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New a posteriori error estimates for hp version of finite element methods of nonlinear parabolic optimal control problems
Journal of Inequalities and Applications20162016:62
https://doi.org/10.1186/s13660-016-0994-3
© Lu et al. 2016
- Received: 17 June 2015
- Accepted: 1 February 2016
- Published: 13 February 2016
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.
Keywords
- residual-based a posteriori error estimates
- nonlinear parabolic optimal control problems
- hp version of finite element method
MSC
- 49J20
- 65N30
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:
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 \(\phi^{\prime}(y)\geq0\).
$$\begin{aligned}& \min_{u\in K} \biggl\{ \frac{1}{2} \int_{0}^{T}\bigl(\Vert y-y_{d}\Vert ^{2}_{L^{2}(\Omega)}+\Vert u\Vert ^{2}_{L^{2}(\Omega_{U})} \bigr)\,dt \biggr\} , \end{aligned}$$
(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)
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
Then a weak formula of the convex nonlinear parabolic optimal control problems (2.1)-(2.4) reads
where \(y\in W\), \(u\in X\), \(u(t)\in K\), subject to
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:
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.
$$\begin{aligned} a(v,v)\geq c\Vert v\Vert _{H^{1}(\Omega)}^{2}, \qquad \bigl\vert a(v,w) \bigr\vert \leq C\vert v\vert _{H^{1}(\Omega)}^{2}\vert w\vert _{H^{1}(\Omega)}^{2}, \quad \forall v,w\in Y. \end{aligned}$$
$$\begin{aligned} &\min_{u(t)\in K} \biggl\{ \frac{1}{2} \int_{0}^{T} \bigl(\Vert y-y_{d} \Vert ^{2}_{L^{2}(\Omega )}+\Vert u\Vert ^{2}_{L^{2}(\Omega_{U})} \bigr)\,dt \biggr\} , \end{aligned}$$
(2.5)
$$\begin{aligned}& (y_{t},w)+a(y,w)+\bigl(\phi(y),w\bigr)=(f+Bu,w),\quad \forall w\in Y, t \in(0,T], \end{aligned}$$
(2.6)
$$\begin{aligned}& y(x,0)=y_{0}(x), \quad x\in\Omega. \end{aligned}$$
(2.7)
$$\begin{aligned}& (y_{t},w)+a(y,w)+\bigl(\phi(y),w\bigr)=(f+Bu,w),\quad \forall w\in Y, y(0)=y_{0}(x), \end{aligned}$$
(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)
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
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
\(B^{*}p\).
$$\begin{aligned} u=\max \bigl\{ 0,\overline{B^{*}p} \bigr\} -B^{*}p, \end{aligned}$$
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
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.,
This implies that there exists a constant \(C>0\) that depends solely on γ such that
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].
$$\bigl\{ (x,y)\in\mathbb{R}^{2} : 0< x< 1,0< y< \min(x,1-x) \bigr\} . $$
$$\begin{aligned} h_{\tau}^{-1}\bigl\Vert F_{\tau}^{\prime} \bigr\Vert _{L^{\infty }(\widehat{\tau})}+h_{\tau}\bigl\Vert \bigl(F_{\tau}^{\prime} \bigr)^{-1}\bigr\Vert _{L^{\infty }(\widehat{\tau})}\leq \gamma. \end{aligned}$$
(2.11)
$$\begin{aligned} C^{-1}h_{\tau }\leq h_{\tau^{\prime}}\leq Ch_{\tau}, \quad \tau,\tau^{\prime}\in\mathcal {T} \mbox{ with } \overline{\tau}\cap\overline{\tau}^{\prime}\neq\emptyset, \end{aligned}$$
(2.12)
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 (\(V\in\mathcal{N}(\mathcal{T})\), \(e\in\mathcal{E}(\mathcal{T})\)):
where \(\chi^{0}\) denotes the interior of the set χ. Finally, we denote by \(h_{e}\) the length of the edge e.
$$\begin{aligned}& \mathcal {N}(e)=\bigl\{ V\in\mathcal {N}(\mathcal{T}):V\in\overline{e}\bigr\} , \\& w_{V}=\bigl\{ x\in\Omega:x\in\overline{\tau}\mbox{ and } \overline{\tau}\cap\{V\}\neq\emptyset\bigr\} ^{0}, \\& w^{1}_{e}=\bigcup_{V\in\mathcal{N}(e)}w_{V},\qquad w^{1}_{\tau}=\bigcup_{V\in\mathcal{N}(\tau)}w_{V}, \\& h_{\tau_{U}}=\operatorname {diam}\tau_{U}, \qquad p_{e}=\max\bigl\{ p_{\tau}: e\in\mathcal{E}(\tau)\bigr\} , \end{aligned}$$
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
where we set
We also assume that the polynomial degree vector \(\mathbf{p}_{1}\) satisfies
Then we can introduce the finite dimensional spaces \(K_{hp}=K\cap U^{ \mathbf{p}_{2}}(\mathcal{T}_{U})\), \(V_{hp}=V\cap S^{\mathbf{p}_{1}}(\mathcal{T})\).
