{U}))} \leq& C\eta _{1}^{2}+C\bigl\Vert \tilde{P}_{hp}-p(U_{hp}) \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))} \\ \leq& C\eta_{1}^{2}+C\Vert \tilde{P}_{hp}-P_{hp} \Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))} \\ &{}+C\bigl\Vert P_{hp}-p(U_{hp}) \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))} \\ \leq& C\sum_{i=1}^{6}\eta_{i}^{2}+C \Vert \tilde{P}_{hp}-P_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))}. \end{aligned}$$

(3.35)

Note that A is positive definite, it follows from the Poincaré inequality that

$$\begin{aligned} \Vert \tilde{P}_{hp}-P_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))} \leq& \int_{0}^{T} \int_{\Omega}\bigl\vert A^{*}\nabla( \tilde{P}_{hp}-P_{hp})\bigr\vert ^{2}\,dx\,dt \\ \leq& C\eta_{5}^{2}. \end{aligned}$$

(3.36)

Then it follows from (3.35) and (3.36) that

$$\begin{aligned} \Vert u-U_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega_{U}))}\leq C\sum _{i=1}^{6}\eta_{i}^{2}. \end{aligned}$$

(3.37)

Note that

$$\begin{aligned}& \Vert y-Y_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}\leq\bigl\Vert y-y(U_{hp})\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +\bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}, \end{aligned}$$

(3.38)

$$\begin{aligned}& \Vert p-P_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}\leq\bigl\Vert p-p(U_{hp})\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}, \end{aligned}$$

(3.39)

and

$$\begin{aligned}& \bigl\Vert y-y(U_{hp})\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \leq C\Vert u-U_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega_{U}))}, \end{aligned}$$

(3.40)

$$\begin{aligned}& \bigl\Vert p-p(U_{hp})\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \leq\bigl\Vert y-y(U_{hp})\bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))} \leq C\Vert u-U_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega_{U}))}. \end{aligned}$$

(3.41)

Therefore, we obtain (3.34) from (3.16) and (3.37)-(3.41). □

4 Conclusion and future work

In this paper, we present the hp version of the finite element approximation for the optimal control problems governed by nonlinear parabolic equations. By using the hp finite element approximation for both the state and the co-state variables and the hp discontinuous Galerkin finite element approximation for the control variable, we derive hp residual-based a posteriori error estimates for the nonlinear parabolic optimal control problems. To the best of our knowledge in the context of optimal control problems, these residual-based a posteriori error estimates for the nonlinear parabolic optimal control problems are new.

In future, we shall consider the hp version of the finite element method for hyperbolic optimal control problems. Furthermore, we shall consider a posteriori error estimates and the superconvergence of the hp finite element solutions for hyperbolic optimal control problems.