Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems

1.   Introduction

Since the pioneer work in adaptive finite element methods was proposed by Babu$\mathrm{\breve{s}}$ka and Rheinboldt [2], adaptive finite element methods have been applied successfully in engineering and scientific computations. The adaptive finite element method is based on the error information obtained by the computer to determine whether the solution is accurate enough. Hence the soul of adaptive finite elements is the a posteriori error estimation.

When Dörfler [15] presented a marking strategy aiming at electing the set of elements for refinement, based on the error indicators which was controlled by the control, the state and adjoint state, adaptive finite element algorithm was put on the stage of academic research. He provided a fineness assumption on the initial grid $\mathcal{T}_{h_0}$ which was used to prove the reduction of energy errors while in later investigations, Morin, Nochetto and Siebert [30] removed the assumption. Moreover, they proposed the interior node property in order to obtain the proof for convergence of adaptive finite element methods [31]. More pioneering works on adaptive finite element methods see literatures [5,3,9,13,16,17,22], in which the linear elliptic optimal control problem was mainly been investigated.

Convergence and quasi-optimality are the two key factors of adaptive finite element methods. It was noteworthy that Mekchay and Nochetto [29] extended the convergence result of Morin, Nochetto and Siebert [31] for general second order linear elliptic partial differential equations by introducing a novel concept that was the total error which was the sum of the energy errors adding the oscillations. This provided a valuable empirical basis for future scholars' work on convergence analysis. Meanwhile, Binev, Dahmen and Devore [4] firstly presented the property of optimality. Later, a large number of scholars participate in the study of the property. For example, Carstensen and Hoppe [6] proposed convergence and quasi-optimality which were established for the Raviart Thomas finite element method. Gong and Yan [20] considered the convergence analysis of adaptive finite element method for elliptic optimal control problems with pointwise control constraints.

As to our best knowledge, nonlinear optimal control problems have gradually penetrated into many fields of scientific research and engineering technology. Chen and Lu [11] investigated adaptive fully-discrete finite element methods for semilinear parabolic quadratic boundary optimal control problems. Gaevskaya, Hoppe, Iliash and Kieweg [17] developed an adaptive finite element method for a class of distributed optimal control problems with control constraints, and found requirement $h_0\ll1$ on the initial grid $\mathcal{T}_{h_0}$ is not restrictive for the convergence analysis of adaptive finite element for nonlinear problems. Chen, Gong, He and Zhou [10] studied an adaptive finite element method for a class of a nonlinear eigenvalue problems that may be of nonconvex energy functional and prove the convergence of adaptive finite element approximations.

Leng and Chen [23] proved the convergence and the quasi-optimality of an adaptive element method with integral control constraints while we extend the result of [23] to a nonlinear optimal control problem with integral control constraint on $L^2-$norms in this paper. We follow the idea of [27] to derive reliable and efficient posteriori error estimations and the idea of [19] to prove the posteriori upper and the global lower bound of the errors, in which the bubble function matters. Moreover, a contraction for an adaptive finite element method is obtained based on a mild assumption on initial mesh $\mathcal{T}_{h_0}$ which can be seen in [10,14,20,21,23,24,25,29,31]. Furthermore, we propose the optimal convergence rate. However, the quasi-optimality is the best obstacle mission for us to prove. Therefore, we continue to use the idea of [14,21,23]. Finally, we provide some numerical experiments to verify our theoretical analysis.

Here are some notations will be used in this paper. Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^2$ and $\partial\Omega$ denote the boundary of $\Omega$. We use the standard notation $W^{m,q}(\omega)$ with norm $\|\cdot\|_{m,q,\omega}$ and seminorm $|\cdot|_{m,q,\omega}$ to express the standard Sobolev space for $\omega\subset\Omega$. Moreover, we will omit the subscription if $\omega = \Omega$. For $q = 2$, we denote $W^{m,2}(\Omega)$ by $H^m(\Omega)$ and $\|\cdot\|_m = \|\cdot\|_{m,2}$. Also for $m = 0 \ \mathrm{and}\ q = 2$, we denote $W^{0,2}(\omega) = L^2(\omega)$ and $\|\cdot\|_{0,2,\omega} = \|\cdot\|_{0,\omega}$. For $m = 0$ and $q = \infty$, we denote $W^{0,\infty}(\omega) = L^\infty(\omega)$ and $||\cdot||_{0,\infty,\omega} = \max\limits_{\omega}|\cdot| = ||\cdot||_{\infty,\omega}$. Additionally, we observe that $H^1_0(\Omega) = \{v\in H^1(\Omega):v = 0\ \mathrm{on}\ \partial\Omega\}$. Beyond that, let $c$ and $C$ are the constants which independent of grids size, then we use $A\approx B$ to represent $cA\leq B\leq CA$. In addition, $(\cdot, \cdot)$ denotes the $L^2$ inner product.