$$\begin{aligned}& S^{ \mathbf{p}_{1}}(\mathcal{T})= \bigl\{ v\in C(\Omega):v|_{\tau}\circ F_{\tau}\in\Pi_{p_{\tau}}(\widehat{\tau}) \bigr\} , \\& U^{ \mathbf{p}_{2}}(\mathcal{T}_{U})= \bigl\{ v\in L^{2}( \Omega_{U}):v|_{\tau_{U}}\circ F_{\tau_{U}}\in \Pi_{p_{\tau_{U}}}(\widehat{\tau}) \bigr\} , \end{aligned}$$
$$\Pi_{k}(\widehat{\tau})= \textstyle\begin{cases} {P}_{k}=\operatorname {span}\{x^{i}y^{j}:0\leq i+j\leq k\},& \mbox{if } \widehat{\tau}=T,\\ {Q}_{k}=\operatorname {span}\{x^{i}y^{j}:0\leq i,j\leq k\}, & \mbox{if } \widehat{\tau}=S. \end{cases} $$
$$\begin{aligned} \gamma^{-1}p_{\tau}\leq p_{\tau^{\prime}}\leq \gamma p_{\tau}, \quad \tau,\tau^{\prime}\in\mathcal{T} \mbox{ with } \overline{\tau}\cap\overline{\tau}^{\prime}\neq\emptyset. \end{aligned}$$
(2.13)
The semidiscrete hp finite element approximation of (2.1)-(2.4) is as follows:
where \(y_{hp}\in H^{1}(0,T;V_{hp})\) and \(y_{0}^{hp}\in V_{hp}\) is an finite element approximation of \(y_{0}\).
$$\begin{aligned}& \min_{u_{hp}\in K_{hp}} \biggl\{ \frac{1}{2} \int_{0}^{T}\bigl(\Vert y_{hp}-y_{d} \Vert ^{2}_{L^{2}(\Omega )}+\Vert u_{hp}\Vert ^{2}_{L^{2}(\Omega_{U})}\bigr)\,dt \biggr\} , \end{aligned}$$
(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)
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
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:
$$\begin{aligned}& \min_{u_{hp}^{i}\in K^{i}_{hp}} \Biggl\{ \sum _{i=1}^{M} \biggl(\frac{1}{2}\bigl\Vert y_{hp}^{i}-y_{d}(x,t_{i})\bigr\Vert ^{2}_{L^{2}(\Omega)}+\frac{1}{2}\bigl\Vert u_{hp}^{i} \bigr\Vert ^{2}_{L^{2}(\Omega_{U})} \biggr) \Biggr\} , \end{aligned}$$
(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)
$$\begin{aligned}& \biggl(\frac{Y_{hp}^{i}-Y_{hp}^{i-1}}{k_{i}},w_{hp} \biggr)+a\bigl(Y_{hp}^{i},w_{hp} \bigr)+\bigl(\phi \bigl(Y_{hp}^{i}\bigr),w_{hp} \bigr)=\bigl(f(x,t_{i})+BU_{hp}^{i},w_{hp} \bigr), \end{aligned}$$
(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
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
The following lemmas [10, 16, 19] are important in deriving a posteriori error estimates of residual type.
$$\begin{aligned}& Y_{hp}|_{(t_{i-1},t_{i}]}= \bigl((t_{i}-t)Y_{hp}^{i-1}+(t-t_{i-1})Y_{hp}^{i} \bigr)/k_{i}, \\& P_{hp}|_{(t_{i-1},t_{i}]}= \bigl((t_{i}-t)P_{hp}^{i-1}+(t-t_{i-1})P_{hp}^{i} \bigr)/k_{i}, \\& U_{hp}|_{(t_{i-1},t_{i}]}=U_{hp}^{i}. \end{aligned}$$
$$\begin{aligned}& \biggl(\frac{\partial Y_{hp}}{\partial t},w_{hp} \biggr)+a(\hat {Y}_{hp},w_{hp})+\bigl(\phi(\hat{Y}_{hp}),w_{hp} \bigr)=(\hat {f}+BU_{hp},w_{hp}), \end{aligned}$$
(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)
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:
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.
$$\bigl\Vert v-\pi_{p_{\tau_{U}}}^{h_{\tau_{U}}}\bigr\Vert _{L^{2}(\tau _{U})}\leq C\frac{h_{\tau_{U}}}{p_{\tau_{U}}}\Vert v\Vert _{H^{1}(\tau_{U})}, $$
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
where
are bounded functions in Ω̄ [22].
