\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 267, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2017/267\hfil Finite volume methods of elliptic optimal control]
{A priori error estimates of finite volume methods for general
elliptic optimal \\ control problems}
\author[Y. Feng, Z. Lu, L. Cao, L. Li, S. Zhang \hfil EJDE-2017/267\hfilneg]
{Yuming Feng, Zuliang Lu, Longzhou Cao, Lin Li, Shuhua Zhang}
\address{Yuming Feng \newline
Key Laboratory of Intelligent Information Processing and Control,
Chongqing Three Gorges University, Wanzhou, Chongqing, 404100, China. \newline
Chongqing Engineering Research Center of Internet of Things
and Intelligent Control Technology,
Chongqing Three Gorges University,
Wanzhou, Chongqing, 404100, China}
\email{yumingfeng25928@163.com}
\address{Zuliang Lu (corresponding author)\newline
Key Laboratory for Nonlinear Science and System Structure,
Chongqing Three Gorges University,
Chongqing 404100, China. \newline
Research Center for Mathematics and Economics,
Tianjin University of Finance and Economics,
Tianjin 300222, China}
\email{zulianglux@126.com}
\address{Longzhou Cao \newline
Key Laboratory for Nonlinear Science and System Structure,
Chongqing Three Gorges University,
Chongqing 404100, China}
\email{caolongzhou@126.com}
\address{Lin Li \newline
Key Laboratory for Nonlinear Science and System Structure,
Chongqing Three Gorges University,
Chongqing 404100, China}
\email{linligx@126.com}
\address{Shuhua Zhang \newline
Research Center for Mathematics and Economics,
Tianjin University of Finance and Economics,
Tianjin 300222, China}
\email{szhang@tjufe.edu.cn}
\dedicatory{Communicated by Goong Chen}
\thanks{Submitted August 9, 2017. Published October 27, 2017.}
\subjclass[2010]{49J20, 65N30}
\keywords{A priori error estimates; general elliptic optimal control problems;
\hfill\break\indent finite volume methods; optimal-order}
\begin{abstract}
In this article, we establish a priori error estimates for the finite
volume approximation of general elliptic optimal control problems.
We use finite volume methods to discretize the state and adjoint
equation of the optimal control problems. For the variational
inequality, we use the variational discretization methods to
discretize the control. We show the existence and the uniqueness of
the solution for discrete optimality conditions. Under some
reasonable assumptions, we obtain some optimal order error estimates
for the state, costate and control variables. On one hand, the
convergence rate for the state, costate and control variables is
$O(h^2)$ or $O(h^2 \sqrt{|\log(\frac{1}{h})|})$ in the sense of $L^2$
norm or $L^{\infty}$ norm. On the other hand, the convergence rate
for the state and costate variables is $O(h)$ or $O(h{|\log (\frac{1}{h})|})$
in the sense of $H^1$ norm or $W^{1,\infty}$ norm.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks
\section{Introduction}
In recent years, optimal control problems have attracted substantial
interest 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. Finite
element methods are an important numerical method for the problems
of partial differential equations and widely used in the numerical
solution of optimal control problems. There have been extensive
studies in convergence of finite element approximation for optimal
control problems. Let us mention two early papers devoted to linear
optimal control problems by Falk \cite{F} and Geveci \cite{TG}. A
systematic introduction of finite element method for optimal control
problems can be found in \cite{ChenHuang, YL1, YL2, YLH, RWHT, JL,
WY, LC, LC1, LCZ}, but there are very less published results on this
topic for finite volume methods for optimal control problems.
Recently, the adaptive finite element method has been
investigated extensively and become one of the most popular methods
in the scientific computation and numerical modeling. In
\cite{hiis}, 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 \cite{KV, KL}. Some related works can also be found in
\cite{Tingwen1,Tingwen2}.
