Open Access

Residual-based a posteriori error estimates for hp finite element solutions of semilinear Neumann boundary optimal control problems

Boundary Value Problems20162016:59

https://doi.org/10.1186/s13661-016-0562-2

Received: 26 August 2015

Accepted: 15 February 2016

Published: 2 March 2016

Abstract

In this paper, we investigate residual-based a posteriori error estimates for the hp finite element approximation of semilinear Neumann boundary elliptic optimal control problems. By using the hp finite element approximation for both the state and the co-state and the hp discontinuous Galerkin finite element approximation for the control, we derive a posteriori error bounds in \(L^{2}\)-\(H^{1}\) norms for the Neumann boundary optimal control problems governed by semilinear elliptic equations. We also give \(L^{2}\)-\(L^{2}\) a posteriori error estimates for the optimal control problems. Such estimates, which are apparently not available in the literature, can be used to construct reliable adaptive finite element approximations for the semilinear Neumann boundary optimal control problems.

Keywords

residual-based a posteriori error estimates semilinear Neumann boundary elliptic optimal control problems hp finite element methods hp discontinuous Galerkin finite element methods

MSC

49J20 65N30

1 Introduction

In this paper, we study residual-based a posteriori error estimates for the hp finite element approximation of semilinear Neumann boundary optimal control problems. We consider the following semilinear elliptic optimal control problems:
$$\begin{aligned}& \min_{u \in K\subset U}\bigl\{ g(y)+j(u)\bigr\} , \end{aligned}$$
(1.1)
$$\begin{aligned}& -\operatorname{div}(A\nabla y) + \phi(y)=f,\quad \mbox{in } \Omega, \end{aligned}$$
(1.2)
$$\begin{aligned}& (A\nabla y)\cdot n|_{\partial\Omega}=u+z_{0}, \end{aligned}$$
(1.3)
where the bounded open set \(\Omega\subset{\mathbb{R}^{2}}\) is a convex polygon with the boundary Ω, \(K=\{u\in U=L^{2}(\partial\Omega):\int_{\partial\Omega}u\, dx \geq0\}\), \(f\in L^{2}(\Omega)\), \(z_{0}\in L^{2}(\partial\Omega)\), n is the outward normal on Ω. For \(1\leq p<\infty\) and m any nonnegative integer let \(W^{m,p}(\Omega)=\{v\in L^{p}(\Omega); D^{\alpha}v\in L^{p}(\Omega) \text{ if } |\alpha|\leq m\} \) denote the Sobolev spaces endowed with the norm \(\| v \|_{m,p}^{p}=\sum\limits_{|\alpha|\leq m}\| D^{\alpha}v\|_{L^{p}(\Omega)}^{p}\), and the semi-norm \(| v|_{m,p}^{p}=\sum\limits_{|\alpha|= m}\| D^{\alpha}v\|_{L^{p}(\Omega)}^{p}\). We set \(W_{0}^{m,p}(\Omega)=\{v\in W^{m,p}(\Omega): v|_{\partial\Omega}=0\}\). For \(p=2\), we denote \(H^{m}(\Omega)=W^{m,2}(\Omega)\), \(H_{0}^{m}(\Omega)=W_{0}^{m,2}(\Omega)\), and \(\|\cdot\|_{m}=\|\cdot\|_{m,2}\), \(\|\cdot\|=\|\cdot\|_{0,2}\). Furthermore, we assume that the coefficient matrix \(A(x)=(a_{i,j}(x))_{2\times2}\in (W^{1,\infty}({\Omega}))^{2\times2}\) is a symmetric positive definite matrix and there is a constant \(c>0\) satisfying for any vector \(\mathbf{X}\in\mathbb{R}^{2}\), \(\mathbf{X}^{t}A\mathbf{X}\geq c\|\mathbf{X}\|_{\mathbb{R}^{2}}^{2}\). The function \(\phi(\cdot)\in W^{1,\infty}(-R,R)\) for any \(R>0\), \(\phi^{\prime}(y)\in L^{2}(\Omega)\) for any \(y\in H^{1}(\Omega)\), and \(\phi^{\prime}\geq0\). Let g and j be strictly convex functions which are continuously differentiable on the space \(L^{2}(\partial\Omega)\), and K be a closed convex set in the control space U. We further assume that \(j(u)\rightarrow+\infty\) as \(\|u\|_{U}\rightarrow\infty\) and \(g^{\prime}(\cdot)\) is a locally Lipschitz continuous function.

Optimal control problems have attracted substantial interest in recent years due to their applications in aero-hydrodynamics, atmospheric, hydraulic pollution problems, combustion, exploration and extraction of oil and gas resources, and engineering. They must be solved successfully with efficient numerical methods. Among these numerical methods, finite element methods are a successful choice for solving the optimal control problems. 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 [312], but there are very less published results for optimal control problems by using hp finite element methods. Recently, the adaptive finite element methods have been investigated extensively and became one of the most popular methods in scientific computation. In [13], the authors studied a posteriori error estimates for adaptive finite element discretizations of boundary control problems. A posteriori error estimates and adaptive finite element approximations for parameter estimation problems have been obtained in [14, 15]. There are three main versions in adaptive finite element approximation, i.e., the p-version, the h-version, and the hp-version. The p-version of finite element methods uses a fixed mesh and improves the approximation of the solution by increasing 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, 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. There have been many theoretical studies as regards the hp finite element method in [16, 17]. An adaptive finite element approximation ensures a higher density of nodes in a certain area of the given domain, where the solution is more difficult to approximate, indicated by a posteriori error estimators. Hence it is an important approach to boost the accuracy and efficiency of finite element discretizations.

Actually, there are many h-versions of adaptive finite element methods for optimal control problems in [1820]. But for a high order element such as a hp-version of the finite element method for optimal control problems they are very few. More recently, in [21], for the constrained optimal control problem governed by linear elliptic equations, the authors have derived a posteriori error estimates for the hp finite element solutions. Inspired by the work of [21], we consider a posteriori error estimates in \(L^{2}\)-\(H^{1}\) norms and \(L^{2}\)-\(L^{2}\) norms for hp finite element solutions of general semilinear Neumann boundary optimal control problems. To the best of our knowledge for optimal control problems, these a posteriori error estimates for the general semilinear boundary optimal control problems are new.

The paper is organized as follows. In Section 2, we discuss the hp finite element approximation for the semilinear Neumann boundary optimal control problems. In Section 3, we derive both \(L^{2}\)-\(H^{1}\) a posteriori upper error bounds for the error estimates of the control, the state, and the co-state. Then we also obtain sharper a posteriori error estimates for the control approximation and error estimates in the \(L^{2}\) norm for the state and co-state on the boundary. Finally, we give a conclusion and some possible future work in Section 4.