$$\begin{aligned} \phi(\varphi)-\phi(\rho)=-\tilde{\phi}^{\prime}(\varphi) ( \rho -\varphi )=-\phi^{\prime}(\rho) (\rho-\varphi) +\tilde{ \phi}^{\prime\prime}(\varphi) (\rho-\varphi)^{2}, \end{aligned}$$
(2.29)
$$\begin{aligned}& \tilde{\phi}^{\prime}(\varphi)= \int_{0}^{1}\phi^{\prime}\bigl(\varphi +s(\rho -\varphi)\bigr)\,ds, \qquad \tilde{\phi}^{\prime\prime}(\varphi)= \int_{0}^{1}(1-s)\phi^{\prime \prime }\bigl(\rho+s( \varphi-\rho)\bigr)\,ds \end{aligned}$$
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
It can be shown [19] that
$$\begin{aligned}& S(u)=\frac{1}{2} \bigl(\Vert y-y_{d}\Vert ^{2}_{L^{2}(\Omega)}+\Vert u\Vert ^{2}_{L^{2}(\Omega_{U})} \bigr), \end{aligned}$$
(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)
$$\begin{aligned}& \bigl(S^{\prime}(u),v\bigr)=\bigl(u+B^{*}p, v \bigr), \end{aligned}$$
(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
and
$$ \min_{u\in K} \biggl\{ \int_{0}^{T}S(u)\,dt \biggr\} $$
(3.6)
$$ \min_{U_{hp}\in K_{hp}} \biggl\{ \int_{0}^{T} S_{hp}(U_{hp})\,dt \biggr\} . $$
(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
Then we have
where
and
\(p(U_{hp})\)
is defined by the following equations:
$$ \bigl\vert \bigl(S_{hp}^{\prime}(U_{hp}),v_{hp}-u \bigr)\bigr\vert \leq C \sum_{\tau\in \mathcal {T}_{h}}h_{\tau} \bigl\Vert S_{hp}^{\prime}(U_{hp})\bigr\Vert _{H^{s}(\tau)}\Vert u-U_{hp}\Vert _{L^{2}(\tau)}^{s}. $$
(3.9)
$$ \Vert u-U_{hp}\Vert _{L^{2}(J;L^{2}(\Omega))}^{2} \leq C\eta_{1}^{2}+C\bigl\Vert p(U_{hp})- \tilde{P}_{hp}\bigr\Vert _{L^{2}(J;L^{2}(\Omega))}^{2}, $$
(3.10)
$$\eta_{1}^{2}= \int_{0}^{T}\sum_{\tau\in \mathcal {T}_{h}}h_{\tau}^{1+s} \bigl\Vert U_{hp}+B^{*}\tilde {P}_{hp}\bigr\Vert _{H^{1}(\tau)}^{1+s}\,dt, $$
$$\begin{aligned}& \biggl(\frac{\partial}{\partial t}y(U_{hp}),w \biggr)+a\bigl(y(U_{hp}),w \bigr)+\bigl(\phi \bigl(y(U_{hp})\bigr),w\bigr)=(f+BU_{hp},w),\quad \forall w\in V, \end{aligned}$$
(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
where \(p(U_{hp})\) is defined in (3.11)-(3.12). Thanks to (3.14), it is easy to derive
$$ S_{hp}^{\prime}(U_{hp})=U_{hp}+B^{*} \tilde {P}_{hp},\qquad S^{\prime}(U_{hp})=U_{hp}+B^{*}p(U_{hp}), $$
(3.14)
$$\begin{aligned} \bigl\Vert S_{hp}^{\prime}(U_{hp})-S^{\prime }(U_{hp}) \bigr\Vert _{L^{2}(\Omega)}=\bigl\Vert p(U_{hp})- \tilde{P}_{hp}\bigr\Vert _{L^{2}(\Omega)}. \end{aligned}$$
(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
where
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
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
\(e\subset\partial\Omega\).
$$\begin{aligned} \bigl\Vert Y_{hp}-y(U_{hp})\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +\bigl\Vert P_{hp}-p(U_{hp}) \bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}\leq C\sum _{i=2}^{6}\eta_{i}^{2}, \end{aligned}$$
(3.16)
$$\begin{aligned}& \eta_{2}^{2}= \int_{0}^{T}\sum_{\tau} \frac{h_{\tau }^{2}}{p_{\tau}^{2}} \int_{\tau} \biggl(\hat{f}+BU_{hp}+\operatorname {div}(A\nabla \hat{Y}_{hp})-\phi(\hat {Y}_{hp})-\frac {\partial Y_{hp}}{\partial t} \biggr)^{2}\,dx\,dt, \\& \eta_{3}^{2}= \int_{0}^{T}\sum_{\tau} \frac{h_{\tau }^{2}}{p_{\tau }^{2}} \int_{\tau} \biggl(\hat{Y}_{hp}-\hat{y}_{d}+ \operatorname {div}\bigl(A^{*}\nabla\tilde{P}_{hp}\bigr)-\phi ^{\prime}(\tilde{Y}_{hp}){\tilde{P}_{hp}}+ \frac{\partial P_{hp}}{\partial t} \biggr)^{2}\,dx\,dt, \\& \eta_{4}^{2}= \int_{0}^{T}\sum_{e} \int_{e}\frac {h_{e}}{p_{e}}\bigl[(A\nabla \hat{Y}_{hp}) \cdot n\bigr]^{2}\,de\,dt+ \int_{0}^{T}\sum_{e} \int_{e}\frac{h_{e}}{p_{e}}\bigl[\bigl(A^{*}\nabla \tilde{P}_{hp}\bigr)\cdot n\bigr]^{2}\,de\,dt, \\& \eta_{5}^{2}= \int_{0}^{T} \int_{\Omega}\bigl\vert A\nabla(\hat{Y}_{hp}-Y_{hp}) \bigr\vert ^{2}\,dx\,dt+ \int_{0}^{T} \int_{\Omega}\bigl\vert A^{*}\nabla( \tilde{P}_{hp}-P_{hp})\bigr\vert ^{2}\,dx\,dt , \\& \eta_{6}^{2}= \Vert Y_{hp}- \tilde{Y}_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}+\Vert Y_{hp}-\hat{Y}_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))}+ \Vert y_{d}-\hat {y}_{d}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))} \\& \hphantom{\eta_{6}^{2}=}{} +\Vert f-\hat{f}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))}+\bigl\Vert y_{0}(x)-Y_{hp}(x,0)\bigr\Vert ^{2}_{L^{2}(\Omega)}, \end{aligned}$$