The rest of our paper is arranged as follows. In Section 2, we give what the optimal control problems we want to investigate and some basic notations must be used. Then the a posteriori error estimation is obtained and an adaptive algorithm is proposed in Section 3. In Section 4, we use quasi-orthogonality and discrete local upper bound to prove the convergence of the adaptive finite element method and so is the quasi-optimality for details in Section 5. In the end, some numerical simulations is given to verify our theoretical analysis.

2.   Nonlinear optimal control problem

In this section we first introduce some basic notations, and then we show what the nonlinear optimal control problem we discussed about.

$\mathcal{T}_h$ is a regular triangulation of $\Omega$ such that $\bar{\Omega} = \cup_{T\in\mathcal{T}_h}\bar{T}$. $T$ is an element of $\mathcal{T}_h$. Let $\mathcal{T}_{h_0}$ be the initial partition of $\bar{\Omega}$ into disjoint triangles. By newest-vertex bisections for $\mathcal{T}_{h_0}$, we can obtain a class $\mathbb{T}$ of conforming partitions. For $\mathcal{T}_h, \tilde{\mathcal{T}}_h \in \mathbb{T}$, we use $\mathcal{T}_h\subset\mathcal{\tilde{T}}_h$ to indicate that $\tilde{\mathcal{T}}_h$ is a refinement of $\mathcal{T}_h$ and $h_{T} = |T|^{1/2}$. According to [14], the continuous piecewise linear mesh function is defined by $h_{\mathcal{T}_h}$. Moreover, $h_{\mathcal{T}_h}(z)$ is the average of the $h_{T'}$ over all $T'\in\mathcal{T}_h$ for any vertex $z$ of $\mathcal{T}_h$ with $z\in T'$. Then we have the following properties via keeping the meshs level low enough [21].

Lemma 2.1. [21] For some constants $c$ and $C$ and fixed constant $\mu$, there holds

where all grids satisfied above are denoted by $\mathbb{T}_\mu$.

In this paper we mainly enter into meaningful discussions with the following nonlinear optimal control problem:

where $y_{d}\in L^2(\Omega),\ U_{ad} = \{v:v\in L^2(\Omega),\ \int_{\Omega}v\geq 0\}$ is a closed convex subset of $U = L^2(\Omega)$ and $\phi(\cdot)\in W^{2,\infty}(-R,R)$ for any $R>0,\ \phi'(y)\in L^2(\Omega)$ for any $y\in H^1(\Omega),\ \phi'\geq0$. Let $V = H_{0}^1(\Omega)$, we give the weak formulation to deal with state equation, namely, find $y\in V$ such that

where

Then the nonlinear optimal control problem can be restated as follows

It is well known [26,27] that the nonlinear optimal control problem has at least one solution $(y,u)$, and that if a pair $(y,u)$ is the solution of the optimal control problem, then there is a co-state $p\in V$ such that the triplet $(y,p,u)$ satisfies the following optimality conditions:

Since the coercivity of $a(\cdot, \cdot)$, we define a solution operator $S:L^2(\Omega)\rightarrow H_{0}^1(\Omega)$ of (4) such that $S(f+u) = y$ and let $S^*$ be the adjoint of $S$ such that $S^*(y-y_{d}) = p$. Suppose $V_h$ is the continuous piecewise linear finite element space with respect to the partition $\mathcal{T}_h\in \mathbb{T}$. For $\mathcal{T}_h\in \mathbb{T}$, we define $U^h$ as the piecewise constant finite element space with respect to $\mathcal{T}_h$. Let $U_{ad}^h = \{v_h\in U^h:\int_{\Omega}v_h\geq 0\}\subset L^2(\Omega)$. Then we derive the standard finite element discretization for the nonlinear optimal control problem as follows:

Similarly the nonlinear optimal control problem (10)-(11) has at least one solution $(y_h,u_h)$, and that if a pair $(y_h,u_h)$ is the solution of (10)-(11), then there is a co-state $p_h \in V_{h}$ such that the triplet $(y_{h},p_{h},u_{h})$ satisfies the following optimality conditions:

Based on [12,14,21,27], we have the following Lemmas in order to derive a $L^2-$norms posteriori error estimation for both the control, the state and adjoint state variables.

Lemma 2.2. [21] Suppose that $\Omega$ is convex such that for any $f\in L^2(\Omega)$, (3)-(4) have at least one solution $y = S(f+u)\in H^2(\Omega)\cap H_0^1(\Omega)$ and

and apparently the assumption is valid for $S^*$.

Lemma 2.3. [12] Assume that $\Omega$ is convex, here holds

for any $(y,p)\in V$ and where

are bounded functions in $\bar{\Omega}$.

Lemma 2.4. [14,27] For all $v\in H^1(\Omega)$, $\mathcal{T}_h\in\mathbb{T}$ and $T\in\mathcal{T}$, we have

3.   A posteriori error estimation

In this section, we will recall a residual-based a posteriori error estimation for nonlinear elliptic equations. For the model problem that we studied in Section 2, a reliable and efficient a posteriori estimation will be obtained. In the end of this section, an adaptive finite element algorithm will be introduced.