2 Finite element methods of boundary optimal control

In this section, we study the hp finite element approximation of semilinear convex optimal control problems where the control appears in the Neumann boundary conditions. To consider the hp finite element approximation of the semilinear boundary optimal control problems, we have to give a weak formula for the state equation. Let the state space be \(V=H^{1}(\Omega)\) and \(H=L^{2}(\Omega)\). Let
$$\begin{aligned}& a(y,w)= \int_{\Omega}(A\nabla y)\cdot\nabla w\, dx,\quad \forall y,w\in V, \\& (f_{1},f_{2})= \int_{\Omega}f_{1}f_{2}\, dx,\quad \forall (f_{1},f_{2})\in H \times H, \\& (u,v)_{U}= \int_{\partial\Omega}uv\, dx,\quad \forall(u,v)\in U\times U. \end{aligned}$$
It follows from the assumptions on A that there are constants c and \(C>0\) such that
$$ a(v,v) \geq c\|v\|_{V}^{2},\qquad \bigl\vert a(v,w)\bigr\vert \leq C\|v\|_{V}\|w\|_{V},\quad \forall v,w\in V. $$
(2.1)
Then the standard weak formula for the state equation reads as follows: find \(y(u)\in V\) such that
$$ a\bigl(y(u),w\bigr)+\bigl(\phi\bigl(y(u)\bigr),w\bigr)=(f,w)+(u+z_{0},w)_{U}, \quad \forall w\in V. $$
(2.2)
Therefore, the above semilinear Neumann boundary optimal control problems can be restated as follows:
$$\begin{aligned}& \min_{u \in K\subset U}\bigl\{ g(y)+j(u)\bigr\} , \end{aligned}$$
(2.3)
$$\begin{aligned}& a(y,w)+\bigl(\phi(y),w\bigr)=(f,w)+(u+z_{0},w)_{U},\quad \forall w\in V. \end{aligned}$$
(2.4)
It is well known (see [20]) that the boundary optimal control problems (2.3)-(2.4) has a solution \((y,u)\) and that if a pair \((y,u)\) is the solution of (2.3)-(2.4), then there is a co-state \(p\in V\) such that the triplet \((y,p,u)\) satisfies the following optimality conditions:
$$\begin{aligned}& a(y,w)+\bigl(\phi(y),w\bigr)=(f,w)+(u+z_{0},w)_{U},\quad \forall w\in V, \end{aligned}$$
(2.5)
$$\begin{aligned}& a(q,p)+\bigl(\phi^{\prime}(y)p,q\bigr)=\bigl(g^{\prime}(y),q\bigr), \quad \forall q\in V, \end{aligned}$$
(2.6)
$$\begin{aligned}& \bigl(j^{\prime}(u)+ p,v-u\bigr)_{U}\geq0, \quad \forall v\in K \subset U. \end{aligned}$$
(2.7)
Now, we consider the hp finite element approximation for the boundary optimal control problem. We consider the triangulation \(\mathcal{T}\) of the set \(\Omega\subset \mathbb{R}^{2}\) which is a collection of elements \(\tau\in \mathcal{T}\) (τ is a triangle); associated with each element τ is an affine element map \(F_{\tau}:\hat{\tau}\rightarrow \tau\), where the reference element is the reference triangle \(T=\{(x,y)\in\mathbb{R}^{2} : 0< x<1,0<y<\min(x,1-x)\}\). We consider the triangulation \(\mathcal{T}\) which satisfies the standard conditions defined in [22]. We write \(h_{\tau}=\operatorname{diam}\tau\). Assume that the triangulation \(\mathcal{T}\) is γ-shape regular, i.e.,
$$ h_{\tau}^{-1}\bigl\Vert F_{\tau}^{\prime} \bigr\Vert _{L^{\infty}(\hat{\tau})}+h_{\tau}\bigl\Vert \bigl(F_{\tau}^{\prime} \bigr)^{-1}\bigr\Vert _{L^{\infty}(\hat{\tau})}\leq \gamma. $$
(2.8)
This implies (see [22]) that there exists a constant \(C>0\) that depends solely on γ such that
$$ C^{-1}h_{\tau}\leq h_{\tau^{\prime}}\leq Ch_{\tau},\quad \tau,\tau^{\prime}\in\mathcal{T} \mbox{ with } \bar{\tau}\cap\bar{\tau}^{\prime}\neq\emptyset, $$
(2.9)
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 further assume the triangulation \(\mathcal{T}\) satisfies the relation between the patch and the reference patch. Let \(\mathcal{T}_{U}\) be a partition of Ω into disjoint regular 1-simplices s, so that \(\partial\Omega=\bigcup\limits_{s\in\mathcal{T}_{U}}\bar{s}\). Associated with every s is an affine map \(F_{s}:\hat{s}\rightarrow s\), where \(\hat{s}=[-1,1]\). Assume that and \(\bar{s}^{\prime}\) have either only one common vertex or are disjoint if s and \(s^{\prime}\in\mathcal{T}_{U}\).
For each element \(\tau\in\mathcal{T}\), we denote \(\mathcal {E}(\tau)\) the set of edges of τ and by \(\mathcal{N}(\tau)\) 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 \(s\in\mathcal{T}_{U}\), we choose a polynomial degree vector \(\mathbf{p}_{2}=(p_{s})_{s\in\mathcal{T}_{U}}\) (\(p_{s}\in\mathbb{N}\)). \(\mathcal{N}(\mathcal{T})\) 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 (\(V\in\mathcal{N}(\mathcal{T})\), \(e\in\mathcal {E}(\mathcal{T})\)):
$$\begin{aligned}& \mathcal {N}(e)=\bigl\{ V\in\mathcal{N}(\mathcal{T}):V\in\bar{e}\bigr\} ,\qquad w_{V}=\bigl\{ x\in\Omega:x\in\bar{\tau} \mbox{ and } \bar{\tau}\cap\{V \}\neq\emptyset\bigr\} ^{0}, \\& w^{1}_{e}=\bigcup_{V\in\mathcal{N}(e)}w_{V}, \qquad w^{1}_{\tau}=\bigcup_{V\in\mathcal{N}(\tau)}w_{V}, \qquad p_{e}=\max\bigl\{ p_{\tau}: e\in\mathcal {E}(\tau)\bigr\} , \end{aligned}$$
where \(\chi^{0}\) denotes the interior of the set χ. We denote by \(h_{e}\) (\(h_{s}\)) the length of the edge e (s). Additionally, c or C denotes a general positive constant independent of \(h_{\tau}\), \(p_{\tau}\), \(h_{e}\), \(p_{e}\), \(h_{s}\), and \(p_{s}\).
Next, we define the hp-FEM space \(S^{\mathbf{p}_{1}}(\mathcal {T})\subset H^{1}(\Omega)\) and the hp-DGFEM space \(U^{\mathbf{p}_{2}}(\mathcal{T}_{U})\subset L^{2}(\partial\Omega)\) by
$$\begin{aligned}& S^{\mathbf{p}_{1}}(\mathcal{T})=\bigl\{ v\in C(\Omega):v|_{\tau}\circ F_{\tau}\in P_{p_{\tau}}(\hat{\tau})\bigr\} , \\& U^{\mathbf{p}_{2}}(\mathcal{T}_{U})=\bigl\{ v\in L^{2}( \partial\Omega):v|_{s}\circ F_{s}\in P_{p_{s}}( \hat{s})\bigr\} , \end{aligned}$$
where \({P}_{p_{\tau}}(\hat{\tau}):=\operatorname{span}\{ x^{i}y^{j}:0\leq i+j\leq p_{\tau}\}\), \({P}_{p_{s}}(\hat{s}):=\operatorname{span}\{x^{i}:0\leq i\leq p_{s}\}\). We assume that the polynomial degree vector \(\mathbf{p}_{1}\) satisfies
$$ \gamma^{-1}p_{\tau}\leq p_{\tau^{\prime}}\leq \gamma p_{\tau},\quad \tau,\tau^{\prime}\in\mathcal{T} \mbox{ with } \bar{\tau}\cap\bar{\tau}^{\prime}\neq\emptyset. $$
(2.10)
Let \(K_{hp}=K\cap U^{\mathbf{p}_{2}}(\mathcal{T}_{U})\) and \(V_{hp}=S^{\mathbf{p}_{1}}(\mathcal{T})\), then for the finite element approximation of (2.3)-(2.4):
$$\begin{aligned}& \min_{u_{hp}\in K_{hp}}\bigl\{ g(y_{hp})+j(u_{hp})\bigr\} , \end{aligned}$$
(2.11)
$$\begin{aligned}& a(y_{hp},w_{hp})+\bigl(\phi (y_{hp}),w \bigr)=(f,w_{hp})+(u_{hp}+z_{0},w_{hp})_{U}, \quad \forall w_{hp}\in V_{hp}. \end{aligned}$$
(2.12)
It is well known that the boundary optimal control problem (2.11)-(2.12) has a solution \((y_{hp},u_{hp})\) and that if a pair \((y_{hp},u_{hp})\in V_{hp}\times K_{hp}\) is the solution of (2.11)-(2.12), 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:
$$\begin{aligned}& a(y_{hp},w_{hp})+\bigl(\phi(y_{hp}),w_{hp} \bigr)=(f,w_{hp})+(u_{hp}+z_{0},w_{hp})_{U}, \quad \forall w_{hp}\in V_{hp}\subset V, \end{aligned}$$
(2.13)
$$\begin{aligned}& a(q_{hp},p_{hp})+\bigl(\phi^{\prime}(y_{hp})p_{hp},q_{hp} \bigr)=\bigl(g^{\prime }(y_{hp}),q_{hp}\bigr), \quad \forall q_{hp}\in V_{hp}\subset V, \end{aligned}$$
(2.14)
$$\begin{aligned}& \bigl(j^{\prime}(u_{hp})+ p_{hp},v_{hp}-u_{hp} \bigr)_{U}\geq0, \quad \forall v_{hp}\in K_{hp} \subset U. \end{aligned}$$
(2.15)
The following lemmas are important in deriving hp a posteriori error estimates of residual type.

Lemma 2.1

There exist a constant \(C>0\) independent of v, \(h_{s}\), and \(p_{s}\) and a mapping \(\pi_{p_{s}}^{h_{s}}:H^{1}(s)\rightarrow P_{p_{s}}(s)\) such that \(\forall v\in H^{1}(s)\), \(s\in\mathcal{T}_{U}\) the following inequality is valid:
$$ \bigl\Vert v-\pi_{p_{s}}^{h_{s}}v\bigr\Vert _{L^{2}(s)} \leq C\frac{h_{s}}{p_{s}}|v|_{H^{1}(s)}, $$
where \(P_{p_{s}}(s):=\operatorname{span}\{x^{i}y^{j}:0\leq i+j\leq p_{s}\}\).

Proof

It follows easily from Proposition A.2 in [22] and the scaling argument. □

Lemma 2.2

[22]

Let \(\mathbf{p}_{1}\) be an arbitrary polynomial degree distribution satisfies (2.10). Then there exists a linear operator \(E_{1}:H^{1}(\Omega)\rightarrow S^{\mathbf{p}_{1}}(\mathcal{T})\), and there exists a constant \(C>0\) depending solely on γ such that for every \(v\in H^{1}(\Omega)\) and all elements \(\tau\in\mathcal{T}\) and all edges \(e\in\mathcal{E}(\mathcal{T})\),
$$\begin{aligned}& \|v-E_{1}v\|_{L^{2}(\tau)}+\frac {h_{\tau}}{p_{\tau}}\bigl\Vert \nabla(v-E_{1}v)\bigr\Vert _{L^{2}(\tau)}\leq C \frac{h_{\tau}}{p_{\tau}}\|\nabla v\|_{L^{2}(w_{\tau}^{1})}, \end{aligned}$$
(2.16)
$$\begin{aligned}& \|v-E_{1}v\|_{L^{2}(e)}\leq C \biggl( \frac{h_{e}}{p_{e}} \biggr)^{\frac{1}{2}}\|\nabla v\|_{L^{2}(w_{e}^{1})}. \end{aligned}$$
(2.17)