$$\begin{aligned}& \bigl[(A\nabla\hat{Y}_{hp})\cdot n\bigr]_{e}=(A\nabla \hat{Y}_{hp}|_{\tau_{e}^{1}}-A\nabla\hat {Y}_{hp}|_{\tau _{e}^{2}}) \cdot n, \\& \bigl[\bigl(A^{*}\nabla\tilde{P}_{hp}\bigr)\cdot n \bigr]_{e}=\bigl(A^{*}\nabla\tilde{P}_{hp}|_{\tau_{e}^{1}}-A^{*} \nabla\tilde {P}_{hp}|_{\tau_{e}^{2}}\bigr)\cdot n, \end{aligned}$$
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
By using the assumption of ϕ, (2.29), and (3.17), we obtain
Connecting (2.27), (3.12), and (3.18), we have
Then we have
It follows from Lemma 2.3 that
Similarly,
For \(I_{3}\)-\(I_{5}\), we have
and
Similarly, we can obtain
and
Then let δ be small enough, from (3.20)-(3.27), we derive
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
Thus
From (2.29), it is easy to see that
By using (3.29) and (3.30), we have
Connecting (2.26), (2.29), (3.11), and (3.31), we obtain
Hence, let δ be small enough, we have
Then (3.16) follows from (3.28) and (3.33). □
$$\begin{aligned} \int_{0}^{T}- \biggl(\frac{\partial (p(U_{hp})-P_{hp})}{\partial t},r_{hp} \biggr)\,dt\geq0. \end{aligned}$$
(3.17)
$$\begin{aligned} &c\Vert r_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad \leq \int_{0}^{T}a\bigl(r_{hp},p(U_{hp})-P_{hp} \bigr)\,dt+ \int_{0}^{T}\bigl(\phi ^{\prime } \bigl(y(U_{hp})\bigr) \bigl(p(U_{hp})-P_{hp} \bigr),p(U_{hp})-P_{hp}\bigr)\,dt \\ &\quad \leq \int_{0}^{T}\bigl(\nabla r_{hp},A^{*} \nabla\bigl(p(U_{hp})-P_{hp}\bigr)\bigr)\,dt- \int_{0}^{T} \biggl(\frac{\partial(p(U_{hp})-P_{hp})}{\partial t},r_{hp} \biggr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr) \bigl(p(U_{hp})-\tilde {P}_{hp}\bigr),p(U_{hp})-P_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr) (\tilde {P}_{hp}-{P}_{hp}),p(U_{hp})-P_{hp} \bigr)\,dt \\ &\quad = \int_{0}^{T}\bigl(\nabla r_{hp},A^{*} \nabla\bigl(p(U_{hp})-\tilde{P}_{hp}\bigr)\bigr)\,dt- \int_{0}^{T} \biggl(\frac{\partial(p(U_{hp})-P_{hp})}{\partial t},r_{hp} \biggr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr)p(U_{hp})-\phi^{\prime }(\tilde {Y}_{hp})\tilde{P}_{hp},p(U_{hp})-P_{hp} \bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\tilde{\phi}^{\prime\prime}( \tilde{Y}_{hp}) \bigl(\tilde {Y}_{hp}-y(U_{hp}) \bigr)\tilde{P}_{hp},p(U_{hp})-P_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr) (\tilde {P}_{hp}-{P}_{hp}),p(U_{hp})-P_{hp} \bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\nabla r_{hp},A^{*} \nabla(\tilde{P}_{hp}-P_{hp})\bigr)\,dt. \end{aligned}$$
(3.18)
$$\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)
$$\begin{aligned} &c\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad \leq \int_{0}^{T} \biggl(\hat{Y}_{hp}- \hat{y}_{d}+\operatorname {div}\bigl(A^{*}\nabla \tilde {P}_{hp} \bigr)-\phi^{\prime}(\tilde{Y}_{hp})\tilde{P}_{hp}+ \frac {\partial P_{hp}}{\partial t},r_{hp}-E_{1}r_{hp} \biggr)\,dt \\ &\qquad {}+ \int_{0}^{T}\sum_{e} \int_{e}\bigl[\bigl(A^{*}\nabla \tilde{P}_{hp}\bigr)\cdot n\bigr](r_{hp}-E_{1}r_{hp}) \,de\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(y(U_{hp})- \hat{Y}_{hp},r_{hp}\bigr)\,dt+ \int_{0}^{T}(\hat{y}_{d}-y_{d},r_{hp}) \,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\nabla r_{hp},A^{*} \nabla(\tilde{P}_{hp}-P_{hp})\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\tilde{\phi}^{\prime\prime}( \tilde{Y}_{hp}) \bigl(\tilde {Y}_{hp}-y(U_{hp}) \bigr)\tilde{P}_{hp},p(U_{hp})-P_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr) (\tilde {P}_{hp}-{P}_{hp}),p(U_{hp})-P_{hp} \bigr)\,dt \\ &\quad \equiv\sum_{i=1}^{7}I_{i}. \end{aligned}$$