Here we define some error indicators. $\eta(\cdot)$ are error indicators and $osc(\cdot)$ represent the data oscillations. For $\mathcal{T}_h\in \mathbb{T}, \ T\in \mathcal{T}_h$, we define

where $u_{h}\in U_{ad}^{h}$, $y_{h},p_{h}\in V_{h}$, $(\nabla y_h)\cdot\mathbf{n}$ denotes the jump of $\nabla y_h$, and $\mathbf{n}$ denotes the outward normal oriented to $\partial T\backslash\partial\Omega$, and where $f_{T}$ is $L^2$-projection of $f$ onto piecewise constant space on $T$ and $f_T = \frac{\int_Tf}{|T|}$. For $\omega\subset \mathcal{T}_h$, we have

Similarly, we have $\eta_{2,\mathcal{T}_h}^2(u_{h},y_{h},\omega),\ \eta_{3,\mathcal{T}_h}^2(y_{h},p_{h},\omega)$ and $osc^2_{\mathcal{T}_h}(y_{h}-y_{d},\omega)$.

Lemma 3.1. Let $\mathcal{T}_h\in\mathbb{T}_\mu$ under the conditions of Lemma 2.2, we have

for sufficiently small $\mu$.

Proof. Suppose that $y^h,p^h\in V$ are intermediate variables satisfying the equations as follows

Employing the Galerkin orthogonality, the approximation properties, Lemma 2.2, and $||\nabla h_{\mathcal{T}}||_\infty\leq\mu$, there exist similar results, which resemble to Lemma 3.1 in [21], for nonlinear elliptic optimal control problems with sufficiently small $\mu$ that

It has been proved $|||y-y^h|||\leq C||y-y^h||_1\leq C||u-u_h||_0$ in the Theorem 3.1 of [28] for nonlinear elliptic optimal control problems. Then associated with the Lemma 4.4 in [7], we deduce similar conclusions for nonlinear optimal control problems that

By using the triangle inequality, we have

In connection with what we discussed above, the triangle inequality and Lemma 2.1, it is easy to prove the prevenient results in Lemma 3.1.

Now we are in the position to derive a posteriori error estimation for both the control, the state and adjoint state variables.

Theorem 3.2. Let $(u,y,p)\in U_{ad}\times H_0^1(\Omega)\times H_0^1(\Omega)$ be the solution of (7)-(9) and $(u_h,y_h,p_h)\in U_{ad}^h\times V_h\times V_h$ be the solution of (12)-(14). Then we have a posteriori error upper bound

and apparently a global lower bound

where $c$ and $C$ only depend on the shape regularity of $\mathcal{T}_h$.

Proof. In view of Lemma 7.3.1 in [27], we similarly derive that

where $p^h$ is the solution of (3.2). Then we shall to deal with $||p_h-p^h||_0^2$.

Let $\xi = p^h-p_h$, and $\xi_I = \pi_h\xi$ where $\pi_h$ is the standard Lagrange interpolation operator of $\xi$, then it follows from Lemma 2.1, Lemma 2.3 and (8) that

where $\phi(\cdot)\in W^{2,\infty}(\Omega)$, the embedding $||v||_{0,\infty}\leq C||v||_2$ and $||p^h||_0\leq C$ have been adopted and $\delta$ is positive. Then choosing $\delta = \frac{1}{2C}$, we obtain

Similarly there is going to be proved by letting $\tilde{\xi} = y^h-y_h$ that

where $\phi(\cdot)\in W^{2,\infty}(\Omega)$ have been applied and hence we have

It is easy to derive the expect upper bound by combining with (20)-(22).

Next we are going to deduce the global lower error bound through the standard bubble function [1,19]. Similar to Lemma 3.7 in [19], it can be similarly proved that there exists polynomial $w_T\in H_0^2(T)$ such that

and apparently

Then it follows from (23) and (24) that

Combining with (2.8) and Lemma 2.4, there holds

where $\phi(\cdot)\in W^{2,\infty}(\Omega)$ have been used, the embedding $||v||_{0,\infty,T}\leq C||v||_{2,T}$ and the property $||p_h||\leq C$ have been adopted.

Similarly, we have

Hence by using the Cauchy inequality with the help of (25)-(27), we obtain

Then it brings about

Then we need to use the new bubble functions defined in [19] to both in deal with the jump. Similar to [18,19], it can be similarly proved that there exists polynomial $w_{\partial T}\in H_0^2(T)$ such that

and apparently

And then it follows from (29) and(30) that

Similarly, it can be deduced that

Combining with (13) and Lemma 2.4, there holds

Hence by using the Cauchy inequality with the help of (29)-(30), we have

where $\phi(\cdot)\in W^{2,\infty}(\Omega)$ and $w_{\partial T}\in H_0^2(\Omega)$ have been used.

In connection with (28) and (32), it is easy to get that

It can also be deduced that

Above-mentioned results tell the proof of Theorem 3.2 is accomplished.

