# DBCP

sudiptoc506  Sudipto Chowdhury , Divay Garg  Department of Mathematics, The
LNM Institute of Information Technology Jaipur, Rajasthan- 302031, India
(sudipto.choudhary@lnmiit.ac.in)Department of Mathematics, Indian Institute of
Technology Delhi, New Delhi- 110016, India (divaygarg2@gmail.com)[The work of
the author is granted by CSIR Research Fellowship]

(August 2023)

# Non-Conforming Finite Element Method For Constrained Dirichlet Boundary
Control Problem

sudiptoc506  Sudipto Chowdhury , Divay Garg  Department of Mathematics, The
LNM Institute of Information Technology Jaipur, Rajasthan- 302031, India
(sudipto.choudhary@lnmiit.ac.in)Department of Mathematics, Indian Institute of
Technology Delhi, New Delhi- 110016, India (divaygarg2@gmail.com)[The work of
the author is granted by CSIR Research Fellowship]

(August 2023)

###### Abstract

This article examines the Dirichlet boundary control problem governed by the
Poisson equation, where the control variables are square integrable functions
defined on the boundary of a two dimensional bounded, convex, polygonal
domain. It employs an ultra weak formulation and utilizes Crouzeix-Raviart
finite elements to discretize the state variable, while employing piecewise
constants for the control variable discretization. The study demonstrates that
the energy norm of an enriched discrete optimal control is uniformly bounded
with respect to the discretization parameter. Furthermore, it establishes an
optimal order _a priori_ error estimate for the control variable.

Keywords: Optimal control problem, Crouziex-Raviart, Finite element method, A
priori error analysis, Control constraints
AMS subject classification: 65N30, 65N15, 65N12, 65K10

##  1 Introduction

We consider the control constrained Dirichlet boundary control problem on a
convex and bounded polygonal domain
$\displaystyle\Omega\subseteq\mathbb{R}^{2}$. We denote the polygonal boundary
of the domain $\displaystyle\Omega$ by
$\displaystyle\Gamma:=\cup_{j=1}^{k}\Gamma_{j}$, where
$\displaystyle\Gamma_{j}$ is a straight line segment for all $\displaystyle
1\leq j\leq k$. The minimization problem seeks a pair
$\displaystyle(\bar{y},\bar{u})\in L_{2}(\Omega)\times U_{ad}$ such that

$\displaystyle\displaystyle J(\bar{y},\bar{u})=\min_{(y,u)\in
L_{2}(\Omega)\times U_{ad}}J(y,u),$ (1.1)

subject to the following second order elliptic equation:

$\displaystyle\displaystyle\begin{cases}-\Delta
y&=f\quad\text{in}\quad\Omega,\\\ \hskip
11.38092pty&=u\quad\text{on}\quad\Gamma.\end{cases}$ (1.2)

The given function $\displaystyle f\in L_{2}(\Omega)$ denotes the force which
acts on the system externally and $\displaystyle U_{ad}\subseteq
L_{2}(\Gamma)$ is the set of admissible controls which will be defined later.
The quadratic cost functional $\displaystyle
J(\cdot,\cdot):L_{2}(\Omega)\times L_{2}(\Gamma)\rightarrow\mathbb{R}$ is
defined in the following manner:

$\displaystyle\displaystyle
J(y,u):=\frac{1}{2}\int_{\Omega}|y-y_{d}|^{2}\,dx+\frac{\alpha}{2}\int_{\Gamma}|u|^{2}\,ds,$
(1.3)

where $\displaystyle y_{d}\in L_{2}(\Omega)$ is the given desired state and
$\displaystyle\alpha>0$ is known parameter which is introduced for the
regularization. The state equation (1.2) under the standard weak formulation
is not well posed for $\displaystyle u\in L_{2}(\Gamma)$ as the weak solution
of (1.2) lie in $\displaystyle H^{1}(\Omega)$ space and therefore by trace
theorem [7], the first trace of it must lie in $\displaystyle H^{1/2}(\Gamma)$
which is not the case in this formulation. There are multiple techniques
available in the literature to overcome this variational difficulty, one of
which is to use the method of transposition which is also known as the ultra
weak formulation [11, 24]. In brevity, we discuss the method of transposition.
Define a test function space as follows:

$\displaystyle\displaystyle X:=\\{\phi\in H^{1}_{0}(\Omega):\Delta\phi\in
L_{2}(\Omega)\\}.$

By elliptic regularity theory on convex polygonal domains, we have
$\displaystyle X=H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$. Now, we obtain the
ultra weak formulation of the state equation (1.2). Multiplying
$\displaystyle\phi\in X$ in both sides of (1.2) and then integrate both the
sides to find

$\displaystyle\displaystyle-\int_{\Omega}\Delta y\,\phi\,dx$
$\displaystyle\displaystyle=\int_{\Omega}f\,\phi\,dx.$

A use of integration by parts in the above equation yields

$\displaystyle\displaystyle\int_{\Omega}f\,\phi\,dx$
$\displaystyle\displaystyle=\int_{\Omega}\nabla
y\cdot\nabla\phi\,dx-\int_{\Gamma}\phi\,\frac{\partial y}{\partial n}\,ds$
$\displaystyle\displaystyle=\int_{\Omega}\nabla y\cdot\nabla\phi\,dx,$

where we have used the fact that $\displaystyle\phi\in H^{1}_{0}(\Omega)$ to
get the last equality. The idea in the method of transposition is to apply the
integration by parts once more. By doing so, we get

$\displaystyle\displaystyle\int_{\Omega}f\,\phi\,dx$
$\displaystyle\displaystyle=-\int_{\Omega}y\,\Delta\phi\,dx+\int_{\Gamma}y\,\frac{\partial\phi}{\partial
n}\,ds.$

Thus, the ultra weak formulation of (1.2) can be stated as: Given
$\displaystyle f\in L_{2}(\Omega)$ and $\displaystyle u\in L_{2}(\Gamma)$,
find $\displaystyle y\in L_{2}(\Omega)$ such that

$\displaystyle\displaystyle(y,\Delta\phi)$
$\displaystyle\displaystyle=-(f,\phi)+\langle u,\partial\phi/\partial
n\rangle\quad\forall~{}\phi\in X,$ (1.4)

where $\displaystyle(\cdot,\cdot)$ and
$\displaystyle\langle\cdot,\cdot\rangle$ denotes the $\displaystyle L_{2}$
inner product on the domain $\displaystyle\Omega$ and its boundary
$\displaystyle\Gamma$ respectively. Another approach to study second order
Dirichlet boundary control problem is to transform the Dirichlet boundary
condition into Robin’s type boundary condition [12]. Other method is to employ
the energy space based approach in which the control is penalized in the
energy space $\displaystyle H^{1/2}(\Gamma)$ [13, 26]. In this article the
control variable is discretized with piecewise constant finite elements while
the state and adjoint states are discretized using first-order non conforming
Crouzeix Raviart finite elements.

In the finite element analysis for the Dirichlet boundary control problems
addressed so far in the literature polynomial degrees used to discretize the
state variable or the adjoint state variable and the control variable happens
to be the same. But in this article we have discretized the control variable
with piecewise constant finite elements whereas the state and adjoint
variables are discretized using Crouzeix Raviart finite elements. Therefore
the main challenge is to associate the discrete state variables with the
discrete control variables via an invertible bounded linear operator such that
the operator norm of the inverse remains uniformly bounded with respect to the
discretization parameter $\displaystyle h>0$ (to be defined later). In order
to construct this operator in (4.21) a mild restriction on the triangulation
is imposed (discussed in Section 4 ). This constitutes one of the main
novelties of this article. Apart from this an enrichment operator is
introduced satisfying the orthogonality described in Theorem 3.4, which plays
a crucial role in the entire analysis. Moreover in Theorem 4.1, 4.2,4.3, 5.2
and 4.5 several super convergence results are derived, which also contributes
significantly to the novelty of this article.