(3.20)
$$\begin{aligned} I_{1}={}& \int_{0}^{T} \biggl(\hat{Y}_{hp}- \hat{y}_{d}+\operatorname {div}\bigl(A^{*}\nabla \tilde {P}_{hp} \bigr)-\phi^{\prime}(\tilde{Y}_{hp})\tilde{P}_{hp}+ \frac {\partial P_{hp}}{\partial t},r_{hp}-E_{1}r_{hp} \biggr)\,dt \\ \leq{}& C(\delta) \int_{0}^{T}\sum_{\tau} \frac{h_{\tau}^{2}}{p_{\tau }^{2}} \int _{\tau} \biggl(\hat{Y}_{hp}-\hat{y}_{d}+ \operatorname {div}\bigl(A^{*}\nabla\tilde {P}_{hp}\bigr)-\phi ^{\prime}(\tilde{Y}_{hp})\tilde{P}_{hp}+ \frac{\partial P_{hp}}{\partial t} \biggr)^{2}\,dx\,dt \\ &{}+\delta \int_{0}^{T}\Vert r_{hp}\Vert _{H^{1}(\Omega)}^{2}\,dt \\ ={}&C(\delta)\eta^{2}_{3}+\delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}. \end{aligned}$$
(3.21)
$$\begin{aligned} I_{2}&= \int_{0}^{T}\sum_{e} \int_{e}\bigl[\bigl(A^{*}\nabla \tilde{P}_{hp}\bigr)\cdot n\bigr](r_{hp}-E_{1}r_{hp}) \,de\,dt \\ &\leq C(\delta) \int_{0}^{T}\sum_{e} \int_{e}\frac {h_{e}}{p_{e}}\bigl[\bigl(A^{*}\nabla \tilde{P}_{hp}\bigr)\cdot n\bigr]^{2}\,de\,dt+\delta \int_{0}^{T}\Vert r_{hp}\Vert _{H^{1}(\Omega)}^{2}\,dt \\ &\leq C(\delta)\eta_{4}^{2}+\delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}. \end{aligned}$$
(3.22)
$$\begin{aligned}& \begin{aligned}[b] I_{3}={}& \int_{0}^{T}\bigl(y(U_{hp})- \hat{Y}_{hp},r_{hp}\bigr)\,dt \\ \leq{}& C(\delta) \int_{0}^{T} \int_{\Omega}\bigl\vert y(U_{hp})-\hat{Y}_{hp} \bigr\vert ^{2}\,dx\,dt +\delta\bigl\Vert p(U_{hp})-P_{hp} \bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ \leq{}&C(\delta)\bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +C(\delta)\Vert Y_{hp}- \hat{Y}_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))} \\ &{}+ \delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ \leq{}& C(\delta)\eta_{6}^{2}+C(\delta)\bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}+ \delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}, \end{aligned} \end{aligned}$$
(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)
$$\begin{aligned} I_{5}&= \int_{0}^{T}\bigl(\nabla r_{hp},A^{*} \nabla(\tilde{P}_{hp}-P_{hp})\bigr)\,dt \\ &\leq C(\delta) \int_{0}^{T} \int_{\Omega}\bigl\vert A^{*}\nabla( \tilde{P}_{hp}-P_{hp})\bigr\vert ^{2}\,dx\,dt+ \delta \int_{0}^{T} \int_{\Omega} \vert \nabla r_{hp}\vert ^{2} \,dx\,dt \\ &\leq C(\delta)\eta_{5}^{2}+\delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}. \end{aligned}$$
(3.25)
$$\begin{aligned} I_{6}={}& \int_{0}^{T}\bigl(\tilde{\phi}^{\prime\prime }( \tilde {Y}_{hp}) \bigl(\tilde{Y}_{hp}-y(U_{hp}) \bigr)\tilde{P}_{hp},p(U_{hp})-P_{hp}\bigr)\,dt \\ \leq{}& C(\delta) \int_{0}^{T} \int_{\Omega}\bigl\vert y(U_{hp})-\tilde{Y}_{hp} \bigr\vert ^{2}\,dx\,dt +\delta\bigl\Vert p(U_{hp})-P_{hp} \bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ \leq{}&C(\delta)\bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +C(\delta)\Vert Y_{hp}- \tilde{Y}_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))} \\ &{}+ \delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ \leq{}& C(\delta)\eta_{6}^{2}+C(\delta)\bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}+ \delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \end{aligned}$$
(3.26)