Theorem 3.2 gives a reliable and efficient posteriori error estimations for the sum of the $L^2-$norms errors for the control, the state and the co-state variables. Then we introduce an adaptive finite element algorithm to explain what we mainly investigate in this paper.

Algorithm 3.1. Adaptive finite element algorithm for nonlinear optimal control problems:

(o) Given an initial mesh $\mathcal{T}_{h_0}$ and construct finite element space $U_{ad}^{h_0}$ and $V_{h_0}$. Select marking parameter $0<\theta\leq 1$ and set $k: = 0$.

(1) Solve the discrete nonlinear optimal control problem (12)-(14), then obtain approximate solution $(u_{h_k},y_{h_k},p_{h_k})$ with respect to $\mathcal{T}_{h_k}$.

(2) Compute the local error estimator $\eta_{\mathcal{T}_{h_k}}(T)$ for all $T\in\mathcal{T}_{h_k}$.

(3) Select a minimal subset $\mathcal{M}_{h_k}$ of $\mathcal{T}_{h_k}$ such that

where $\eta_{\mathcal{T}_{h_k}}^2(\omega) = \eta_{1,\mathcal{T}_h}^2(p_h,\omega)+ \eta_{2,\mathcal{T}_h}^2(u_{h},y_{h},\omega)+\eta_{3,\mathcal{T}_h}^2(y_{h},p_{h},\omega)$ for all $\omega\subset\mathcal{T}_{h_k}$.

(4) Refine $\mathcal{M}_{h_k}$ by bisecting $b\geq 1$ times in passing from $\mathcal{T}_{h_k}$ to $\mathcal{T}_{h_{k+1}}$ and generally additional elements are refined in the process in order to ensure that $\mathcal{T}_{h_{k+1}}$ is conforming.

(5) Solve the discrete nonlinear optimal control problem (12)-(14), then obtain approximate solution $(u_{h_{k+1}}, y_{h_{k+1}}, p_{h_{k+1}})$ with respect to $\mathcal{T}_{h_{k+1}}$.

(6) Set $k = k+1$ and go to step (2).

4.   Convergence analysis

In this section, we will do our best to demonstrate the convergence while we first give some properties which take vital significance to the proof of the convergence and even the quasi-optimality for the error indicators and the data oscillations before we begin to show the convergence analysis.

Lemma 4.1. Let $(u,y,p)\in U_{ad}\times H_0^1(\Omega)\times H_0^1(\Omega)$ be the solution of (7)-(9) and $(u_h,y_h,p_h)\in U_{ad}^h\times V_h\times V_h$ be the solution of (12)-(14). Then we have a posteriori upper bound

where $C$ only depends on the shape regularity of $\mathcal{T}_h$.

Proof. By applying (18), the triangle inequality, and Lemma 2.1, we obtain

Similar to the proof of Lemma 3.3 in [21], we deduce that

Analogously, the following conclusions can be drawn

and apparently

It is obvious to get the expected result in Lemma 4.1 via using (33)-(36), and the upper bound in Theorem 3.2.

Next, we gives a stability result for error indicators which is can be found in Lemma 3.4 in [21], Lemma 4.1 in [23], and even Proposition 3.3 in [8], and so on.

Lemma 4.2. For $T\in\mathcal{T}_h,\ \mathcal{T}_h\in\mathbb{T}$, let $u_{h_1},u_{h_2}\in U_{ad}^h$, $y_{h_1},y_{h_2}\in V_h$ and $p_{h_1},p_{h_2}\in V_h$, we have

where $\omega_T$ denotes the patch of elements that share an edge with $T$.

Proof. We first prove (37) while (40) can be just proved similarly. Consulting the literatures [1,21,25], namely the trace inequality, there exists $T\in\mathcal{T}_h,\mathcal{T}_h\in\mathbb{T}$ such that

for arbitrary $v\in H^1(\Omega)$. In connection with the inverse estimates and (41), we have

Recalling (1) in Lemma 1, we know that

Recalling the definition of $\eta_{1,\mathcal{T}_h}(p_{h},T)$, we employ the triangle inequality to calculate for $T\in\mathcal{T}_{h_k}$ that

Then it is easy to derive the desired result (37) by adopting (42)-(44).

Next we are to prove (38) while (39) can be proved similarly. We calculate while applying the inequality (15) in Lemma 2.4 for the edge $T\cap T'$ to obtain that

Recalling the definition $\eta_{2,\mathcal{T}_h}(u_{h},y_h,T)$, Lemma 2.3 and Lemma 3.1, we adopt the triangle inequality to calculate for $T\in\mathcal{T}_{h_k}$ that

Then it is easy to deduce the expected result (38) by connecting (45) into (46).

Lemma 4.3. For $\mathcal{T}_h\in\mathbb{T}$, let $\mathcal{M}_h\subset\mathcal{T}_h$ be the set of marked elements and let $\tilde{\mathcal{T}}_h\in\mathbb{T}$ be the refinement of $\mathcal{T}_h$ so that we have

where $u_h\in U_{ad}^h,\tilde{u}_h\in U_{ad}^{\tilde{h}},y_h,p_h\in V_h, \tilde{y}_h,\tilde{p}_h\in V_{\tilde{h}},\sigma\in(0,1]$ and $\lambda = 1-2^{-\frac{3b}{2}}$.