Lemma 2.3

Let \(\mathbf{p}_{1}\) be an arbitrary polynomial degree distribution satisfying (2.10) and \(p_{\tau}\geq2\), \(\forall \tau\in\mathcal{T}\). Then there exists a bounded linear operator \(E_{2}:H^{2}(\Omega)\rightarrow S^{\mathbf{p}_{1}}(\mathcal{T})\), and there exists a constant \(C>0\) that depends solely on γ such that for every \(v\in H^{2}(\Omega)\) and all elements \(\tau\in\mathcal {T}\) and all edges \(e\in\mathcal{E}(\mathcal{T})\),
$$\begin{aligned}& \|v-E_{2}v\|_{L^{2}(\tau)}+\frac{h_{\tau}}{p_{\tau }}\bigl\Vert \nabla(v-E_{2}v)\bigr\Vert _{L^{2}(\tau)}\leq C \biggl( \frac{h_{\tau}}{p_{\tau}} \biggr)^{2}|v|_{H^{2}(w_{\tau }^{1})}, \end{aligned}$$
(2.18)
$$\begin{aligned}& \|v-E_{2}v\|_{L^{2}(e)}\leq C \biggl(\frac{h_{e}}{p_{e}} \biggr)^{\frac{3}{2}}|v|_{H^{2}(w_{e}^{1})}. \end{aligned}$$
(2.19)
For \(\varphi\in W_{h}\), we shall write
$$ \phi(\varphi)-\phi(\rho)=-\tilde{\phi}^{\prime}(\varphi) ( \rho -\varphi)=-\phi^{\prime}(\rho) (\rho-\varphi) +\tilde{ \phi}^{\prime\prime}(\varphi) (\rho-\varphi)^{2}, $$
(2.20)
where
$$\begin{aligned}& \tilde{\phi}^{\prime}(\varphi)= \int_{0}^{1}\phi^{\prime}\bigl(\varphi +s( \rho-\varphi)\bigr)\, ds, \\& \tilde{\phi}^{\prime\prime}(\varphi)= \int_{0}^{1}(1-s)\phi^{\prime \prime}\bigl(\rho+s( \varphi-\rho)\bigr)\, ds \end{aligned}$$
are bounded functions in Ω̄ [23].

3 Residual-based a posteriori error estimators

In this section, we discuss residual-based a posteriori error estimates for the semilinear Neumann boundary optimal control problems. First of all, we use the \(L^{2}\) norm for estimating the control approximation error on the boundary, and the \(H^{1}\) norm for the state and co-state approximation error on the domain. For simplicity of presentation, let
$$ J(u)=g\bigl(y(u)\bigr)+j(u),\qquad J_{hp}(u_{hp})=g \bigl(y(u_{hp})\bigr)+j(u_{hp}). $$
Then the optimal control problems of (2.3) and (2.11) read
$$ \min_{u\in K}\bigl\{ J(u)\bigr\} $$
(3.1)
and
$$ \min_{u_{hp}\in K_{hp}}\bigl\{ J_{hp}(u_{hp})\bigr\} . $$
(3.2)
It can be shown that
$$\begin{aligned}& \bigl(J^{\prime}(u),v\bigr)_{U}=\bigl(j^{\prime}(u)+ p,v \bigr)_{U}, \\& \bigl(J^{\prime}_{hp}(u_{hp}),v\bigr)_{U}= \bigl(j^{\prime }(u_{hp})+p_{hp},v\bigr)_{U}, \\& \bigl(J^{\prime}(u_{hp}),v\bigr)_{U}= \bigl(j^{\prime }(u_{hp})+p(u_{hp}),v \bigr)_{U}, \end{aligned}$$
where \(p(u_{hp})\) is the solution of the auxiliary equations:
$$\begin{aligned}& a\bigl(y(u_{hp}),w\bigr)+\bigl(\phi \bigl(y(u_{hp})\bigr),w \bigr)=(f,w)+(u_{hp}+z_{0},w)_{U},\quad \forall w \in V, \end{aligned}$$
(3.3)
$$\begin{aligned}& a\bigl(q,p(u_{hp})\bigr)+\bigl(\phi^{\prime} \bigl(y(u_{hp})\bigr)p(u_{hp}) ,q\bigr)=\bigl(g^{\prime} \bigl(y(u_{hp})\bigr),q\bigr), \quad \forall q\in V. \end{aligned}$$
(3.4)
In order to estimate the control u, we introduce the \(L^{2}(\partial\Omega)\)-projection of u into \(U^{\mathbf {p}_{2}}(\mathcal{T}_{U})\), i.e., let \(P_{hp}u\in U^{\mathbf{p}_{2}}(\mathcal {T}_{U})\) be the function defined by
$$ (u-P_{hp}u,w_{hp})_{U}=0,\quad \forall w_{hp}\in U^{\mathbf{p}_{2}}(\mathcal{T}_{U}). $$
(3.5)

Theorem 3.1

Let \((y,u)\) and \((y_{hp},u_{hp})\) be the solutions of (2.3)-(2.4) and (2.11)-(2.12). Let p and \(p_{hp}\) be the solutions of the co-state equations (2.6) and (2.14), respectively. Assume that
$$ \bigl(J^{\prime}(u)-J^{\prime}(v),u-v\bigr)_{U}\geq c \|u-v\|_{L^{2}(\Omega_{U})}^{2},\quad \forall u,v\in U. $$
(3.6)
Moreover, we assume \(j^{\prime}(u_{hp})+p_{hp}\in H^{1}(\Omega)\). Then we have
$$ \|u-u_{hp}\|_{L^{2}(\partial\Omega)}^{2}\leq C\eta_{1}^{2}+C \bigl\Vert p_{hp}-p(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega )}^{2}, $$
(3.7)
where
$$ \eta_{1}^{2}=\sum_{s} \frac {h_{s}^{2}}{p_{s}^{2}}\bigl\vert j^{\prime}(u_{hp})+ p_{hp} \bigr\vert _{H^{1}(s)}^{2}, $$
and \(p(u_{hp})\) is the solution of the system (3.3)-(3.4).

Proof

It follows from (2.7), (2.15), and (3.6) that
$$\begin{aligned}& c\|u-u_{hp}\|_{L^{2}(\partial\Omega)}^{2} \\& \quad\leq \bigl(J^{\prime }(u),u-u_{hp}\bigr)_{U}- \bigl(J^{\prime}(u_{hp}),u-u_{hp}\bigr)_{U} \\& \quad \leq -\bigl(J^{\prime}(u_{hp}),u-u_{hp} \bigr)_{U} \\& \quad \leq -\bigl(J^{\prime}(u_{hp}),u-u_{hp} \bigr)_{U}+\bigl(j^{\prime }(u_{hp})+p_{hp},v_{hp}-u_{hp} \bigr)_{U} \\& \quad = \bigl(J^{\prime}_{hp}(u_{hp}),u_{hp}-u \bigr)_{U}+\bigl(J^{\prime }_{hp}(u_{hp})-J^{\prime}(u_{hp}),u-u_{hp} \bigr)_{U} \\& \qquad {} +\bigl(j^{\prime}(u_{hp})+p_{hp},v_{hp}-u_{hp} \bigr)_{U} \\& \quad = \bigl(j^{\prime}(u_{hp})+p_{hp},u_{hp}-u \bigr)_{U}+\bigl( p_{hp}-p(u_{hp}),u-u_{hp} \bigr)_{U} \\& \qquad {} +\bigl(j^{\prime}(u_{hp})+p_{hp},v_{hp}-u_{hp} \bigr)_{U} \\& \quad = \bigl(j^{\prime}(u_{hp})+ p_{hp},v_{hp}-u \bigr)_{U}+\bigl(p_{hp}-p(u_{hp}),u-u_{hp} \bigr)_{U} \\& \quad \leq \bigl(j^{\prime}(u_{hp})+ p_{hp},v_{hp}-u \bigr)_{U}+C\bigl\Vert p_{hp}-p(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega)}^{2} \\& \qquad {}+\frac{c}{4}\|u-u_{hp}\|_{L^{2}(\partial\Omega)}^{2}. \end{aligned}$$
(3.8)
Setting \(w_{hp}=1\) in (3.5), we have \(\int_{\partial\Omega}P_{hp}u=\int_{\partial\Omega}u\geq0\). Thus, we have \(P_{hp}u\in K_{hp}\). Let \(v_{hp}=P_{hp}u\in K_{hp}\). It follows from (3.5) and Lemma 2.1 that
$$\begin{aligned}& \bigl(j^{\prime}(u_{hp})+ p_{hp},v_{hp}-u \bigr)_{U} \\& \quad = \bigl(j^{\prime}(u_{hp})+ p_{hp},P_{hp}u-u \bigr)_{U} \\& \quad = \sum_{s}\bigl(j^{\prime}(u_{hp})+ p_{hp}-\pi_{p_{s}}^{h_{s}}\bigl(j^{\prime }(u_{hp})+ p_{hp}\bigr),P_{hp}u-u\bigr)_{s} \\& \quad \leq C\sum_{s}\frac{h_{s}}{p_{s}}\bigl\vert j^{\prime}(u_{hp})+ p_{hp}\bigr\vert _{H^{1}(s)} \|P_{hp}u-u\|_{L^{2}(s)} \\& \quad \leq C\sum_{s}\frac{h_{s}^{2}}{p_{s}^{2}}\bigl\vert j^{\prime}(u_{hp})+ p_{hp}\bigr\vert _{H^{1}(s)}^{2} +\frac{c}{4}\|P_{hp}u-u \|_{L^{2}(\partial\Omega)}^{2} \\& \quad \leq C\sum_{s}\frac{h_{s}^{2}}{p_{s}^{2}}\bigl\vert j^{\prime}(u_{hp})+ p_{hp}\bigr\vert _{H^{1}(s)}^{2} +\frac{c}{4}\|u-u_{hp} \|_{L^{2}(\partial\Omega)}^{2}. \end{aligned}$$
(3.9)
By using (3.8) and (3.9), we have
$$ \|u-u_{hp}\|_{L^{2}(\partial\Omega)}^{2} \leq C\sum _{s}\frac{h_{s}^{2}}{p_{s}^{2}}\bigl\vert j^{\prime}(u_{hp})+ p_{hp}\bigr\vert _{H^{1}(s)}^{2} +C\bigl\Vert p_{hp}-p(u_{hp}) \bigr\Vert _{L^{2}(\partial\Omega)}^{2}. $$
(3.10)
Then (3.7) follows from (3.10). □

In the following theorem we estimate \(\|p_{hp}-p(u_{hp})\|_{H^{1}(\Omega)}^{2}\) and then obtain the desired hp a posteriori error estimates.