$$\begin{aligned} I_{7}&= \int_{0}^{T}\bigl(\phi^{\prime} \bigl(y(U_{hp})\bigr) (\tilde {P}_{hp}-{P}_{hp}),p(U_{hp})-P_{hp} \bigr)\,dt \\ &\leq C(\delta)\Vert \tilde{P}_{hp}-{P}_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}+\delta\bigl\Vert p(U_{hp})-P_{hp} \bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega ))} \\ &\leq C(\delta) \int_{0}^{T} \int_{\Omega}\bigl\vert A^{*}\nabla(\tilde {P}_{hp}-P_{hp})\bigr\vert ^{2}\,dx\,dt+\delta \bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\leq C(\delta)\eta_{5}^{2}+\delta\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}. \end{aligned}$$
(3.27)
$$\begin{aligned} &\bigl\Vert p(U_{hp})-P_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega ))}\leq C(\delta)\sum_{i=2}^{6} \eta_{i}^{2}+C(\delta)\bigl\Vert y(U_{hp})-Y_{hp} \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))}. \end{aligned}$$
(3.28)
$$\begin{aligned} & \int_{0}^{T} \biggl(\frac{\partial (y(U_{hp})-Y_{hp})}{\partial t},e_{hp} \biggr)\,dt\\ &\quad = \int_{0}^{T} \int_{\Omega}\frac{\partial (y(U_{hp})-Y_{hp})}{\partial t}e_{hp}\,dx\,dt \\ &\quad = \int_{\Omega}\bigl(\bigl(y(U_{hp})-Y_{hp}\bigr) (x,T)\bigr)^{2}\,dx- \int_{\Omega}\bigl(\bigl(y(U_{hp})-Y_{hp}\bigr) (x,0)\bigr)^{2}\,dx \\ &\qquad {}- \int_{0}^{T} \biggl(\frac{\partial (y(U_{hp})-Y_{hp})}{\partial t},e_{hp} \biggr)\,dt. \end{aligned}$$
$$\begin{aligned} \int_{0}^{T} \biggl(\frac{\partial (y(U_{hp})-Y_{hp})}{\partial t},e_{hp} \biggr)\,dt+\frac{1}{2}\bigl\Vert y_{0}(x)-Y_{hp}(x,0) \bigr\Vert ^{2}_{L^{2}(\Omega)}\geq 0. \end{aligned}$$
(3.29)
$$\begin{aligned} \bigl({\phi}\bigl(y(U_{hp})\bigr)-\phi(Y_{hp}),e_{hp} \bigr)=\bigl(\tilde{\phi}^{\prime }\bigl(y(U_{hp})\bigr) \bigl(y(U_{hp})-Y_{hp}\bigr),e_{hp}\bigr) \geq0. \end{aligned}$$
(3.30)
$$\begin{aligned} &c\Vert e_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad \leq \int_{0}^{T}a\bigl(y(U_{hp})-Y_{hp},e_{hp} \bigr)\,dt+ \int_{0}^{T}\bigl({\phi }\bigl(y(U_{hp}) \bigr)-\phi(Y_{hp}),e_{hp}\bigr)\,dt \\ &\quad \leq \int_{0}^{T}a\bigl(y(U_{hp})-Y_{hp},e_{hp} \bigr)\,dt+ \int_{0}^{T}\bigl({\phi}\bigl(y(U_{hp}) \bigr)-{\phi}(\hat{Y}_{hp}),e_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T} \biggl(\frac{\partial (y(U_{hp})-Y_{hp})}{\partial t},e_{hp} \biggr)\,dt+ \int_{0}^{T}\bigl({\phi}(\hat{Y}_{hp})- \phi (Y_{hp}),e_{hp}\bigr)\,dt \\ &\qquad {}+\frac{1}{2}\bigl\Vert y_{0}(x)-Y_{hp}(x,0)\bigr\Vert ^{2}_{L^{2}(\Omega)} \\ &\quad = \int_{0}^{T}\bigl(A\nabla\bigl(y(U_{hp})- \hat{Y}_{hp}\bigr),\nabla e_{hp}\bigr)\,dt+ \int_{0}^{T}\bigl({\phi}\bigl(y(U_{hp}) \bigr)-\phi(\hat{Y}_{hp}),e_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T} \biggl(\frac{\partial(y(U_{hp})-Y_{hp})}{\partial t},e_{hp} \biggr)\,dt+ \int_{0}^{T}\bigl(A\nabla(\hat{Y}_{hp}-Y_{hp}), \nabla e_{hp}\bigr)\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl({\phi}(\hat{Y}_{hp})- \phi(Y_{hp}),e_{hp}\bigr)\,dt \\ &\qquad {}+\frac{1}{2}\bigl\Vert y_{0}(x)-Y_{hp}(x,0)\bigr\Vert ^{2}_{L^{2}(\Omega)}. \end{aligned}$$
(3.31)
$$\begin{aligned} &c\Vert e_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad \leq \int_{0}^{T} \biggl(\hat{f}+BU_{hp}+\operatorname {div}(A \nabla\hat {Y}_{hp})-\phi (\hat{Y}_{hp})-\frac{\partial Y_{hp}}{\partial t},e_{hp}-E_{1}e_{hp} \biggr)\,dt \\ &\qquad {}+ \int_{0}^{T}\sum_{e} \int_{e}\bigl[(A\nabla\hat{Y}_{hp})\cdot n \bigr](e_{hp}-E_{1}e_{hp})\,de\,dt+ \int_{0}^{T}(f-\hat{f},e_{hp})\,dt \\ &\qquad {}+ \int_{0}^{T}\bigl(A\nabla(\hat{Y}_{hp}-Y_{hp}), \nabla e_{hp}\bigr)\,dt+\frac{1}{2}\bigl\Vert y_{0}(x)-Y_{hp}(x,0)\bigr\Vert ^{2}_{L^{2}(\Omega )} \\ &\qquad {}+ \int_{0}^{T}\bigl(\tilde{\phi}^{\prime}( \hat{Y}_{hp}) (\hat {Y}_{hp}-Y_{hp}),e_{hp} \bigr)\,dt \\ &\quad \leq C(\delta) \int_{0}^{T}\sum_{\tau} \frac{h_{\tau}^{2}}{p_{\tau }^{2}} \int_{\tau} \biggl(\hat{f}+BU_{hp}+\operatorname {div}(A\nabla \hat{Y}_{hp})-\phi(\hat {Y}_{hp})-\frac {\partial Y_{hp}}{\partial