Proof. We just prove (49) and (50). The proofs of (47) and (48) are similar with (49). Employing the Young's inequality with parameter $\sigma$ and (39), we obtain

For a marked element $T'\in\mathcal{M}_h\subset\mathcal{T}_h$, let $\tilde{\mathcal{T}}_{h_{T'}} = \{T\in\tilde{\mathcal{T}}_{h}:T\subset T'\}$. Note that the jump $[\nabla p] = 0$ as $u\in U_{ad}^h\subset U_{ad}^{\tilde{h}},\ \mathrm{and}\ y,p\in V_{\mathcal{T}_h}\subset V_{\tilde{\mathcal{T}}_h}$. According to the definition of $h_T$, we can obtain $h_T = |T|^{\frac{1}{2}}\leq(2^{-b}|T'|)^{\frac{1}{2}}\leq2^{-\frac{b}{2}}h_T'$, inferring that

For $T'\in\mathcal{T}_h\backslash\mathcal{M}_h$, we deduce that

which can use to derive that

Then adding (52) into (51) and rearranging the terms can obtain the expected result (49).

Next, for arbitrary $T\in\mathcal{T}_h\cap\tilde{\mathcal{T}}_h$ by using (50) in Lemma 4.2, we have

where the Young's inequality have been applied and $osc_{\mathcal{T}_h}(y-y_d,T) = osc_{\tilde{\mathcal{T}}_h}(y-y_d,T)$. Then summing over $T\in\mathcal{T}_h\cap\tilde{\mathcal{T}}_h$ for (53), we can easy to derive the desired result (50).

In order to facilitate computation, we introduce the following new notation

for $\omega\in\mathcal{T}_{h_k}$. As to the proof of the convergence, one of the main obstacle is that there do not have the orthogonality while it is vital of the proof for the convergence. Thus getting back to the second place we transfer proof of the quasi-orthogonality. The latter is popularly adopted in the adaptive mixed and the nonconforming adaptive finite element methods [24]. Apparently it is true for the following basic relationships with $\mathcal{T}_{h_k},\mathcal{T}_{h_{k+1}}\in\mathbb{T}$ and $\mathcal{T}_{h_k}\subset\mathcal{T}_{h_{k+1}}$ that

so that we obtain the quasi-orthogonality in Lemma 4.4 by estimating the last term of (54).

Lemma 4.4. For any $\epsilon>0,\mathcal{T}_{h_k},\mathcal{T}_{h_{k+1}}\in\mathbb{T}$, there holds

where $h_{0} = \max\limits_{{T\in \mathcal{T}_{h_{0}}}}h_{T}$.

Proof. We just prove (55) while (56) can be proved in a similar way. Let $y^{h_{k+1}}$ satisfying (16) with $f+u_{h_{k+1}}$, then we have

Similar to the proof of Lemma 4.3 in [23], we can estimate the second term of the right side of (57) as shown below

Next we will subdivide the proof. By applying (2), (18), and the triangle inequality, we have

in which we use the same way to derive that

It follows from the triangle inequality that

and apparently

Then in connection with what we discuss above to deduce that

For the first term of the right side of (57), we obtain

Contacting (57), (58) and (59) to gain

Because of $h_{\mathcal{T}_{h_{k+1}}}\leq h_{\mathcal{T}_{h_{k}}}$, we have the following result

such that

Then we can obtain (55) by combing with (54), (60), and (61).

Theorem 4.5. Let $(u,y,p)\in U_{ad}\times H_0^1(\Omega)\times H_0^1(\Omega)$ be the solution of (7)-(9) and $(u_h,y_h,p_h)\in U_{ad}^h\times V_h\times V_h$ be the solution of (12)-(14) generated by the adaptive finite element algorithm 3.1. Then exist $\gamma>0$ and $\alpha\in(0,1]$ depending only on the shape of regularity of initial $\mathcal{T}_{h_{0}},b,\Omega$ and the marking parameter $\theta\in(0,1]$ such that

apparently providing $h_0\ll1$ and sufficiently small $\mu$.

Proof. Taking Lemma 3.1, Lemma 4.3, and Lemma 4.4 into account, we can easy to derive that

Simplifying (64) and (65) by employing Lemma 3.1 and Lemma 4.1, we have

Multiplying (67) with $\gamma_1 = \frac{1}{C(1+\sigma^{-1})}$ and adding the result to (66), we can easy to deduce that

where $\gamma_2 = 1+\frac{C}{\epsilon}$. Applying the marking strategy in Algorithm in 3.1 and (63), we have

where $\gamma_3\in(0,1)$. By simple calculation and letting

where $\alpha_1\in(0,1)$ if choosing $\epsilon,\sigma$ small enough and where $\alpha_2\in(0,1)$ if electing $\mu,h_0$ sufficiently small, then here holds

which tells a contraction property, namely (62), in Algorithm 3.1 if selecting $\alpha = \max\{\alpha_1,\alpha_2\}$.