Finite volume methods have a long history as a class of important
numerical tools for solving differential equations. Because of their local
conservative property and other attractive properties such as the
robustness with the unstructured meshes, the finite volume methods
are widely used in computational fluid dynamics. In general, two
different functional spaces are used in the finite volume methods,
one for the trial space and one for the test space. Owing to the two
different spaces, the numerical analysis of the finite volume
methods is more difficult than that of the finite element methods
and finite difference methods. So, the analysis of finite volume
methods lags far behind that of finite element and finite difference
methods. Early work for the finite volume methods can be found in
\cite{Bank, Cai, Chat, ChenLiZhou, ChouLi, Ewing}. In \cite{Bank},
Bank and Rose obtain the result that the finite volume approximation
is comparable with the finite element approximation in $H^1$ norm.
The optimal $L^2$ error estimate is obtained in \cite{ChenLiZhou}
under the assumption that $f\in H^1$. In \cite{Ewing}, Ewing obtain
the $H^1$ norm and maximum-norm error estimates. In \cite{Chat}, the
author proposes a nonconforming finite volume element method and
obtains the $L^2$ norm and $H^1$ norm error estimates. Chou and Ye
propose a discontinuous finite volume element method. Unified error
analysis for conforming, nonconforming and discontinuous finite
volume method is presented in \cite{ChouYe}. High order finite
volume methods can be found in \cite{ChenLong, ChenWuXu}. For other
recently development, we refer reader to see \cite{Cartensen, DuJu,
Kumar, ShuYuHuang}.
For optimal control problems, the state and costate variables are
discretized by continuous linear elements and the control variable
by piecewise constant or piecewise linear polynomials in most
references. The convergence rate of the control variable is $O(h)$
or $O(h^{3/2})$ in the sense of $L^2$ norm or $L^{\infty}$ norm in
\cite{Meyer2}. In \cite{Hinze}, Hinze proposes a variational
discretization methods for optimal control problems with control
constraints. With the variational discretization concept, the
control variable is not discretized directly, but discretized by a
projection of the discrete costate variable. The convergence rate of
the control variable is $O(h^2)$. There are two approaches to find
the approximate solution of the optimal control problems governed by
partial differential equation. One is of the
optimize-then-discretize type. One first applies the Lagrange
multiplier methods to obtain an optimal system, at the continuous
level, consisting of the state equation, an adjoint equation and an
optimal condition. Then one use some numerical method to discretize
the resulting system. The other is of the discretize-then-optimize
type. One first discretizes the optimal control problems by some
means and then applies the Lagrange multiplier rule to the resulting
discrete optimization problem. The two discrete systems, determined
by the two approaches, are the same when finite element method is
used. In general, these discrete systems are not the same. In
\cite{YanZhou}, the authors also use the optimize-then-discretize
approach to solve the optimal control problem governed by convection
dominated diffusion equation.
Recently, in \cite{LCH}, the authors discussed distributed optimal
control problems governed by elliptic equations by using the finite
volume element methods. The objective functional was
$\frac{1}{2}||y-
y_{d}||^2_{L^2(\Omega)}+\frac{1}{2}||u||^2_{L^2(\Omega)}$.
They used finite volume methods to discretize the state and adjoint
equation of the optimal control problems. Under some reasonable
assumptions, they obtained some error estimates. In this paper, we
will use the optimize-then-discretize methods to discretize general
elliptic optimal control problems. We consider the elliptic optimal
control with objective functional $g(y)+j(u)$. We show the existence
and the uniqueness of the solution for discrete optimality
conditions. Finally, we obtain some optimal order error estimates
for the state, costate and control variables.