t} \biggr)^{2}\,dx\,dt \\ &\qquad {}+C(\delta) \int_{0}^{T}\sum_{e} \int_{e}\frac {h_{e}}{p_{e}}\bigl[(A\nabla \hat{Y}_{hp}) \cdot n\bigr]^{2}\,de\,dt +C(\delta)\Vert f-\hat{f}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))} \\ &\qquad {}+C(\delta ) \int_{0}^{T} \int_{\Omega}\bigl\vert A\nabla(\hat{Y}_{hp}-Y_{hp}) \bigr\vert ^{2}\,dx\,dt+C(\delta)\Vert \hat{Y}_{hp}-Y_{hp} \Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\qquad {}+\frac{1}{2}\bigl\Vert y_{0}(x)-Y_{hp}(x,0) \bigr\Vert ^{2}_{L^{2}(\Omega )}+\delta\bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \\ &\quad \leq C(\delta)\sum_{i=2}^{6} \eta_{i}^{2}+\delta\bigl\Vert y(U_{hp})-Y_{hp} \bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}. \end{aligned}$$
(3.32)
$$\begin{aligned} \bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega ))}\leq C(\delta)\sum_{i=1}^{6} \eta_{i}^{2}. \end{aligned}$$
(3.33)
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
where
\(\eta_{i}^{2}\), \(i=1,\ldots,6\)
are defined in Theorem
3.1
and Theorem
3.2.
$$\begin{aligned} &\Vert y-Y_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}+\Vert p-P_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +\Vert u-U_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega_{U}))} \\ &\quad \leq C\sum _{i=1}^{6}\eta_{i}^{2}, \end{aligned}$$
(3.34)
Proof
It follows from Theorem 3.1 and Theorem 3.2 that
Note that A is positive definite, it follows from the Poincaré inequality that
Then it follows from (3.35) and (3.36) that
Note that
and
Therefore, we obtain (3.34) from (3.16) and (3.37)-(3.41). □
$$\begin{aligned} \Vert u-U_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega_{U}))} \leq& C\eta _{1}^{2}+C\bigl\Vert \tilde{P}_{hp}-p(U_{hp}) \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))} \\ \leq& C\eta_{1}^{2}+C\Vert \tilde{P}_{hp}-P_{hp} \Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))} \\ &{}+C\bigl\Vert P_{hp}-p(U_{hp}) \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))} \\ \leq& C\sum_{i=1}^{6}\eta_{i}^{2}+C \Vert \tilde{P}_{hp}-P_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))}. \end{aligned}$$
(3.35)
$$\begin{aligned} \Vert \tilde{P}_{hp}-P_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega))} \leq& \int_{0}^{T} \int_{\Omega}\bigl\vert A^{*}\nabla( \tilde{P}_{hp}-P_{hp})\bigr\vert ^{2}\,dx\,dt \\ \leq& C\eta_{5}^{2}. \end{aligned}$$
(3.36)
$$\begin{aligned} \Vert u-U_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega_{U}))}\leq C\sum _{i=1}^{6}\eta_{i}^{2}. \end{aligned}$$
(3.37)
$$\begin{aligned}& \Vert y-Y_{hp}\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}\leq\bigl\Vert y-y(U_{hp})\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} +\bigl\Vert y(U_{hp})-Y_{hp}\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))}, \end{aligned}$$
(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)
$$\begin{aligned}& \bigl\Vert y-y(U_{hp})\bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega))} \leq C\Vert u-U_{hp}\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega_{U}))}, \end{aligned}$$
(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)
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.
Declarations
Acknowledgements
The authors express their thanks to the referees for their helpful suggestions, which led to improvements of the presentation. This work is supported by National Science Foundation of China (11201510), China Postdoctoral Science Foundation (2015M580197), Chongqing Research Program of Basic Research and Frontier Technology (cstc2015jcyjA20001), Ministry of education Chunhui projects (Z2015139) and Science and Technology Project of Wanzhou District of Chongqing (2013030050).