Theorem 4.6. Let $(u,y,p)\in U_{ad}\times H_0^1(\Omega)\times H_0^1(\Omega)$ be the solution of (7)-(9) and $(u_h,y_h,p_h)\in U_{ad}^h\times V_h\times V_h$ be the solution of (12)-(14) generated by the adaptive finite element algorithm 3.1 with the other conditions being same with Theorem 4.5, then there holds

Proof. It follows from Lemma 4.1 and Theorem 4.5 that we can obviously get

Then combining with Lemma 3.1, it is distinct to get the desired result in Theorem 4.6.

5.   Quasi-optimality analysis

In this section, we consider the quasi-optimality for the adaptive finite element method. Firstly we give the notations interpretation. For any $\mathcal{T}_h,\tilde{\mathcal{T}}_h\in\mathbb{T}$, let $\#\mathcal{T}_h$ be the number of elements in $\mathcal{T}_h$, and $\mathcal{T}_{h}\oplus\tilde{\mathcal{T}}_{h}$ be the smallest common conforming refinement of $\mathcal{T}_{h}$ and $\tilde{\mathcal{T}}_{h}$ satisfying [15,30,31] the property

According to [14,21,23,24], we need to define a function approximation class

where

and

We need a local upper bound for the distance between two nested solutions consulting [8] in order to illustrate the quasi-optimality of an adaptive finite element method due to the errors here can only be estimated by using refined element indices without buffer.

Lemma 5.1. Let $(u,y,p)\in U_{ad}\times H_0^1(\Omega)\times H_0^1(\Omega)$ be the solution of (7)-(9). Given sufficiently small $\mu$, let $\mathcal{T}_h\in\mathbb{T}_\mu$ and $\mathcal{T}_h\subset\tilde{\mathcal{T}}_{h}\in\mathbb{T}$, $(u_h,y_h,p_h)\in U_{ad}^h\times V_h\times V_h$ and $(\tilde{u}_h,\tilde{y}_h,\tilde{p}_h)\in U_{ad}^{\tilde{h}}\times V_{\tilde{h}}\times V_{\tilde{h}}$ be the solution of (12)-(14) on $\mathcal{T}_h$ and $\tilde{\mathcal{T}}_{h}$. Then there holds

where $\mathcal{R}_h: = \mathcal{R}_{\mathcal{T}_h\rightarrow\tilde{\mathcal{T}}_h}$ is the subset of elements that are refined from $\mathcal{T}_h$ to $\tilde{\mathcal{T}}_h$.

Proof. According to the optimal condition (14), we obtain

thus getting

For $(\tilde{p}_h-p_h,u_h-\tilde{u}_h)$ of (70), we have

In order to estimate the right side of the above inequality, we suppose that $\varphi\in H_0^1(\Omega)$ is the solution of the following equation

By applying the duality arguments, we can gain that

in which $\xi_{\mathcal{T}_h}$ is the standard Lagrange interpolator onto $V_h$. For the first term of right side of (71), we have

Similar to Lemma 2 in [14], we infer that

By using the similar way, we deduce that

Combining (72)-(76) and Lemma 2.2 with $H^2$-regularity, we derive that

Next we are going to estimate the second term on the right side of (70). Assume that

Then we set $v_h = \pi_{\mathcal{T}_{h}}\tilde{u}_h$ for which $\pi_{\mathcal{T}_{h}}$ is $L^2-$projection onto $\mathbb{P}_0(\mathcal{T}_h)$, thus obtaining

for arbitrary $T\in\mathcal{T}_{h}$ and where $I_{\mathcal{T}_{h}}$ is the identical $L^2-$projection onto $\mathbb{P}_0(\mathcal{T}_h)$. Owing to $T\in\mathcal{T}_h\backslash\mathcal{R}_h\subset\mathcal{T}_h$ such that $(\pi_{\mathcal{T}_{h}}-I_{\mathcal{T}_{h}})(\tilde{u}_h-u_h) = 0$, then here holds

In connection with (70), (77) and (78), we infer that

Employing (73)-(76) and the triangle inequality, we deduce that

It is similar to Lemma 2 in [14] that we infer that

To sum up, the proof is finished by adopting (79)-(81).

Next lemma tells the Dörfler property on the set $\mathcal{R}_h = \mathcal{R}_{\mathcal{T}_{h_k}\rightarrow\mathcal{T}_{h}}$ in order to bound the number of marked elements.