Theorem 3.2

Let \((y,p,u)\) and \((y_{hp},p_{hp},u_{hp})\) be the solutions of (2.5)-(2.7) and (2.13)-(2.15), respectively. Assume that all the conditions in Theorem  3.1 hold. Then we have
$$ \|u-u_{hp}\|_{L^{2}(\partial\Omega)}^{2}+\|y-y_{hp}\| _{H^{1}(\Omega)}^{2} +\|p-p_{hp}\|_{H^{1}(\Omega)}^{2} \leq C\sum_{i=1}^{7}{\eta}_{i}^{2}, $$
(3.11)
where \(\eta_{1}^{2}\) is defined in Theorem  3.1 and
$$\begin{aligned}& {\eta}_{2}^{2}=\sum_{\tau} \int_{\tau }\frac{h_{\tau}^{2}}{p_{\tau}^{2}}\bigl(\operatorname{div}(A\nabla p_{hp})-\phi^{\prime}(y_{hp})p_{hp} \bigr)^{2}, \\& {\eta}_{3}^{2}=\sum_{e\subset\partial\Omega} \int _{e}\frac{h_{e}}{p_{e}}\bigl(A\nabla p_{hp}\cdot n-g^{\prime}(y_{hp})\bigr)^{2}, \\& {\eta}_{4}^{2}=\sum_{e\cap\partial\Omega=\emptyset} \int _{e}\frac{h_{e}}{p_{e}}\bigl[(A\nabla p_{hp}\cdot n)\bigr]^{2}, \\& {\eta}_{5}^{2}=\sum_{\tau} \int_{\tau}\frac{h_{\tau }^{2}}{p_{\tau}^{2}}\bigl(f+\operatorname{div}(A\nabla y_{hp})-\phi(y_{hp})\bigr)^{2}, \\& {\eta}_{6}^{2}=\sum_{e\subset\partial\Omega} \int _{e}\frac{h_{e}}{p_{e}}(A\nabla y_{hp}\cdot n-u_{hp}-z_{0})^{2}, \\& {\eta}_{7}^{2}=\sum_{e\cap\partial\Omega=\emptyset} \int _{e}\frac{h_{e}}{p_{e}}\bigl[(A\nabla y_{hp}\cdot n)\bigr]^{2}. \end{aligned}$$

Proof

Let \(e_{p}=p_{hp}-p(u_{hp})\) and \(E_{1}\) be the linear operator defined in Lemma 2.2, we have
$$\begin{aligned} c\|e_{p}\|_{H^{1}(\Omega )}^{2} \leq& a(e_{p},e_{p})+\bigl(\phi^{\prime}\bigl(y(u_{hp}) \bigr)e_{p},e_{p}\bigr) \\ =&a(e_{p}-E_{1}e_{p},e_{p})+a(E_{1}e_{p},e_{p})+ \bigl(\phi^{\prime }(y_{hp})p_{hp}-\phi^{\prime} \bigl(y(u_{hp})\bigr)p(u_{hp}),e_{p}\bigr) \\ &{}-\bigl(\bigl(\phi^{\prime}(y_{hp})-\phi^{\prime } \bigl(y(u_{hp})\bigr)\bigr)p_{hp},e_{p}\bigr) \\ =&a(e_{p}-E_{1}e_{p},e_{p})+\bigl( \phi^{\prime}(y_{hp})p_{hp}-\phi^{\prime } \bigl(y(u_{hp})\bigr)p(u_{hp}),e_{p}-E_{1}e_{p} \bigr) \\ &{}+a(E_{1}e_{p},e_{p})+\bigl( \phi^{\prime}(y_{hp})p_{hp}-\phi^{\prime } \bigl(y(u_{hp})\bigr)p(u_{hp}),E_{1}e_{p} \bigr) \\ &{}-\bigl(\bigl(\phi^{\prime }(y_{hp})-\phi ^{\prime} \bigl(y(u_{hp})\bigr)\bigr)p_{hp},e_{p}\bigr) \\ =&\sum_{\tau} \int_{\tau}A\nabla \bigl(p_{hp}-p(u_{hp}) \bigr)\cdot\nabla(e_{p}-E_{1}e_{p}) \\ &{}+\bigl(\phi^{\prime}(y_{hp})p_{hp}- \phi^{\prime }\bigl(y(u_{hp})\bigr)p(u_{hp}), e_{p}-E_{1}e_{p}\bigr) \\ &{}+\bigl(g^{\prime }(y_{hp})-g^{\prime} \bigl(y(u_{hp})\bigr),E_{1}e_{p}\bigr)-\bigl(\bigl( \phi^{\prime }(y_{hp})-\phi^{\prime}\bigl(y(u_{hp}) \bigr)\bigr)p_{hp},e_{p}\bigr). \end{aligned}$$
(3.12)
It follows from (2.1), (2.14), (2.16)-(2.17), and (3.12) that
$$\begin{aligned} c\|e_{p}\|_{H^{1}(\Omega)}^{2} \leq&\sum _{\tau} \int_{\tau}\bigl(-\operatorname{div}(A\nabla p_{hp})+ \phi^{\prime}(y_{hp})p_{hp}\bigr) (e_{p}-E_{1}e_{p})- \bigl(g^{\prime }\bigl(y(u_{hp})\bigr),e_{p}-E_{1}e_{p} \bigr) \\ &{}+\sum_{\tau} \int_{\partial\tau}(A\nabla p_{hp})\cdot n(e_{p}-E_{1}e_{p}) \\ &{}+\bigl(g^{\prime}(y_{hp})-g^{\prime} \bigl(y(u_{hp})\bigr),E_{1}e_{p}\bigr)-\bigl(\bigl( \phi ^{\prime}(y_{hp})-\phi^{\prime}\bigl(y(u_{hp}) \bigr)\bigr)p_{hp},e_{p}\bigr) \\ =&\sum_{\tau} \int_{\tau}\bigl(-\operatorname{div}(A\nabla p_{hp})+ \phi^{\prime}(y_{hp})p_{hp}\bigr) (e_{p}-E_{1}e_{p})+ \bigl(g^{\prime }(y_{hp})-g^{\prime}\bigl(y(u_{hp}) \bigr),e_{p}\bigr) \\ &{}+\sum_{\tau} \int_{\partial\tau}(A\nabla p_{hp})\cdot n(e_{p}-E_{1}e_{p}) -\bigl(\bigl(\phi^{\prime}(y_{hp})-\phi^{\prime } \bigl(y(u_{hp})\bigr)\bigr)p_{hp},e_{p}\bigr) \\ &{}+\sum_{e\cap\partial\Omega=\emptyset} \int_{e}\bigl[(A\nabla p_{hp}\cdot n) \bigr](e_{p}-E_{1}e_{p}) \\ &{}+\sum_{e\subset\partial\Omega} \int_{e}\bigl(A\nabla p_{hp}\cdot n-g^{\prime}(y_{hp})\bigr) (e_{p}-E_{1}e_{p}) \\ \leq& C\sum_{\tau}\frac{h_{\tau}^{2}}{p_{\tau}^{2}} \int_{\tau }\bigl(\operatorname{div}(A\nabla p_{hp})- \phi^{\prime}(y_{hp})p_{hp}\bigr)^{2}+C\sum _{e\cap \partial\Omega=\emptyset}\frac{h_{e}}{p_{e}} \int_{e}\bigl[(A\nabla p_{hp}\cdot n) \bigr]^{2} \\ &{}+C\sum_{e\subset\partial\Omega}\frac{h_{e}}{p_{e}} \int _{e}\bigl(A\nabla p_{hp}\cdot n-g^{\prime}(y_{hp})\bigr)^{2} \\ &{}+C\bigl\Vert \phi^{\prime}(y_{hp})-\phi^{\prime} \bigl(y(u_{hp})\bigr)\bigr\Vert _{L^{2}(\partial\Omega)}^{2} \\ &{}+C\bigl\Vert g^{\prime}(y_{hp})-g^{\prime } \bigl(y(u_{hp})\bigr)\bigr\Vert _{L^{2}(\partial\Omega)}^{2}+ \frac{c}{2}\|e_{p}\| _{H^{1}(\Omega)}^{2}. \end{aligned}$$
(3.13)
Therefore, noting that \(g^{\prime}\) is locally Lipschitz continuous, we have
$$ \bigl\Vert p_{hp}-p(u_{hp})\bigr\Vert _{H^{1}(\Omega)}^{2}\leq C\sum_{i=2}^{4}{ \eta}_{i}^{2}+C\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\partial \Omega)}^{2}. $$
(3.14)
Similarly, it can be proved that
$$ \bigl\Vert y_{hp}-y(u_{hp})\bigr\Vert _{H^{1}(\Omega)}^{2}\leq C\sum_{i=5}^{7}{ \eta}_{i}^{2}. $$
(3.15)
It follows from (3.14), (3.15), and the trace theorem that
$$\begin{aligned} \bigl\Vert p_{hp}-p(u_{hp})\bigr\Vert _{H^{1}(\Omega)}^{2}&\leq C\sum_{i=2}^{4}{ \eta}_{i}^{2}+C\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\partial \Omega)}^{2} \\ &\leq C\sum_{i=2}^{4}{ \eta}_{i}^{2}+C\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{H^{1}(\Omega )}^{2} \\ &\leq C\sum_{i=2}^{7}{ \eta}_{i}^{2} . \end{aligned}$$
(3.16)
Note that
$$\begin{aligned}& \Vert p-p_{hp}\Vert _{H^{1}(\Omega)}\leq\bigl\Vert p-p(u_{hp})\bigr\Vert _{H^{1}(\Omega)}+\bigl\Vert p_{hp}-p(u_{hp})\bigr\Vert _{H^{1}(\Omega)}, \end{aligned}$$
(3.17)
$$\begin{aligned}& \Vert y-y_{hp}\Vert _{H^{1}(\Omega)}\leq\bigl\Vert y-y(u_{hp})\bigr\Vert _{H^{1}(\Omega)}+\bigl\Vert y_{hp}-y(u_{hp})\bigr\Vert _{H^{1}(\Omega)}, \end{aligned}$$
(3.18)
and
$$\begin{aligned}& \bigl\Vert y-y(u_{hp})\bigr\Vert _{H^{1}(\Omega)}\leq C\Vert u-u_{hp}\Vert _{L^{2}(\partial\Omega)}, \end{aligned}$$
(3.19)
$$\begin{aligned}& \bigl\Vert p-p(u_{hp})\bigr\Vert _{H^{1}(\Omega)}\leq C\bigl\Vert y-y(u_{hp})\bigr\Vert _{H^{1}(\Omega)}\leq C\Vert u-u_{hp} \Vert _{L^{2}(\partial\Omega)} . \end{aligned}$$
(3.20)
Combining (3.7), (3.14)-(3.16), and (3.17)-(3.20), we derive
$$\begin{aligned}& \Vert u-u_{hp}\Vert _{L^{2}(\partial\Omega)}^{2}+\Vert y-y_{hp}\Vert _{H^{1}(\Omega)}^{2} +\Vert p-p_{hp}\Vert _{H^{1}(\Omega)}^{2} \\& \quad \leq \Vert u-u_{hp}\Vert _{L^{2}(\partial\Omega)}^{2}+\bigl\Vert y-y(u_{hp})\bigr\Vert _{H^{1}(\Omega)}^{2} \\& \qquad {} +\bigl\Vert y_{hp}-y(u_{hp})\bigr\Vert _{H^{1}(\Omega)}^{2} +\bigl\Vert p-p(u_{hp})\bigr\Vert _{H^{1}(\Omega)}^{2}+\bigl\Vert p_{hp}-p(u_{hp}) \bigr\Vert _{H^{1}(\Omega )}^{2} \\& \quad \leq \Vert u-u_{hp}\Vert _{L^{2}(\partial\Omega)}^{2}+\bigl\Vert y-y(u_{hp})\bigr\Vert _{H^{1}(\Omega)}^{2}+\bigl\Vert p-p(u_{hp})\bigr\Vert _{H^{1}(\Omega)}^{2} \\& \quad \leq C\sum_{i=1}^{7}{ \eta}_{i}^{2}. \end{aligned}$$
(3.21)
Then we have proved (3.11). □