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.
Authors’ Affiliations
(1)
Key Laboratory of Signal and Information Processing, Chongqing Three Gorges University, Chongqing, 404000, P.R. China
(2)
Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing, 404000, P.R. China
(3)
Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin, 300222, P.R. China
(4)
Chongqing Wanzhou Long Bao Middle School, Chongqing, 404001, P.R. China
(5)
Huashang College, Guangdong University of Finance, Guangzhou, 511300, P.R. China
References
- Falk, FS: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44, 28-47 (1973) View ArticleMathSciNetMATHGoogle Scholar
- Geveci, T: On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO. Anal. Numér. 13, 313-328 (1979) MathSciNetMATHGoogle Scholar
- Chen, Y, Lu, Z, Huang, Y: Superconvergence of triangular Raviart-Thomas mixed finite element methods for bilinear constrained optimal control problem. Comput. Math. Appl. 66, 1498-1513 (2013) View ArticleMathSciNetGoogle Scholar
- Chen, Y, Yi, N, Liu, W: A Legendre-Galerkin spectral method for optimal control problems governed by elliptic equations. SIAM J. Numer. Anal. 46, 2254-2275 (2008) View ArticleMathSciNetMATHGoogle Scholar
- Lu, Z: A residual-based posteriori error estimates for hp finite element solutions of general bilinear optimal control problems. J. Math. Inequal. 9, 665-682 (2015) MathSciNetGoogle Scholar
- Hoppe, RHW, Iliash, Y, Iyyunni, C, Sweilam, NH: A posteriori error estimates for adaptive finite element discretizations of boundary control problems. J. Numer. Math. 14, 57-82 (2006) View ArticleMathSciNetMATHGoogle Scholar
- Kröner, A, Vexler, B: A priori error estimates for elliptic optimal control problems with a bilinear state equation. J. Comput. Appl. Math. 230, 781-802 (2009) View ArticleMathSciNetMATHGoogle Scholar
- Kunisch, K, Liu, W, Chang, Y, Yan, N, Li, R: Adaptive finite element approximation for a class of parameter estimation problems. J. Comput. Math. 28, 645-675 (2001) MathSciNetGoogle Scholar
- Melenk, JM, Wohlmuth, B: On residual-based a posteriori error estimation in hp-FEM. Adv. Comput. Math. 15, 311-331 (2001) View ArticleMathSciNetMATHGoogle Scholar
- Babuska, I, Suri, M: The hp-version of the finite element method with quasiuniform meshes. Modél. Math. Anal. Numér. 21, 199-238 (1987) MathSciNetMATHGoogle Scholar
- Oden, JT, Demkowicz, L, Rachowicz, W, Estermann, TAW: Toward a universal h-p adaptive finite element strategy, part 2. A posteriori error estimation. Comput. Methods Appl. Mech. Eng. 77, 113-180 (1989) View ArticleMATHGoogle Scholar
- Babuska, I, Suri, M: The p- and h-p version of the finite element method, an overview. Comput. Methods Appl. Mech. Eng. 80, 5-26 (1990) View ArticleMathSciNetMATHGoogle Scholar
- Babuska, I, Guo, B, Stephan, EP: The h-p version of the finite element method, an overview. Comput. Methods Appl. Mech. Eng. 80, 319-325 (1990) View ArticleMathSciNetMATHGoogle Scholar
- Babuska, I, Guo, B: Approximation properties of the h-p version of the finite element method. Comput. Methods Appl. Mech. Eng. 133, 319-346 (1996) View ArticleMathSciNetMATHGoogle Scholar
- Guo, B, Cao, W: An additive Schwarz method for the hp version of the finite element method in three dimensions. SIAM J. Numer. Anal. 35, 632-654 (1998) View ArticleMathSciNetMATHGoogle Scholar
- Melenk, JM: hp-interpolation of nonsmooth functions and an application to hp-a posteriori error estimation. SIAM J. Numer. Anal. 43, 127-155 (2005) View ArticleMathSciNetMATHGoogle Scholar
- Liu, W: Adaptive multi-meshes in finite element approximation of optimal control. Contemp. Math. 383, 113-132 (2005) View ArticleGoogle Scholar
- Liu, W, Yan, N: A posteriori error estimates for optimal control problems governed by parabolic equations. Numer. Math. 93, 497-521 (2003) View ArticleMathSciNetMATHGoogle Scholar
- Chen, Y, Lin, Y: A posteriori error estimates for hp finite element solutions of convex optimal control problems. J. Comput. Appl. Math. 235, 3435-3454 (2011) View ArticleMathSciNetMATHGoogle Scholar
- Lions, JL, Magenes, E: Non Homogeneous Boundary Value Problems and Applications. Springer, Berlin (1972) View ArticleGoogle Scholar
- Lions, JL: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971) View ArticleMATHGoogle Scholar
- Milner, FA: Mixed finite element methods for quasilinear second-order elliptic problems. Math. Comput. 44, 303-320 (1985) View ArticleMathSciNetMATHGoogle Scholar
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