Lemma 5.2. We assume that the marking parameter $\theta\in(0,\theta^*)$ with $\theta^* = \frac{C}{1+4C}$ and let $(u_{h},y_h,p_h)\in U_{ad}^h\times V_h\times V_h$ and $(u_{h_k},y_{h_k},p_{h_k})\in U_{ad}^{h_k}\times V_{h_k}\times V_{h_k}$ be the solution of (12)-(14) on $\mathcal{T}_h$ and $\mathcal{T}_{h_k}$. There exists a constant $\delta = \frac{1}{2}(1-\frac{\theta}{\theta^*})$ such that

where $e_{\mathcal{T}_{h}}^2 = ||u-u_h||_0^2+||y-y_h||_0^2+||p-p_h||_0^2$, similarly for $e_{\mathcal{T}_{h_k}}^2$.Then there holds

Proof. Combining with (82) and the upper bound in Theorem 4.5, we can obtain

Employing the triangle inequality and the Young's inequality, here holds

Thus obtaining the result

with Lemma 5.1. By applying the dominance property which is similar to Remark 2.1 in [8], we infer that

for $T\in\mathcal{R}_h$ and apparently

for $T\in\mathcal{T}_h\backslash\mathcal{R}_h$. For $T\in \mathcal{T}_h\cap\tilde{\mathcal{T}}_h$, we can get the following result by employing (50) of Lemma 4.3 and the inverse estimates

Then it can be derived via using (83), (84) and (86) that

which tells the proof.

Lemma 5.3. Assume that the marking parameter $\theta$ satisfies the the conditions in Lemma 5.2. Let $(u,y,p)\in U_{ad}\times H_0^1(\Omega)\times H_0^1(\Omega)$ be the solution of (7)-(9) and $(u_{h_k},y_{h_k},p_{h_k})\in U_{ad}^{h_k}\times V_{h_k}\times V_{h_k}$ be the solution of (12)-(14) generated by Algorithm 3.1. Then the number of marked elements $\mathcal{M}_{h_k}\subset\mathcal{T}_{h_{k}}$ satisfies

if $(u,y,p,y_d)\in\mathcal{A}^s$ for $\mu$ being small enough.

Proof. Let $\nu^2 = \delta\beta^{\frac{1}{2s}}(e_{\mathcal{T}_{h_k}}^2+osc_{\mathcal{T}_{h_k}}^2(\mathcal{T}_{h_k}))$, where $\delta$ is defined in Lemma 5.2 and $\beta$ is to be defined as follows. Then there exists a $\mathcal{T}_{h_\nu}\in\mathbb{T}$ and $(u_{h_{\nu}},y_{h_{\nu}},p_{h_{\nu}})\in U_{ad}^{h_{\nu}}\times V_{h_{\nu}}\times V_{h_{\nu}}$ such that

for any $(u,y,p,y_d)\in\mathcal{A}^s$. Next we suppose $\mathcal{T}_{h_\epsilon} = \mathcal{T}_{h_\nu}\oplus\mathcal{T}_{h_k}$ is the smallest common refinement of $\mathcal{T}_{h_\nu}$ and $\mathcal{T}_{h_k}$, and let $(u_{h_\epsilon},y_{h_\epsilon},p_{h_\epsilon})$ be the solution of (12)-(14). Then we give the inequalities as follows in order to deduce the expect result we wanted

where there are some notations been defined by

and apparently $e_{\mathcal{T}_{h_\epsilon}}^2$ and $osc_{\mathcal{T}_{h_\epsilon}}^2(\mathcal{T}_{h_\epsilon})$ can be defined similarly. Just as obviously, here holds

for all $v\in U_{ad},v_{h_\nu}\in U_{ad}^{h_\nu},\ \mathrm{and}\ v_{h_\epsilon}\in U_{ad}^{h_\epsilon}$. Applying the Young's inequality and (91), we have

and apparently in the same way

Hence combining (91)-(94) to get

For all $T'\in\mathcal{T}_{h_\nu}$, assuming that $\mathcal{T}_{h_{T'}}: = \{T\in\mathcal{T}_{h_\epsilon}:T\in T'\}$, then we derive that

which tells that

By using the same way, we have

Therefore, we can gain (90) via employing (95)-(97). Based on the definition of $\nu^2$ and (89), we deduce that

According to Lemma 5.2, we find that the subset $\mathcal{R}_{\mathcal{T}_{h_k}\rightarrow\mathcal{T}_{h_\epsilon}}$ verifies the marking property for $\theta\leq\theta^*$, then we infer that

Hence we can obtain the desired result by contacting with (88), (98), and the definition of $\nu^2$.

Next we derive a equivalent property of the indicator dominates oscillation by concluding Theorem 3.2, Lemma 3.1, and Lemma 4.1 as follows

which is of vital importance for the proof of quasi-optimality.

Theorem 5.4. Assume that $\mathcal{T}_{h_0}$ satisfies the condition ($b$) of Section 4 in [32]. Let $(u,y,p)\in U_{ad}\times H_0^1(\Omega)\times H_0^1(\Omega)$ be the solution of (7)-(9) and $(u_{h_k},y_{h_k},p_{h_k})\in U_{ad}^{h_k}\times V_{h_k}\times V_{h_k}$ be the solution of (12)-(14) generated by Algorithm 3.1. Then there holds

provided $h_0\ll1$ if $(u,y,p,y_d)\in\mathcal{A}^s$ for $\mu$ being small enough.