##  2 Continuous Problem

In this segment, we discuss the variational setting of the investigated
problem and derive the first order necessary optimality conditions. Before, we
enter into the analysis part, we define the notations which are used
throughout the article. The notations $\displaystyle H^{m}(\Omega)$,
$\displaystyle H_{0}^{m}(\Omega)$, and $\displaystyle W^{m,p}(\Omega)$
represent the standard Sobolev spaces on $\displaystyle\Omega$, where
$\displaystyle m$ is a non negative integer. The Sobolev spaces of non integer
order $\displaystyle s$, $\displaystyle H^{s}(\Omega)$ and $\displaystyle
W^{s,p}(\Omega)$, are defined using interpolation, as explained in [7]. On a
Lipschitz domain, this definition is equivalent to the definition that uses
double-integral norms, which is discussed in [7, Theorem 14.2.3]. The norms of
$\displaystyle H^{s}(\Gamma)$ and $\displaystyle W^{s,p}(\Gamma)$,
$\displaystyle 0\leq s\leq 1$ and $\displaystyle 1<p<\infty$, are defined on
the boundary $\displaystyle\Gamma$ using charts, which is equivalent to using
double-integral norms on $\displaystyle\Gamma$, according to [7, 25]. The
outwards unit normal vector to $\displaystyle\Gamma$ is denoted by
$\displaystyle n$. The dual space of $\displaystyle H^{1/2}(\Gamma)$ is
denoted by $\displaystyle H^{-1/2}(\Gamma)$ equipped with the operator norm
$\displaystyle\|\cdot\|_{H^{-1/2}(\Gamma)}$ defined as:

$\displaystyle\displaystyle\|\mu\|_{H^{-1/2}(\Gamma)}:=\sup\limits_{v\in
H^{1/2}(\Gamma),v\neq
0}\frac{\langle\mu,v\rangle}{\|v\|_{H^{1/2}(\Gamma)}},\quad\mu\in
H^{-1/2}(\Gamma).$

Any generic member of $\displaystyle\mathbb{R}^{2}$ is denoted by
$\displaystyle x=(x_{1},x_{2})$, where $\displaystyle x_{1}$, $\displaystyle
x_{2}\in\mathbb{R}$. In the article, $\displaystyle C$ denotes a positive
constant, which is generic in nature and does not depend on the solutions or
the mesh-size.

######  Remark 2.1.

When $\displaystyle v$ belongs to $\displaystyle H^{2}(\Omega)$, the trace of
its gradient, denoted by $\displaystyle\nabla v|_{\Gamma}$, lies in
$\displaystyle H^{1/2}(\Gamma)^{2}$. If $\displaystyle\Gamma$ has a smooth
boundary, the outwards unit normal vector $\displaystyle n$ is continuous,
allowing the definition of the normal derivative
$\displaystyle\partial_{n}v=n\cdot\nabla v$ to be well defined. Thus
$\displaystyle\partial_{n}v\in H^{1/2}(\Gamma)$ for $\displaystyle v\in
H^{2}(\Omega)$ and the following estimate holds:

$\displaystyle\displaystyle\|\partial_{n}v\|_{H^{1/2}(\Gamma)}\leq
C\|v\|_{H^{2}(\Omega)},\quad v\in H^{2}(\Omega).$

However, this estimate is not applicable for polygonal boundaries
$\displaystyle\Gamma$, which are only Lipschitz continuous. Nonetheless, for
$\displaystyle v\in H^{2}(\Omega)$, we still have
$\displaystyle\partial_{n}v|_{\Gamma_{j}}\in H^{1/2}(\Gamma_{j})$ for each
straight component $\displaystyle\Gamma_{j}$, $\displaystyle 1\leq j\leq k$ of
$\displaystyle\Gamma$. To accommodate this, we introduce the space
$\displaystyle\tilde{H}^{1/2}(\Gamma):=\\{v\in
L_{2}(\Gamma),v|_{\Gamma_{j}}\in H^{1/2}(\Gamma_{j})\quad\forall~{}1\leq j\leq
m\\}$. We denote by $\displaystyle\tilde{H}^{-1/2}(\Gamma)$ the completion of
$\displaystyle L_{2}(\Gamma)$ with respect to the following dual norm on
$\displaystyle L_{2}(\Gamma)$