For $1\leq p<\infty$ and $m$ a
nonnegative integer let
$W^{m,p}(\Omega)=\{v\in L^{p}(\Omega);\ D^{\alpha}v\in L^{p}(\Omega)
\ \textrm{if}\ |\alpha|\leq m\} $ denote the Sobolev spaces endowed
with the norm $\| v \|_{m,p}^{p}=\sum_{\mid\alpha\mid\leq
m}\| D^\alpha v\|_{L^{p}(\Omega)}^{p},$ and the semi-norm $\mid
v\mid_{m,p}^{p}=\sum_{\mid\alpha\mid= m}\| D^\alpha
v\|_{L^{p}(\Omega)}^{p}$. We set $W_0^{m,p}(\Omega)=\{v\in
W^{m,p}(\Omega):
v\mid_{\partial \Omega}=0\}$. 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}.$
We consider the general elliptic optimal control problems
\begin{gather}
\min_{u\in U}\{g(y)+j(u)\},\label{1e1}\\
-\operatorname{div}(A\nabla y)=f+u,\quad \text{in }\Omega,\label{1e2}\\
y=0,\quad \text{on }\partial\Omega,\label{1e3}
\end{gather}
where $\Omega\subset \mathbb{R}^2$ is a convex bounded polygon with boundary
$\partial\Omega$, $g$ and $j$ are convex functionals,
$f\in H^1(\Omega)$, $U$ is denoted by $U=\{u\in L^2(\Omega): a\leq
u(x)\leq b,\ a.e.\ {\rm in}\ \Omega,\ a,b\in \mathbb{R}\}$. Furthermore, we
assume that the coefficient matrix $A(x)=(a_{i,j}(x))_{2\times 2}\in
(W^{2,\infty}({\Omega}))^{2\times 2}$ is a symmetric positive
definite matrix and there is a constant $c>0$ satisfying for any
vector $\mathbf{X}\in \mathbb{R}^2$, $\mathbf{X}^{t}A\mathbf{X}\geq c\|
\mathbf{X}\|_{\mathbb{R}^2}^2$.
This article is organized as follows.
In next section, we describe the
finite volume methods briefly and apply the piecewise linear finite
volume elements to the optimal control problems
\eqref{1e1}-\eqref{1e3}.
In Section 3, we prove the existence and
the uniqueness of the solutions for discrete optimality conditions.
And then the optimal order error estimates in $L^2$ norm are derived
for the state, costate and control variables in Second 4. We
estimate the error of the numerical solutions of control, state and
costate in $L^{\infty}$ norm. Finally we estimate $W^{1,\infty}$ and
$H^1$ errors for the state and costate variables in Second 5.
\section{Finite volume element methods}
For the convex polygon $\Omega$, we consider a quasi-uniform
triangulation $\mathcal {T}_h$ consisting of closed triangle
elements $K$ such that $\bar{\Omega}=\cup_{K\in\mathcal
{T}_h}K$. We use $N_h$ to denote the set of all nodes or vertices of
$\mathcal {T}_h$. To define the dual partition $\mathcal {T}_h^*$ of
$\mathcal {T}_h$, we divide each $K\in\mathcal {T}_h$ into three
quadrilaterals by connecting the barycenter $C_K$ of $K$ with line
segments to the midpoints of edges of $K$ as is shown in Figure \ref{fig1}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig1} % TriDu-2.eps
\end{center}
\caption{Dual partition of a triangular $K$.}
\label{fig1}
\end{figure}
The control volume $V_i$ consists of the quadrilaterals sharing the
same vertex $z_i$ as is shown in Figure \ref{fig2}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig2} % Vi-1.eps
\end{center}
\caption{Control volume $V_i$ sharing the same vertex $z_i$.}
\label{fig2}
\end{figure}
The dual partition $\mathcal {T}_h^*$ consists of the union of the
control volume $V_i$. Let $h=\max\{h_K \}$, where $h_K$ is the
diameter of the triangle $K$. As is shown in \cite{Ewing}, the dual
partition $\mathcal {T}_h^*$ is also quasi-uniform, i.e., there
exists a positive constant $C$ such that
\[
C^{-1}h^2\leq \operatorname{meas}(V_i)\leq Ch^2,\quad \forall V_i \in\mathcal
{T}_h^*.
\]
We define the finite dimensional space $V_h$ associated with
$\mathcal {T}_h$ for the trial functions by
\begin{equation*}
V_h=\{v\in C(\Omega): v|_K\in P_1(K),\; \forall K\in \mathcal
{T}_h,\; v|_{\partial\Omega}=0 \},
\end{equation*}
and define the finite dimensional space $Q_h$ associated with the
dual partition $\mathcal {T}_h^*$ for the test functions by
\begin{equation*}
Q_h=\{q\in L^2(\Omega): q|_V\in P_0(V),\; \forall V\in \mathcal
{T}_h^*;\; q|_{V_z}=0,\; z\in\partial\Omega\},
\end{equation*}
where $P_l(K)$ or $P_l(V)$ consists of all the polynomials with
degree less than or equal to $l$ defined on $K$ or $V$.