Proof. Similar to the Theorem 4 in [14], we infer that

Employing (100) and Lemma 5.3 to get that

where $\chi = \beta^{\frac{1}{2s}}|(u,y,p,y_d)|_s^{\frac{1}{s}}\delta^{-\frac{1}{2s}}$. Combining with (99), (101), and Theorem 4.5, we deduce that

which yields the proof of Theorem 5.4.

6.   Numerical examples

In this section, we firstly present the adaptive iteration method where the purpose is to provide empirical analysis for our theory.

Algorithm 6.1. Given an initial control $u_h^0\in U_{ad}^h$, then for $k = 1,2,\cdots$, seek $(y_h^k,p_h^k,u_h^k)$ such that

and apparently

where $\mathcal{P}_h$ is the $L^2$-projection from $L^2(\Omega)$ to $U^h$ and $\bar{p}_{h}^k = \frac{\int_\Omega p_h^k}{|\Omega|}$.

Example 1. We consider the nonlinear optimal control problem subject to the state equation

where we choose $\alpha = 0.5$ and $\Omega = [0,1]\times[0,1]$ and apparently exact solution

By simple calculation we have $\int_\Omega pdx = -\frac{4}{\pi}$ which satisfies $u\in U_{ad}$.

In terms of the same error and actuarial accuracy, the adaptive refinement process saves time than the uniform refinement process. In Figures 1-2, we provide the profiles of the exact state variables, the numerical state variables, the exact co-state and the co-state on adaptively refined grids with $\theta = 0.3$ and 15 adaptive loops for Example 1 generated by Algorithm 3.1 and then we plot the adaptive girds after 5 steps and 13 steps with $\theta = 0.3$ and 15 adaptive iterations for Example 1 in Figure 3. It is easy to observe that lager gradients exist in some certain regions but the solutions are smooth. Moreover, the grid refinement at the center of the domain, while the solutions may have lager gradients near the boundary.

In Figure 4, we plot the profiles of the numerical state and the co-state variables on uniformly refined grids ($\theta = 1$) and 15 adaptive loops for Example 1 generated by Algorithm 3.1, and then the adaptively refined triangulations after 5 adaptive iterations with $\theta = 0.4$ and uniformly refined triangulations after 5 uniform iterations of 15 loops are performed in Figure 5. Obviously, it is not difficult to find that the uniformly refined grids seem to have a better encryption effect, but in connection with Figures 1-2 and Figure 4 to observe that the adaptive finite element method may deliver even much smaller errors compared to uniformly refined method. Combining Figure 3 and Figure 5, the adaptive iteration is more efficient and effective than the uniform iteration whereas the uniform iteration needs to be solved at a higher cost of grids, thus increasing iteration time. So the adaptive finite element method has some advantages in the numerical approximation process.

In Figure 6, we show the convergence history of the total error estimate indicators, where we plot the adaptive triangle iterations of 15 adaptive loops with the coefficient $\theta = 0.3$ and $\theta = 0.4$, even more there provides a convergence of the total error estimation indicators for the uniform triangle iterations ($\theta = 1$). We can see an error reduction with slope -1 that is the optimal convergence rate what we expect via applying linear finite elements from the upper two pictures in Figure 6. Meanwhile, we give the comparisons of the error estimations, for which we know that the optimal second order convergence for the error reductions of error estimations.

Example 2. We consider the same nonlinear optimal control problem as Example 1 with $\alpha = 0.1$, $\Omega = (0,1)\times(0,1)$, and apparently the exact solution

where $m = (x_1-0.2)^2+(x_2-0.6)^2-0.04$ and absolutely $\int_\Omega udx>0$ via simply calculating.

In Figure 7, we plot the numerical state and the co-state on adaptively refined grids with $\theta = 0.3$ and 27 adaptive loops for Example 2 generated by Algorithm 3.1. Obviously, we can observe the lager gradients concentrate on the certain regions as the adaptive finite element method may deliver much smaller errors compared with the uniform refinement. In Figure 8, we present the adaptive grids after 15 and 25 adaptively refined by choosing D$\mathrm{\ddot{o}}$rfler's marking parameter $\theta = 0.3$ with 27 adaptive loops. Apparently, the grids gather around the regions where there exist much lager gradients. Therefor they just validate the phenomenon in Figure 7. In addition, the grides focus on the points as $f$ and $y_d$ have singularities near these points. Hence when dealing with singular points, adaptive encryption has better effect.

We give the comparisons of convergence history of Example 2 in Figure 9. The left plot in Figure 9 is adaptively refined with D$\mathrm{\ddot{o}}$rfler's marking parameter $\theta = 0.3$ and 27 adaptive loops while the right one is uniform refinement $(\theta = 1)$. With the optimal $L^2-$norms convergence we desired, we can see the errors reduction for adaptive refinement. Moreover, the reduced orders only can be found in uniform refinement because of the singularity of the solutions.

Acknowledgments

The authors express their thanks to the referees for their helpful suggestions, which led to improvements of the presentation.