$\displaystyle\displaystyle\|\mu\|_{\tilde{H}^{-1/2}(\Gamma)}:=\sup\limits_{v\in
X\setminus\\{0\\}}\frac{\langle\mu,\partial_{n}v\rangle}{\|v\|_{H^{2}(\Omega)}}\leq\sup_{v\in\tilde{H}^{1/2}(\Gamma),v\neq
0}\frac{\langle\mu,v\rangle}{\|v\|_{\tilde{H}^{1/2}(\Gamma)}}.$

Note that $\displaystyle\tilde{H}^{-1/2}(\Gamma)$ is not in general the dual
space of $\displaystyle\tilde{H}^{1/2}(\Gamma)$. In the case of a smooth
boundary $\displaystyle\Gamma$, the mapping
$\displaystyle\partial_{n}:X\rightarrow H^{1/2}(\Gamma)$ is surjective, so
$\displaystyle\tilde{H}^{-1/2}(\Gamma)=H^{-1/2}(\Gamma)$.

For any $\displaystyle u\in\tilde{H}^{-1/2}(\Gamma)$, the harmonic extension
of $\displaystyle u$ is denoted by $\displaystyle y_{u}\in L_{2}(\Omega)$
which satisfies the following equation:

$\displaystyle\displaystyle(y_{u},\Delta\phi)$
$\displaystyle\displaystyle=\langle
u,\partial_{n}\phi\rangle\quad\forall~{}\phi\in X.$ (2.1)

The function $\displaystyle u\in\tilde{H}^{-1/2}(\Gamma)$ is known as the
ultra weak trace of $\displaystyle y_{u}\in L_{2}(\Omega)$ and the following
trace estimate holds true:

$\displaystyle\displaystyle\|u\|_{\tilde{H}^{-1/2}(\Gamma)}=\sup\limits_{\phi\in
X\setminus\\{0\\}}\frac{(y_{u},\Delta\phi)}{\|\phi\|_{H^{2}(\Omega)}}\lesssim\|y_{u}\|_{L_{2}(\Omega)}.$

Furthermore, if $\displaystyle u\in H^{1/2}(\Gamma)$, then $\displaystyle
y_{u}\in H^{1}(\Omega)$ becomes the standard harmonic extension of
$\displaystyle u$ such that it satisfies the following equation:

$\displaystyle\displaystyle(\nabla y_{u},\nabla\phi)=0\quad\forall~{}\phi\in
V.$

In this case $\displaystyle u\in H^{1/2}(\Gamma)$ is simply the trace of
$\displaystyle y_{u}$ in the sense that $\displaystyle y_{u}|_{\Gamma}=u$.

###  2.1 Optimality System

In this section, we write the model Dirichlet boundary optimal control problem
and derive the corresponding optimality system by applying the first order
necessary optimality conditions. For $\displaystyle(f,u)\in
L_{2}(\Omega)\times L_{2}(\Gamma)$, define $\displaystyle(y_{f},y_{u})\in
L_{2}(\Omega)\times L_{2}(\Omega)$ to be the solutions of the following
equations:

$\displaystyle\displaystyle(y_{f},\Delta\phi)$
$\displaystyle\displaystyle=-(f,\phi)\quad\forall~{}\phi\in X,$ (2.2)
$\displaystyle\displaystyle(y_{u},\Delta\phi)$
$\displaystyle\displaystyle=\langle
u,\partial_{n}\phi\rangle\quad\forall~{}\phi\in X.$ (2.3)

In view of (2.2)-(2.3), the ultra weak solution $\displaystyle y\in
L_{2}(\Omega)$ of (1.4) can be written as $\displaystyle y=(y_{f}+y_{u})$. In
addition, if the given data satisfy $\displaystyle(f,u)\in L_{2}(\Omega)\times
H^{1/2}(\Gamma)$, then we can recover the standard weak formulation, i.e. the
following holds:

$\displaystyle\displaystyle\begin{cases}y&=y_{f}+y_{u},~{}y_{f}\in
H^{1}_{0}(\Omega),~{}y_{u}\in H^{1}(\Omega),\\\
y_{u}&=u\quad\text{on}~{}\Gamma,\\\
a(y_{f},v)&=(f,v)-a(y_{u},v)\quad\forall~{}v\in H^{1}_{0}(\Omega),\end{cases}$
(2.4)

where $\displaystyle a(\cdot,\cdot):H^{1}(\Omega)\times
H^{1}(\Omega)\rightarrow\mathbb{R}$ is a symmetric bilinear form given by:

$\displaystyle\displaystyle a(v,w)$
$\displaystyle\displaystyle=\int_{\Omega}\nabla v\cdot\nabla
w\,dx\quad\forall~{}v,w\in H^{1}(\Omega).$

The bilinear form $\displaystyle a(\cdot,\cdot)$ defines a continuous mapping
on $\displaystyle H^{1}(\Omega)$ and coercive map on $\displaystyle
H^{1}_{0}(\Omega)$. Set $\displaystyle V=H^{1}_{0}(\Omega)$ and $\displaystyle
Q=H^{1}(\Omega)$. Given that $\displaystyle\Omega$ is a convex polygonal
domain, for $\displaystyle u=0$, the weak solution $\displaystyle y$ of (1.2)
belongs to $\displaystyle H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$. Additionally,
it satisfies the following _a priori_ bound.

$\displaystyle\displaystyle\|y\|_{H^{2}(\Omega)}\leq C\|f\|_{L_{2}(\Omega)}.$

The subsequent lemma affirms the well-posed-ness of the boundary value problem
(1.2) in its ultra weak form, and as a particular instance, it ensures the
presence of the ultra weak harmonic extension of the general boundary data
$\displaystyle u\in\tilde{H}^{-1/2}(\Gamma)$.

######  Lemma 2.2.

For any given $\displaystyle u\in\tilde{H}^{-1/2}(\Gamma)$ and $\displaystyle
f\in H^{-2}(\Omega)$, the state equation in its ultra weak form (1.4) has a
unique solution $\displaystyle y\in L_{2}(\Omega)$ such that the following _a
priori_ estimate is satisfied:

$\displaystyle\displaystyle\|y\|_{L_{2}(\Omega)}\leq
C\Big{(}\|f\|_{H^{-2}(\Omega)}+\|u\|_{\tilde{H}^{-1/2}(\Gamma)}\Big{)},$ (2.5)

where $\displaystyle H^{-2}(\Omega)$ is the dual space of $\displaystyle X$.

###### Proof.

For $\displaystyle f\in H^{-1}(\Omega)$ and $\displaystyle u\in
H^{1/2}(\Gamma)$, there exists a unique weak solution $\displaystyle y\in
H^{1}(\Omega)$ of (1.2) satisfying (2.4). An application of integration by
parts in (2.4) yields

$\displaystyle\displaystyle-(y,\Delta\phi)$
$\displaystyle\displaystyle=(f,\phi)-\langle
u,\partial_{n}\phi\rangle\quad\forall~{}\phi\in X,$ (2.6)

which implies that $\displaystyle y$ is the solution of (1.4). We apply
duality argument to prove the _a priori_ error estimate (2.5). Consider
$\displaystyle z\in V$ to be the weak solution of the following Poisson
problem:

$\displaystyle\displaystyle\begin{cases}-\Delta
z&=y\quad\text{in}\quad\Omega,\\\ \hskip
11.38092ptz&=0\quad\text{on}\quad\Gamma.\end{cases}$

By elliptic regularity theory, $\displaystyle z\in H^{2}(\Omega)$ and
$\displaystyle\|z\|_{H^{2}(\Omega)}\leq C\|y\|_{L_{2}(\Omega)}$. Thus, using
(2.6) for $\displaystyle\phi=z$, we get