To connect the trial space and test space, we define a transfer
operator $I_h: V_h\to Q_h$ as follows:
\begin{equation*}
I_hv_h=\sum_{z_i\in N_h} v_h(z_i)\chi_i ,\quad
I_hv_h|_{V_i}=v_h(z_i),\quad \forall V_i \in\mathcal {T}_h^*,
\end{equation*}
where $\chi_i$ is the characteristic function of $V_i$. For the
operator $I_h$, it is well known that there exists a positive
constant $C$ such that for all $v\in V_h$,
\begin{equation}
\|v-I_hv\|_{0,\Omega}\leq C h \|v\|_{1,\Omega}.\label{2e1}
\end{equation}
To address the finite volume methods clearly, we consider the
problem
\begin{gather}
-\operatorname{div}(A\nabla \varphi)=f,\quad \text{in }\Omega,\label{2e2}\\
\varphi=0,\quad \text{on }\partial\Omega, \label{2e3}
\end{gather}
where $A$, $\Omega$, $\partial\Omega$ are the same as in
\eqref{1e2}-\eqref{1e3}, $f\in L^2(\Omega)$ or $H^1(\Omega)$.
The finite volume approximation $\varphi_h$ of
\eqref{2e2}-\eqref{2e3} is defined as the solution of the problem:
find $\varphi_h\in V_h$ such that
\begin{equation}
a(\varphi_h,I_hv_h)=(f,I_hv_h),\quad \forall v_h\in V_h,\label{2e4}
\end{equation}
where the bilinear form $a(\varphi_h,I_hv_h)$ is defined by
\begin{equation*}
a(\varphi,I_hv)=-\sum_{z_i\in N_h}v(z_i)\int_{\partial V_i}A
\nabla\varphi\cdot \mathbf{n}ds,\quad \varphi,v\in H^1_0(\Omega),
\end{equation*}
where $\mathbf{n}$ is the unit outward normal vector to $\partial
V_i$. The bilinear form $a(\cdot,\cdot)$ is not symmetric though the
problem is self-adjoint. Then for all $w_h,v_h\in V_h$, there exist
positive constants $C$ and $h_0\geq 0$ \cite{ChouLi} such that for
all $00.\label{czwy5}
\end{equation}
Let $T(z_h)=P_k\Phi(z_h)$, then the existence and uniqueness of
\eqref{3e31}-\eqref{3e33} is to show that $T(z_h)$ is a contractive
mapping. It follows from \eqref{czwy2} that for all
$z'_h,z''_h\in L^2(\Omega)$,
\begin{align*}
\|T(z'_h)-T(z''_h)\|_{0,\Omega}^2
&=\|P_k(\Phi(z'_h))-P_k(\Phi(z''_h))\|_{0,\Omega}^2\\
&\leq \|\Phi(z'_h)-\Phi(z''_h)\|_{0,\Omega}^2
=(\Phi(z'_h)-\Phi(z''_h),\Phi(z'_h)-\Phi(z''_h)).
\end{align*}
Note that
\begin{align*}
&(\Phi(z'_h)-\Phi(z''_h),\Phi(z'_h)-\Phi(z''_h))\\
&=(1-2\rho)(z'_h-z''_h,z'_h-z''_h)
-2\rho(z'_h-z''_h,p_h(z'_h)-p_h(z''_h))\\
&\quad +\rho^2\|z'_h-z''_h+p_h(z'_h)-p_h(z''_h)\|_{0,\Omega}^2.
\end{align*}
Then we have
\begin{equation}
\begin{aligned}
&\|T(z'_h)-T(z''_h)\|_{0,\Omega}^2\\
&=\leq(1-2\rho)(z'_h-z''_h,z'_h-z''_h)
-2\rho(z'_h-z''_h,p_h(z'_h)-p_h(z''_h)) \\
&\quad+\rho^2\|z'_h-z''_h+p_h(z'_h)-p_h(z''_h)\|_{0,\Omega}^2.