Next, we shall derive sharper a posteriori error estimates for the control approximation and error estimates in the \(L^{2}\) norm for the state and co-state on the boundary. We introduce a subset of Ω: \(\Omega_{d}=\{\tau\in\mathcal {T}:\bar{\tau}\cap\bar{\Omega}_{d}^{-}\neq\emptyset\}\), where \(\Omega_{d}^{-}=\{x\in\Omega:\operatorname{dist}(x,\partial\Omega)< d\}\) and d is a constant independent of \(h_{\tau}\), \(p_{\tau}\), \(h_{e}\), and \(p_{e}\). Then we have the following improved residual-based a posteriori error estimates.

Theorem 3.3

Let \((y,p,u)\) and \((y_{hp},p_{hp},u_{hp})\) be the solutions of (2.5)-(2.7) and (2.13)-(2.15), respectively. Assume that \(j^{\prime}(u_{hp})+p_{hp}\in H^{1}(\Omega)\) and \((J^{\prime}(u)-J^{\prime}(v),u-v)_{U}\geq c\|u-v\|_{L^{2}(\Omega_{U})}^{2}\), \(\forall u,v\in U\). Moreover, \(p_{\tau}\geq2\), \(\forall\tau\in\mathcal{T}\) and \(g^{\prime}(\cdot)\) is locally Lipschitz continuous. Then
$$ \|u-u_{hp}\|_{L^{2}(\partial\Omega)}^{2}+\|y-y_{hp}\| _{L^{2}(\partial\Omega)}^{2}+\|p-p_{hp}\|_{L^{2}(\partial\Omega )}^{2} \leq C\sum_{i=1}^{13}\kappa_{i}^{2}, $$
(3.22)
where
$$\begin{aligned}& \kappa_{1}^{2}=\sum_{s} \frac{h_{s}^{2}}{p_{s}^{2}}\bigl|j^{\prime}(u_{hp})+ p_{hp}\bigr|_{H^{1}(s)}^{2}, \\& \kappa_{2}^{2}=\sum_{\tau\subset\Omega _{d}} \int_{\tau}\frac{h_{\tau}^{2}}{p_{\tau}^{2}}\bigl(\operatorname {div}(A\nabla p _{hp})-\phi^{\prime}(y_{hp})p_{hp} \bigr)^{2}, \\& \kappa_{3}^{2}=\sum_{e\cap\partial\Omega =\emptyset ,e\in\Omega _{d}} \int_{e}\frac{h_{e}}{p_{e}}\bigl[(A\nabla p _{hp}\cdot n)\bigr]^{2}, \\& \kappa_{4}^{2}=\sum_{e\subset\partial\Omega } \int _{e}\frac{h_{e}}{p_{e}}\bigl(A\nabla p _{hp}\cdot n-g^{\prime}(y_{hp})\bigr)^{2}, \\& \kappa_{5}^{2}=\sum_{\tau} \int_{\tau}\frac{h_{\tau }^{4}}{p_{\tau}^{4}}\bigl(\operatorname{div}(A\nabla p _{hp})-\phi^{\prime}(y_{hp})p_{hp} \bigr)^{2}, \\& \kappa_{6}^{2}=\sum_{e\cap\partial\Omega =\emptyset } \int_{e}\frac{h_{e}^{3}}{p_{e}^{3}}\bigl[(A\nabla p _{hp} \cdot n)\bigr]^{2}, \\& \kappa_{7}^{2}=\sum_{e\subset\partial\Omega} \int _{e}\frac{h_{e}^{3}}{p_{e}^{3}}\bigl(A\nabla p _{hp} \cdot n-g^{\prime}(y_{hp})\bigr)^{2}, \end{aligned}$$
and
$$\begin{aligned}& \kappa_{8}^{2}=\sum_{\tau\subset\Omega _{d}} \int_{\tau}\frac{h_{\tau}^{2}}{p_{\tau}^{2}}\bigl(f+\operatorname {div}(A\nabla y_{hp})-\phi(y_{hp})\bigr)^{2}, \\& \kappa_{9}^{2}=\sum_{e\cap\partial\Omega =\emptyset ,e\in\Omega _{d}} \int_{e}\frac{h_{e}}{p_{e}}\bigl[(A\nabla y _{hp}\cdot n)\bigr]^{2}, \\& \kappa_{10}^{2}=\sum_{e\subset\partial\Omega } \int _{e}\frac{h_{e}}{p_{e}}(A\nabla y _{hp}\cdot n-u_{hp}-z_{0})^{2}, \\& \kappa_{11}^{2}=\sum_{\tau} \int_{\tau}\frac{h_{\tau }^{4}}{p_{\tau}^{4}}\bigl(f+\operatorname{div}(A\nabla y_{hp})-\phi(y_{hp})\bigr)^{2}, \\& \kappa_{12}^{2}=\sum_{e\cap\partial\Omega =\emptyset } \int_{e}\frac{h_{e}^{3}}{p_{e}^{3}}\bigl[(A \nabla y _{hp} \cdot n)\bigr]^{2}, \\& \kappa_{13}^{2}=\sum_{e\subset\partial\Omega} \int _{e}\frac{h_{e}^{3}}{p_{e}^{3}}(A \nabla y _{hp}\cdot n-u_{hp}-z_{0})^{2}. \end{aligned}$$

Proof

For the proof of this theorem, we estimate (3.22) in the following five parts, respectively.