$\displaystyle\displaystyle\|y\|^{2}_{L_{2}(\Omega)}$
$\displaystyle\displaystyle=(y,-\Delta z)$
$\displaystyle\displaystyle=(f,z)-\langle u,\partial_{n}z\rangle,$

which on using Cauchy-Schwarz inequality yields (2.5). Since $\displaystyle
H^{-1}(\Omega)\subseteq H^{-2}(\Omega)$ and $\displaystyle
H^{1/2}(\Gamma)\subseteq\tilde{H}^{-1/2}(\Gamma)$ are dense, by density
arguments, for $\displaystyle u\in\tilde{H}^{-1/2}(\Gamma)$ and $\displaystyle
f\in H^{-2}(\Omega)$, the equation (1.4) possesses a unique solution
$\displaystyle y\in L_{2}(\Omega)$ satisfying the _a priori_ error estimate
(2.5). ∎

In light of the Lemma 2.2, we define the control to state map which maps the
given data to the unique solution of (1.4).

######  Definition 2.3.

For $\displaystyle(f,u)\in L_{2}(\Omega)\times L_{2}(\Gamma)$, the solution
map $\displaystyle S:L_{2}(\Omega)\times L_{2}(\Gamma)\rightarrow
L_{2}(\Omega)$ is defined as $\displaystyle S(f,u)=y=(y_{f}+y_{u})$ such that
$\displaystyle y_{f},y_{u}$ satisfy (2.2) and (2.3), respectively.

Now, we define the space of admissible controls $\displaystyle U_{ad}$ as
follows:

$\displaystyle\displaystyle U_{ad}:=\\{u\in L_{2}(\Gamma):u_{a}\leq u(x)\leq
u_{b}~{}~{}\text{a.e}~{}~{}x\in\Gamma\\},$

where $\displaystyle u_{a}$, $\displaystyle u_{b}$
$\displaystyle\in\mathbb{R}$ are such that $\displaystyle u_{a}<u_{b}$. The
model Dirichlet boundary optimal control problem reads as: find
$\displaystyle(\bar{y},\bar{u})\in L_{2}(\Omega)\times U_{ad}$ such that

$\displaystyle\displaystyle J(\bar{y},\bar{u})=\min_{(y,u)\in
L_{2}(\Omega)\times U_{ad}}J(y,u),$ (2.7)

subject to the constraint $\displaystyle y=S(f,u)$, where $\displaystyle
J(\cdot,\cdot)$ is the quadratic cost functional defined by (1.3). Next, we
present the following result that addresses the well-posedness of the
underlying optimal control problem (2.7) and the corresponding first-order
optimality system.

######  Theorem 2.4.

The optimal control problem (2.7) possess a unique solution
$\displaystyle(\bar{y},\bar{u})\in L_{2}(\Omega)\times U_{ad}$. Moreover the
unique pair $\displaystyle(\bar{y},\bar{u})$ solves the following first order
necessary optimality conditions:

$\displaystyle\displaystyle\bar{y}=\bar{y}_{f}+y_{\bar{u}},\quad\bar{y}_{f},~{}y_{\bar{u}}\in
L_{2}(\Omega),$ (2.8)
$\displaystyle\displaystyle(\bar{y}_{f},\Delta\phi)=-(f,\phi)\quad\forall~{}\phi\in
X,$ (2.9)
$\displaystyle\displaystyle(y_{\bar{u}},\Delta\phi)=\langle\bar{u},\partial_{n}\phi\rangle\quad\forall~{}\phi\in
X,$ (2.10)
$\displaystyle\displaystyle(\bar{y}-y_{d},y_{u}-y_{\bar{u}})+\alpha\langle\bar{u},u-\bar{u}\rangle\geq
0\quad\forall~{}u\in U_{ad}.$ (2.11)

###### Proof.

By utilizing the solution operator $\displaystyle S(\cdot,\cdot)$, we can
reformulate the model optimal control problem (2.7) into a simplified form,
expressed as follows:

$\displaystyle\displaystyle\min_{u\in U_{ad}}j(u),$ (2.12)

where $\displaystyle j(\cdot):L_{2}(\Gamma)\rightarrow\mathbb{R}$ represents
the reduced cost functional defined as:

$\displaystyle\displaystyle
j(u):=\frac{1}{2}\|S(f,u)-y_{d}\|^{2}_{L_{2}(\Omega)}+\frac{\alpha}{2}\|u\|^{2}_{L_{2}(\Gamma)}.$

The reduced cost functional $\displaystyle j(\cdot)$ defines a strictly convex
functional on a non empty closed convex Hilbert space $\displaystyle U_{ad}$.
Thus, the standard theory of PDE constrained optimal control problem [27]
implies that problem (2.12) has a unique solution say $\displaystyle\bar{u}\in
U_{ad}$. Setting $\displaystyle\bar{y}=S(f,\bar{u})$ as the corresponding
state variable. By definition of $\displaystyle S(\cdot,\cdot)$, we find that
$\displaystyle\bar{y}\in L_{2}(\Omega)$ satisfies (2.8)-(2.10). By applying
the first order necessary optimality conditions, we get for any $\displaystyle
u\in U_{ad}$

$\displaystyle\displaystyle 0\leq j^{\prime}(\bar{u})(u-\bar{u})$
$\displaystyle\displaystyle:=\lim_{t\downarrow
0}\frac{j(\bar{u}+t(u-\bar{u}))-j(\bar{u})}{t}$
$\displaystyle\displaystyle=(\bar{y}-y_{d},y_{u}-y_{\bar{u}})+\alpha\langle\bar{u},u-\bar{u}\rangle,$

which implies that $\displaystyle\bar{u}\in U_{ad}$ solves (2.11). This
completes the proof. ∎

###  2.2 Regularity of the Optimal Control

In this fragment, we obtain the regularity of the optimal variables namely
$\displaystyle\bar{y}$ and $\displaystyle\bar{u}$ which are useful for the
subsequent analysis. Introduce an adjoint state $\displaystyle\theta\in V$
such that it satisfies the following variational formulation: find
$\displaystyle\theta\in V$ such that:

$\displaystyle\displaystyle(\nabla
v,\nabla\theta)=(\bar{y}-y_{d},v)\quad\forall~{}v\in V.$ (2.13)

By elliptic regularity theory for convex polygonal domain [19],
$\displaystyle\theta\in X$ and it satisfies the following _a priori_ estimate:

$\displaystyle\displaystyle\|\theta\|_{H^{2}(\Omega)}$
$\displaystyle\displaystyle\leq C\|\bar{y}-y_{d}\|_{L_{2}(\Omega)}.$

Using integration by parts and density arguments in (2.13), we get

$\displaystyle\displaystyle(v,\Delta\theta)$
$\displaystyle\displaystyle=-(\bar{y}-y_{d},v)\quad\forall~{}v\in
L_{2}(\Omega).$ (2.14)

Since $\displaystyle\theta\in H^{2}(\Omega)$, it implies
$\displaystyle\partial_{n}\theta\in\tilde{H}^{1/2}(\Gamma)$. For any
$\displaystyle u\in U_{ad}$, using (2.14) for $\displaystyle
v=y_{u}-y_{\bar{u}}\in L_{2}(\Omega)$, we get

$\displaystyle\displaystyle(\bar{y}-y_{d},y_{u}-y_{\bar{u}})$
$\displaystyle\displaystyle=-(y_{u}-y_{\bar{u}},\Delta\theta)$
$\displaystyle\displaystyle=-\langle u-\bar{u},\partial_{n}\theta\rangle,$

where we have used (2.1) to observe the last equality. Hence, the inequality
(2.11) reduces to

$\displaystyle\displaystyle\alpha\langle\bar{u},u-\bar{u}\rangle-\langle
u-\bar{u},\partial_{n}\theta\rangle$ $\displaystyle\displaystyle\geq
0\quad\forall~{}u\in U_{ad},$

which implies

$\displaystyle\displaystyle\langle\alpha\bar{u}-\partial_{n}\theta,u-\bar{u}\rangle\geq
0\quad\forall~{}u\in U_{ad}.$ (2.15)