\end{aligned}\label{czwy6}
\end{equation}
For $z'_h,z''_h\in L^2(\Omega)$, it follows from
\eqref{czwy3}-\eqref{czwy4} and \eqref{phi} that
\begin{gather*}
a(y_h(z'_h)-y_h(z''_h),I_hw_h)=(z'_h-z''_h,I_hw_h), \quad\forall w_h\in V_h,\\
a(p_h(z'_h)-p_h(z''_h),I_hq_h)=(\tilde{g}''(y_h(z'_h))(y_h(z'_h)-y_h(z''_h)),
I_hq_h),\quad \forall q_h\in V_h.
\end{gather*}
Let $w_h=p_h(z'_h)-p_h(z''_h)$ and $q_h=y_h(z'_h)-y_h(z''_h)$, we
have
\begin{align*}
&(z'_h-z''_h,p_h(z'_h)-p_h(z''_h))\\
&=(\tilde{g}''(y_h(z'_h))(y_h(z'_h)-y_h(z''_h)), I_h(y_h(z'_h)-y_h(z''_h)))\\
&\quad +a(y_h(z'_h)-y_h(z''_h),I_h(p_h(z'_h)-p_h(z''_h)))\\
&\quad -a(p_h(z'_h)-p_h(z''_h),I_h(y_h(z'_h)-y_h(z''_h)))\\
&\quad +(z'_h-z''_h,(p_h(z'_h)-p_h(z''_h))-I_h(p_h(z'_h)-p_h(z''_h)))\\
&\geq a(y_h(z'_h)-y_h(z''_h),I_h(p_h(z'_h)-p_h(z''_h)))\\
&\quad -a(p_h(z'_h)-p_h(z''_h),I_h(y_h(z'_h)-y_h(z''_h)))\\
&\quad +(z'_h-z''_h,(p_h(z'_h)-p_h(z''_h))-I_h(p_h(z'_h)-p_h(z''_h))),
\end{align*}
where we have used the fact that $(v_h,I_hv_h)\geq 0$.
Using \cite[Lemma 2.4]{ChouLi} and Lemma \ref{lem9}, we have
\begin{equation}
\begin{aligned}
&a\big(y_h(z'_h)-y_h(z''_h),I_h(p_h(z'_h)-p_h(z''_h))\big) \\
&-a\big(p_h(z'_h)-p_h(z''_h), I_h(y_h(z'_h)-y_h(z''_h))\big) \\
&\geq -c_0h\|p_h(z'_h)-p_h(z''_h)\|_{1,\Omega}\cdot
\|y_h(z'_h)-y_h(z''_h)\|_{1,\Omega} \\
&\geq -c_0c_1h\|z'_h-z''_h\|_{0,\Omega}^2.
\end{aligned} \label{czwy8}
\end{equation}
Note that by \eqref{2e1} and Lemma \ref{lem9}, we have
\begin{equation}
\begin{aligned}
&(z'_h-z''_h,(p_h(z'_h)-p_h(z''_h))-I_h(p_h(z'_h)-p_h(z''_h)))
\\
&\geq -c_2h\|p_h(z'_h)-p_h(z''_h)\|_{1,\Omega} \cdot\|z'_h-z''_h\|_{0,\Omega} \\
&\geq -c_2c_3h\|z'_h-z''_h\|_{0,\Omega}^2.
\end{aligned} \label{czwy9}
\end{equation}
Combining \eqref{czwy8} and \eqref{czwy9}, we deduce that
\begin{equation}
(z'_h-z''_h,p_h(z'_h)-p_h(z''_h))
\geq -(c_0c_1+c_2c_3)h \|z'_h-z''_h\|_{0,\Omega}^2. \label{czwy10}
\end{equation}
Now, it is easy to see that
\begin{equation}
\|z'_h-z''_h+p_h(z'_h)-p_h(z''_h)\|_{0,\Omega}^2\leq c_4
\|z'_h-z''_h\|_{0,\Omega}^2.\label{czwy11}
\end{equation}
Then it follows from \eqref{czwy6}, \eqref{czwy10}, and
\eqref{czwy11} that
\begin{equation}
\|T(z'_h)-T(z''_h)\|_{0,\Omega}^2\leq C\|z'_h-z''_h\|_{0,\Omega}^2.
\end{equation}
For sufficiently small $h$ we can ensure $00$ such that for all $00$ such that for all
$00$ such that for all
$00$ such that for all
$00$ such that for all
$0