Part I. First, we estimate \(\|p_{hp}-p(u_{hp})\|_{L^{2}(\partial\Omega)}\). Let \(e_{p}=p_{hp}-p(u_{hp})\) and \(e_{y}=y_{hp}-y(u_{hp})\), then there is some \(\xi\in C^{\infty}(\Omega_{d}^{-})\) satisfying \(\xi=0\) on \(\partial\Omega_{d}^{-}\backslash \partial\Omega\) and \(\xi=1\) on Ω. It follows from the trace theorem that
$$ \|e_{p}\|_{L^{2}(\partial\Omega)}^{2}=\|\xi e_{p} \|_{L^{2}(\partial\Omega)}^{2}\leq C\|\xi e_{p}\|_{H^{1}(\Omega)}^{2}. $$
(3.23)
By using the assumption of A, we have
$$ \int_{\Omega}A\nabla(\xi e_{p})\nabla(\xi e_{p})= \int_{\Omega}A\nabla e_{p}\nabla\bigl(\xi^{2} e_{p}\bigr)+ \int_{\Omega}(e_{p})^{2}A\nabla\xi\nabla\xi. $$
(3.24)
Let \(v^{p}=\xi^{2} e_{p}\), and let \(E_{1}\) be the linear operator defined in Lemma 2.2. It follows from (2.1), (2.14), (3.4), and (3.24) that
$$\begin{aligned} c\|\xi e_{p}\|_{H^{1}(\Omega)}^{2} \leq& a(\xi e_{p},\xi e_{p})=a\bigl(v^{p},e_{p} \bigr)+ \int_{\Omega}(e_{p})^{2}A\nabla\xi\nabla\xi \\ =& a\bigl(v^{p}-E_{1}v^{p},e_{p} \bigr)+a\bigl(E_{1}v^{p},e_{p}\bigr)+ \int_{\Omega }(e_{p})^{2}A\nabla\xi\nabla\xi \\ =& \sum_{\tau} \int_{\tau}\bigl(A\nabla\bigl(p_{hp}-p(u_{hp}) \bigr)\bigr) \cdot \nabla \bigl(v^{p}-E_{1}v^{p} \bigr) \\ &{}+ \int_{\partial\Omega}\bigl(g^{\prime}(y_{hp})-g^{\prime } \bigl(y(u_{hp})\bigr)\bigr)E_{1}v^{p} \\ &{}-\bigl(\phi^{\prime}(y_{hp})p_{hp}- \phi^{\prime }\bigl(y(u_{hp})\bigr)p(u_{hp}),E_{1}v^{p} \bigr)+ \int _{\Omega}(e_{p})^{2}A\nabla\xi\nabla\xi \\ =& \sum_{\tau} \int_{\tau}\bigl(-\operatorname{div}(A\nabla p_{hp})+ \phi^{\prime}(y_{hp})p_{hp}\bigr) \bigl(v^{p}-E_{1}v^{p} \bigr) -\bigl(\phi^{\prime}(y_{hp})p_{hp}, v^{p}-E_{1}v^{p}\bigr) \\ &{}- \int_{\partial\Omega}g^{\prime}\bigl(y(u_{hp})\bigr) \bigl(v^{p}-E_{1}v^{p}\bigr)+\bigl(\phi ^{\prime}\bigl(y(u_{hp})\bigr)p(u_{hp}), v^{p}-E_{1}v^{p}\bigr) \\ &{}+ \int_{\partial\Omega }\bigl(g^{\prime }(y_{hp})-g^{\prime} \bigl(y(u_{hp})\bigr)\bigr)E_{1}v^{p} -\bigl( \phi^{\prime}(y_{hp})p_{hp}-\phi^{\prime } \bigl(y(u_{hp})\bigr)p(u_{hp}),E_{1}v^{p} \bigr) \\ &{}+\sum_{e\cap\partial\Omega=\emptyset} \int_{e}\bigl[(A \nabla p _{hp}\cdot n)\bigr] \bigl(v^{p}-E_{1}v^{p}\bigr) \\ &{}+\sum_{e\subset\partial \Omega } \int_{e}(A\nabla p _{hp})\cdot n \bigl(v^{p}-E_{1}v^{p}\bigr)+ \int_{\Omega}(e_{p})^{2}A\nabla\xi\nabla\xi \\ =& \sum_{\tau} \int_{\tau}\bigl(-\operatorname{div}(A\nabla p_{hp})+ \phi^{\prime}(y_{hp})p_{hp}\bigr) \bigl(v^{p}-E_{1}v^{p} \bigr) \\ &{}- \int_{\partial\Omega}\bigl(g^{\prime}(y_{hp})-g^{\prime } \bigl(y(u_{hp})\bigr)\bigr)v^{p} +\sum_{e\cap\partial\Omega=\emptyset} \int_{e}\bigl[(A \nabla p _{hp}\cdot n)\bigr] \bigl(v^{p}-E_{1}v^{p}\bigr) \\ &{}+\sum_{e\subset\partial \Omega } \int_{e}\bigl((A \nabla p_{hp})\cdot n-g^{\prime}(y_{hp})\bigr) \bigl(v^{p}-E_{1}v^{p} \bigr) \\ &{}-\bigl(\phi^{\prime }(y_{hp})p_{hp}- \phi^{\prime}\bigl(y(u_{hp})\bigr)p(u_{hp}),v^{p} \bigr)+ \int_{\Omega}(e_{p})^{2}A\nabla\xi\nabla\xi. \end{aligned}$$
(3.25)
By using Lemma 2.2, we have
$$\begin{aligned} c\Vert \xi e_{p}\Vert _{H^{1}(\Omega)}^{2} =& \sum _{\tau} \int_{\tau}\bigl(-\operatorname{div}(A\nabla p_{hp})+ \phi^{\prime}(y_{hp})p_{hp}\bigr) \bigl(v^{p}-E_{1}v^{p} \bigr) \\ &{}- \int_{\partial\Omega}\bigl(g^{\prime}(y_{hp})-g^{\prime } \bigl(y(u_{hp})\bigr)\bigr)v^{p}+\sum _{e\cap\partial\Omega=\emptyset } \int_{e}\bigl[(A \nabla p _{hp}\cdot n)\bigr] \bigl(v^{p}-E_{1}v^{p}\bigr) \\ &{}+\sum_{e\subset\partial\Omega} \int_{e}\bigl((A \nabla p_{hp})\cdot n-g^{\prime}(y_{hp})\bigr) \bigl(v^{p}-E_{1}v^{p} \bigr) \\ &{}-\bigl(\phi^{\prime }(y_{hp})e^{p},v^{p} \bigr)-\bigl(\tilde{\phi}^{\prime\prime}(y_{hp})p(u_{hp})e^{y},v^{p} \bigr)+ \int_{\Omega}(e_{p})^{2}A\nabla\xi\nabla\xi \\ \leq& C\sum_{\tau\subset\Omega_{d}}\frac{h_{\tau}^{2}}{p_{\tau }^{2}}\bigl( \operatorname{div}(A\nabla p_{hp})-\phi^{\prime}(y_{hp})p_{hp} \bigr)^{2} \\ &{}+C\sum_{e\cap \partial\Omega=\emptyset,e\in\Omega _{d}} \int_{e}\frac{h_{e}}{p_{e}}\bigl[(A\nabla p _{hp}\cdot n)\bigr]^{2} \\ &{}+C\sum_{e\subset\partial\Omega}\frac{h_{e}}{p_{e}} \int_{e}\bigl(A \nabla p_{hp}\cdot n-g^{\prime}(y_{hp})\bigr)^{2}+C\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\Omega)}^{2} \\ &{}+C\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\partial\Omega)}^{2}+C\Vert e_{p}\Vert _{L^{2}(\Omega)}^{2}+\frac {c}{2}\bigl\Vert v^{p} \bigr\Vert _{H^{1}(\Omega)}^{2}, \end{aligned}$$
where we use the property that \(\xi=0\) on \(\Omega/\Omega_{d}^{-}\). Noting that \(\|v^{p}\|_{H^{1}(\Omega)}^{2}\leq\|\xi e_{p}\|_{H^{1}(\Omega)}^{2}\), we have
$$\begin{aligned}& \bigl\Vert \xi\bigl(p_{hp}-p(u_{hp})\bigr) \bigr\Vert _{H^{1}(\Omega)}^{2} \\& \quad \leq C\sum_{i=2}^{4} \kappa_{i}^{2}+C\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\Omega)}^{2} \\& \qquad {} +C\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\partial\Omega)}^{2}+C\bigl\Vert p_{hp}-p(u_{hp}) \bigr\Vert _{L^{2}(\Omega)}^{2}. \end{aligned}$$
(3.26)
It follows from (3.23) and (3.26) that
$$\begin{aligned}& \bigl\Vert p_{hp}-p(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega )}^{2} \\& \quad \leq C\bigl\Vert \xi \bigl(p_{hp}-p(u_{hp})\bigr) \bigr\Vert _{H^{1}(\Omega)}^{2} \\& \quad \leq C\sum_{i=2}^{4} \kappa_{i}^{2}+C\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\Omega)}^{2} \\& \qquad {}+C\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\partial\Omega)}^{2}+C\bigl\Vert p_{hp}-p(u_{hp}) \bigr\Vert _{L^{2}(\Omega)}^{2}. \end{aligned}$$
(3.27)
Part II. Now, we estimate \(\|p_{hp}-p(u_{hp})\|_{L^{2}(\Omega)}\). Let \(\varphi_{p}\) be the solution of the following equation:
$$ a(\varphi_{p},w)=(e_{p},w), \quad \forall w\in V. $$
(3.28)
Noting that Ω is convex [24], it has been shown that
$$ \|\varphi_{p}\|_{2,\Omega}\leq C\|e_{p} \|_{0,\Omega}. $$
(3.29)
Let \(E_{2}\) be the linear operator defined in Lemma 2.3. It follows from (2.1), (2.14), (3.28)-(3.29), and Lemma 2.3 that
$$\begin{aligned} \|e_{p}\|_{L^{2}(\Omega)}^{2} =& a(\varphi_{p},e_{p})=a( \varphi_{p}-E_{2}\varphi _{p},e_{p})+a(E_{2} \varphi_{p},e_{p}) \\ = &\sum_{\tau} \int_{\tau}\bigl(A\nabla\bigl(p_{hp}-p(u_{hp}) \bigr)\bigr)\cdot \nabla (\varphi_{p}-E_{2} \varphi_{p}) \\ &{} +\bigl(g^{\prime}(y_{hp})-g^{\prime} \bigl(y(u_{hp})\bigr),E_{2}\varphi_{p}\bigr)-\bigl( \phi ^{\prime}(y_{hp})p_{hp}-\phi^{\prime} \bigl(y(u_{hp})\bigr)p(u_{hp}),E_{2}\varphi _{p}\bigr) \\ =& \sum_{\tau} \int_{\tau}\bigl(-\operatorname{div}(A\nabla p_{hp})+ \phi^{\prime}(y_{hp})p_{hp}\bigr) (\varphi_{p}-E_{2} \varphi _{p})-\bigl(\phi^{\prime}(y_{hp})p_{hp}, \varphi_{p}-E_{2}\varphi_{p}\bigr) \\ &{}-\bigl(g^{\prime}\bigl(y(u_{hp})\bigr),\varphi_{p}-E_{2} \varphi_{p}\bigr)+\bigl(\phi^{\prime }\bigl(y(u_{hp}) \bigr)p(u_{hp}), \varphi_{p}-E_{2} \varphi_{p}\bigr) \\ &{} +\bigl(g^{\prime}(y_{hp})-g^{\prime} \bigl(y(u_{hp})\bigr),E_{2}\varphi _{p}\bigr)-\bigl( \phi^{\prime}(y_{hp})p_{hp}-\phi^{\prime } \bigl(y(u_{hp})\bigr)p(u_{hp}),E_{2} \varphi_{p}\bigr) \\ &{}+\sum_{e\cap\partial\Omega=\emptyset} \int_{e}\bigl[(A \nabla p_{hp}\cdot n)\bigr]( \varphi_{p}-E_{2}\varphi_{p})+\sum _{e\subset\partial \Omega} \int_{e}(A \nabla p_{hp})\cdot n( \varphi_{p}-E_{2}\varphi_{p}) \\ =& \sum_{\tau} \int_{\tau}\bigl(-\operatorname{div}(A\nabla p_{hp})+ \phi^{\prime}(y_{hp})p_{hp}\bigr) (\varphi_{p}-E_{2} \varphi_{p}) +\bigl(g^{\prime}(y_{hp})-g^{\prime} \bigl(y(u_{hp})\bigr),\varphi_{p}\bigr) \\ &{} +\sum_{e\cap\partial\Omega=\emptyset} \int_{e}\bigl[(A \nabla p_{hp}\cdot n)\bigr]( \varphi_{p}-E_{2}\varphi_{p}) \\ &{}+\sum _{e\subset\partial \Omega} \int_{e}\bigl((A \nabla p_{hp})\cdot n-g^{\prime}(y_{hp})\bigr) (\varphi_{p}-E_{2} \varphi_{p}) \\ &{} -\bigl(\phi^{\prime}(y_{hp})p_{hp}- \phi^{\prime }\bigl(y(u_{hp})\bigr)p(u_{hp}), \varphi_{p}\bigr). \end{aligned}$$
By using Lemma 2.3, we have
$$\begin{aligned} \Vert e_{p}\Vert _{L^{2}(\Omega)}^{2} =& \sum _{\tau} \int_{\tau }\bigl(-\operatorname{div} (A\nabla p_{hp})+ \phi^{\prime}(y_{hp})p_{hp}\bigr) (\varphi_{p}-E_{2} \varphi _{p})+\bigl(g^{\prime}(y_{hp})-g^{\prime} \bigl(y(u_{hp})\bigr),\varphi _{p}\bigr) \\ &{} +\sum_{e\cap\partial\Omega=\emptyset} \int_{e}\bigl[(A \nabla p_{hp}\cdot n)\bigr]( \varphi_{p}-E_{2}\varphi_{p}) \\ &{}+\sum _{e\subset\partial \Omega} \int_{e}\bigl((A \nabla p_{hp})\cdot n-g^{\prime}(y_{hp})\bigr) (\varphi_{p}-E_{2} \varphi_{p}) \\ &{} -\bigl(\phi^{\prime}(y_{hp})e^{p},v^{p} \bigr)-\bigl(\tilde{\phi}^{\prime \prime }(y_{hp})p(u_{hp})e^{y},v^{p} \bigr) \\ \leq& C(\delta)\sum_{\tau}\frac{h_{\tau}^{4}}{p_{\tau}^{4}} \int _{\tau}\bigl(\operatorname{div}(A\nabla p_{hp})- \phi^{\prime}(y_{hp})p_{hp}\bigr)^{2} \\ &{}+C( \delta)\sum_{e\cap\partial\Omega=\emptyset}\frac{h_{e}^{3}}{p_{e}^{3}} \int_{e}\bigl[(A \nabla p_{hp}\cdot n) \bigr]^{2} \\ &{} +C(\delta)\sum_{e\subset\partial\Omega}\frac {h_{e}^{3}}{p_{e}^{3}} \int_{e}\bigl(A \nabla p_{hp}\cdot n-g^{\prime}(y_{hp})\bigr)^{2}+C(\delta)\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\Omega)}^{2} \\ &{}+C(\delta)\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\partial\Omega)}^{2}+C\delta \Vert \xi \Vert _{H^{2}(\Omega )}^{2} \\ =& C(\delta) \Biggl(\sum_{i=5}^{7} \kappa_{i}^{2}+\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\Omega)}^{2}+\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\partial\Omega)}^{2} \Biggr)+C\delta \Vert \xi \Vert _{H^{2}(\Omega )}^{2}. \end{aligned}$$
Let δ be small enough, it follows from (3.29) that
$$ \bigl\Vert p_{hp}-p(u_{hp})\bigr\Vert _{L^{2}(\Omega)}^{2}\leq C\sum_{i=5}^{7} \kappa_{i}^{2}+C\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\Omega)}^{2}+C\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\partial\Omega)}^{2}. $$
(3.30)
Part III. Next, we estimate \(\|y_{hp}-y(u_{hp})\|_{L^{2}(\partial\Omega)}\). Let \(e_{y}=y_{hp}-y(u_{hp})\) and \(v^{y}=\xi^{2}e_{y}\), by using (2.1), (2.13), (3.3), and Lemma 2.2, then we have
$$\begin{aligned} c\|\xi e_{y}\|_{H^{1}(\Omega)}^{2} \leq& a(\xi e_{y},\xi e_{y})=a\bigl(v^{y},e_{y} \bigr)+ \int_{\Omega}(e_{y})^{2}A\nabla\xi\nabla\xi \\ =& a\bigl(e_{y},v^{y}-E_{1}v^{y} \bigr)+ \int_{\Omega}(e_{y})^{2}A\nabla\xi\nabla \xi \\ =& \sum_{\tau\subset\Omega_{d}} \int_{\tau}A\nabla \bigl(y_{hp}-y(u_{hp}) \bigr)\cdot \bigl(v^{y}-E_{1}v^{y}\bigr)+ \int_{\Omega}(e_{y})^{2}A\nabla\xi\nabla\xi \\ =& \sum_{\tau\subset\Omega_{d}} \int_{\tau}\bigl(-\operatorname {div}(A\nabla y_{hp})+ \phi(y_{hp})-f\bigr) \bigl(v^{y}-E_{1}v^{y} \bigr) \\ &{}+\sum_{e\cap\partial\Omega=\emptyset} \int_{e}\bigl[(A \nabla y _{hp}\cdot n)\bigr] \bigl(v^{y}-E_{1}v^{y}\bigr) \\ &{}+ \int_{\partial\Omega}\bigl((A \nabla y _{hp})\cdot n-u_{hp}-z_{0}\bigr) \bigl(v^{y}-E_{1}v^{y} \bigr) \\ &{}-\bigl(\phi(y_{hp})-\phi\bigl(y(u_{hp}) \bigr),v^{y}-E_{1}v^{y}\bigr)+ \int_{\Omega }(e_{y})^{2}A\nabla \xi\nabla\xi \\ =& \sum_{\tau\subset\Omega_{d}} \int_{\tau}\bigl(-\operatorname {div}(A\nabla y_{hp})+ \phi(y_{hp})-f\bigr) \bigl(v^{y}-E_{1}v^{y} \bigr) \\ &{}+\sum_{e\cap\partial\Omega=\emptyset} \int_{e}\bigl[(A \nabla y _{hp}\cdot n)\bigr] \bigl(v^{y}-E_{1}v^{y}\bigr) \\ &{}+ \int_{\partial\Omega}\bigl((A \nabla y _{hp})\cdot n-u_{hp}-z_{0}\bigr) \bigl(v^{y}-E_{1}v^{y} \bigr) \\ &{}-\bigl(\tilde{\phi}^{\prime }(y_{hp}) \bigl(y_{hp}-y(u_{hp}) \bigr),v^{y}-E_{1}v^{y}\bigr)+ \int_{\Omega }(e_{y})^{2}A\nabla \xi\nabla\xi \\ \leq& C\sum_{\tau\subset\Omega _{d}} \int_{\tau}h_{\tau}^{2}\bigl(f+ \operatorname{div}(A\nabla y_{hp})-\phi(y_{hp}) \bigr)^{2} \\ &{}+C\sum_{e\cap\partial\Omega =\emptyset,e\in\Omega _{d}} \int_{e}h_{e}\bigl[(A\nabla y _{hp}\cdot n)\bigr]^{2} \\ &{}+C\sum_{e\cap\partial\Omega} \int_{e}h_{e}(A\nabla y _{hp}\cdot n-u_{hp}-z_{0})^{2}+C\|e_{y} \|_{L^{2}(\Omega)}^{2}+\frac{c}{2}\|\xi e_{y} \|_{H^{1}(\Omega)}^{2} \\ =& C\sum_{i=8}^{10}\kappa_{i}^{2} +C\bigl\Vert y_{hp}-y(u_{hp})\bigr\Vert _{L^{2}(\Omega)}^{2}+\frac{c}{2}\|\xi e_{y} \|_{H^{1}(\Omega)}^{2}. \end{aligned}$$
(3.31)
Therefore, it follows from (3.31) and the trace theorem that
$$\begin{aligned} \bigl\Vert y_{hp}-y(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega)}^{2} \leq&C\bigl\Vert \xi \bigl(y_{hp}-y(u_{hp}) \bigr) \bigr\Vert _{H^{1}(\Omega)}^{2} \\ \leq& C\sum_{i=8}^{10} \kappa_{i}^{2}+C\bigl\Vert y_{hp}-y(u_{hp}) \bigr\Vert _{L^{2}(\Omega)}^{2}. \end{aligned}$$
(3.32)
Part IV. Furthermore, we estimate \(\|y_{hp}-y(u_{hp}) \|_{L^{2}(\Omega)}\). Let \(\varphi_{y}\) be the solution of the equation
$$ a(w,\varphi_{y})=(e_{y},w), \quad \forall w\in V. $$
Then we have
$$ \|\varphi_{y}\|_{2,\Omega}\leq C\|e_{y} \|_{0,\Omega}. $$
(3.33)
Similarly, we have
$$\begin{aligned} c\|e_{y}\|_{L^{2}(\Omega)}^{2} =& a(e_{y}, \varphi_{y})=a(e_{y},\varphi_{y}-E_{2} \varphi_{y}) \\ =& \sum_{\tau} \int_{\tau}A \nabla\bigl(y_{hp}-y(u_{hp}) \bigr)\cdot \nabla(\varphi_{y}-E_{2}\varphi_{y}) \\ =& \sum_{\tau} \int_{\tau}\bigl(-\operatorname{div}(A\nabla y_{hp})+ \phi(y_{hp})-f\bigr) (\varphi_{y}-E_{2} \varphi_{y}) \\ &{}+\sum_{e\cap\partial\Omega=\emptyset} \int_{e}\bigl[(A \nabla y_{hp}\cdot n)\bigr]( \varphi_{y}-E_{2}\varphi_{y}) \\ &{} + \int_{\partial\Omega}\bigl((A\nabla y _{hp})\cdot n-u_{hp}-z_{0}\bigr) (\varphi_{y}-E_{2} \varphi_{y}) \\ &{}-\bigl(\phi(y_{hp})-\phi \bigl(y(u_{hp}) \bigr),\varphi_{y}-E_{2}\varphi_{y}\bigr) \\ =& \sum_{\tau} \int_{\tau}\bigl(-\operatorname{div}(A\nabla y_{hp})+ \phi(y_{hp})-f\bigr) (\varphi_{y}-E_{2} \varphi_{y}) \\ &{}+\sum_{e\cap\partial\Omega=\emptyset} \int_{e}\bigl[(A \nabla y_{hp}\cdot n)\bigr]( \varphi_{y}-E_{2}\varphi_{y}) \\ &{} + \int_{\partial\Omega}\bigl((A\nabla y _{hp})\cdot n-u_{hp}-z_{0}\bigr) (\varphi_{y}-E_{2} \varphi_{y}) \\ &{}-\bigl(\tilde{\phi}^{\prime }(y_{hp}) \bigl(y_{hp}-y(u_{hp})\bigr),\varphi_{y}-E_{2} \varphi_{y}\bigr) \\ \leq& C(\delta)\sum_{\tau}\frac{h_{\tau}^{4}}{p_{\tau}^{4}} \int _{\tau}\bigl(f+\operatorname{div}(A\nabla p_{hp})- \phi^{\prime}(y_{hp})p_{hp}\bigr)^{2} \\ &{}+ C( \delta)\sum_{e\cap\partial\Omega=\emptyset}\frac {h_{e}^{3}}{p_{e}^{3}} \int_{e}\bigl[(A \nabla p_{hp}\cdot n) \bigr]^{2} \\ &{} +C(\delta)\sum_{e\subset\partial\Omega}\frac {h_{e}^{3}}{p_{e}^{3}} \int_{e}(A\nabla y _{hp}\cdot n-u_{hp}-z_{0})^{2}+ \frac{c}{2}\|e_{y}\|_{L^{2}(\Omega)}^{2}+C\delta\| \varphi_{y}\|_{H^{2}(\Omega)}^{2} \\ =& C(\delta)\sum_{i=11}^{13} \kappa_{i}^{2}+\frac{c}{2}\|e_{y}\| _{L^{2}(\Omega)}^{2}+C\delta\|\varphi_{y} \|_{H^{2}(\Omega)}^{2}. \end{aligned}$$
Let δ be small enough, it follows from (3.33) that
$$ \bigl\Vert y_{hp}-y(u_{hp})\bigr\Vert _{L^{2}(\Omega)}^{2}\leq C\sum_{i=11}^{13} \kappa_{i}^{2}. $$
(3.34)
It follows from (3.7), (3.27), (3.30), (3.32), and (3.34) that
$$ \Vert u-u_{hp}\Vert _{L^{2}(\partial\Omega)}^{2}+\bigl\Vert y(u_{hp})-y_{hp} \bigr\Vert _{L^{2}(\partial\Omega)}^{2}+ \bigl\Vert p(u_{hp})-p_{hp} \bigr\Vert _{L^{2}(\partial\Omega)}^{2}\leq C\sum_{i=1}^{13} \kappa_{i}^{2}. $$
(3.35)
Part V. Finally, it is easy to see that
$$\begin{aligned}& \Vert y-y_{hp}\Vert _{L^{2}(\partial\Omega)}\leq \bigl\Vert y-y(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega)}+\bigl\Vert y(u_{hp})-y_{hp}\bigr\Vert _{L^{2}(\partial\Omega)}, \end{aligned}$$
(3.36)
$$\begin{aligned}& \Vert p-p_{hp}\Vert _{L^{2}(\partial\Omega)}\leq \bigl\Vert p-p(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega)}+\bigl\Vert p(u_{hp})-p_{hp}\bigr\Vert _{L^{2}(\partial\Omega)}, \end{aligned}$$
(3.37)
and
$$\begin{aligned}& \bigl\Vert y-y(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega)}\leq C\Vert u-u_{hp}\Vert _{L^{2}(\partial\Omega)}, \end{aligned}$$
(3.38)
$$\begin{aligned}& \bigl\Vert p-p(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega)}\leq C\bigl\Vert y-y(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega)}\leq C\Vert u-u_{hp} \Vert _{L^{2}(\partial\Omega)}. \end{aligned}$$
(3.39)
It follows from (3.35) and (3.36)-(3.39) that
$$\begin{aligned}& \Vert u-u_{hp}\Vert _{L^{2}(\partial\Omega)}^{2}+\Vert y-y_{hp}\Vert _{L^{2}(\partial\Omega)}^{2} +\Vert p-p_{hp}\Vert _{L^{2}(\partial\Omega)}^{2} \\& \quad \leq \Vert u-u_{hp}\Vert _{L^{2}(\partial\Omega)}^{2}+\bigl\Vert y-y(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega)}^{2} \\& \qquad {} +\bigl\Vert y_{hp}-y(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega)}^{2} +\bigl\Vert p-p(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega)}^{2}+\bigl\Vert p_{hp}-p(u_{hp}) \bigr\Vert _{L^{2}(\partial\Omega)}^{2} \\& \quad \leq \Vert u-u_{hp}\Vert _{L^{2}(\partial\Omega)}^{2}+\bigl\Vert y-y(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega)}^{2}+\bigl\Vert p-p(u_{hp})\bigr\Vert _{L^{2}(\partial\Omega )}^{2} \\& \quad \leq C\sum_{i=1}^{13}{ \eta}_{i}^{2}. \end{aligned}$$
(3.40)
Then (3.22) follows from (3.40). □

4 Conclusion and future work

In this paper, we use 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 residual-based a posteriori error estimates in \(L^{2}\)-\(H^{1}\) norms for the semilinear Neumann boundary optimal control problems. Then we also give sharper a posteriori error estimates for the control approximation and error estimates in the \(L^{2}\) norm for the state and co-state on the boundary. To the best of our knowledge in the context of optimal control problems, these a posteriori error estimates for the semilinear Neumann boundary optimal control problems are new.

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

Declarations

Acknowledgements

This work is supported by National Basic Research Program (2012CB955804), Major Research Plan of National Natural Science Foundation of China (91430108), National Science Foundation of China (11201510, 11171251), China Postdoctoral Science Foundation (2015M580197), Chongqing Research Program of Basic Research and Frontier Technology (cstc2015jcyjA20001), Ministry of education Chunhui projects (Z2015139), Major Program of Tianjin University of Finance and Economics (ZD1302) 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 for Nonlinear Science and System Structure, Chongqing Three Gorges University
(2)
Research Center for Mathematics and Economics, Tianjin University of Finance and Economics
(3)
Huashang College, Guangdong University of Finance
(4)
Chongqing Wanzhou Long Bao Middle School

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© Lu et al. 2016