It follows from (2.15) and standard arguments from [27, Chapter 2] that

$\displaystyle\displaystyle\bar{u}(x)=P_{[u_{a},u_{b}]}\Big{(}\dfrac{1}{\alpha}(\partial_{n}\theta)(x)\Big{)},\quad\text{for
almost every}~{}x\in\Gamma,$ (2.16)

where $\displaystyle P_{[u_{a},u_{b}]}(w):=\min\\{u_{b},\max\\{u_{a},w\\}\\}$
denotes the projection of $\displaystyle\mathbb{R}$ onto
$\displaystyle[u_{a},u_{b}]$. Since $\displaystyle\Omega$ is convex, therefore
it follows from (2.16) that $\displaystyle\bar{u}\in\tilde{H}^{1/2}(\Gamma)$.
As $\displaystyle\theta$ equals zero on $\displaystyle\Gamma$, the derivative
of $\displaystyle\theta$ in the tangential direction also equals zero on
$\displaystyle\Gamma$. Therefore, at the corner points of the domain denoted
as $\displaystyle x$, we have $\displaystyle(\partial_{n}\theta)(x)=0$. Hence,
it follows from (2.16) that $\displaystyle\bar{u}(x)=0$ at all the corner
points $\displaystyle x$ of the domain, which implies that
$\displaystyle\bar{u}\in H^{1/2}(\Gamma)$.

##  3 Preliminaries

We introduce the following notations which are used throughout the article.

  * •

$\displaystyle\mathcal{T}_{h}$ is a regular triangulation of
$\displaystyle\Omega$ into closed triangles $\displaystyle T$.

  * •

$\displaystyle h_{T}:=$ diam ($\displaystyle T$) is the diameter of the
triangle $\displaystyle T\in\mathcal{T}_{h}$.

  * •

$\displaystyle h:=\max_{T\in\mathcal{T}_{h}}h_{T}$ is the mesh parameter.

  * •

$\displaystyle\mathcal{V}_{h}^{i}:=$ the set of all interior vertices of
$\displaystyle\mathcal{T}_{h}$.

  * •

$\displaystyle\mathcal{V}_{h}^{b}:=$ the set of all boundary vertices of
$\displaystyle\mathcal{T}_{h}$.

  * •

$\displaystyle\mathcal{V}_{h}:=\mathcal{V}_{h}^{i}\cup\mathcal{V}_{h}^{b}$ is
the set of all vertices of $\displaystyle\mathcal{T}_{h}$.

  * •

$\displaystyle\mathcal{E}_{h}^{i}:=$ the set of all interior edges of
$\displaystyle\mathcal{T}_{h}$.

  * •

$\displaystyle\mathcal{E}_{h}^{b}:=$ the set of all boundary edges of
$\displaystyle\mathcal{T}_{h}$.

  * •

$\displaystyle\mathcal{E}_{h}:=\mathcal{E}_{h}^{i}\cup\mathcal{E}_{h}^{b}$ is
the set of all edges of $\displaystyle\mathcal{T}_{h}$.

  * •

$\displaystyle m_{e}:=$ mid point of $\displaystyle e\in\mathcal{E}_{h}$.

  * •

$\displaystyle\mathbb{P}_{k}(T):=$ the set of all polynomials of degree atmost
$\displaystyle k\in\mathbb{N}\cup\\{0\\}$ over $\displaystyle
T\in\mathcal{T}_{h}$.

  * •

$\displaystyle H^{k}(\Omega,\mathcal{T}_{h}):=\\{v\in L_{2}(\Omega):v|_{T}\in
H^{k}(T)\quad\forall~{}T\in\mathcal{T}_{h}\\}$, $\displaystyle
k\in\mathbb{N}\cup\\{0\\}$.

From this point forward, the notation $\displaystyle a\lesssim b$ indicates
the existence of a positive constant $\displaystyle C$ that is not dependent
on the mesh parameter $\displaystyle h$, such that a $\displaystyle a\leq Cb$.
It is assumed that the triangulation $\displaystyle\mathcal{T}_{h}$ is
regular, meaning that any two distinct simplices in
$\displaystyle\mathcal{T}_{h}$ with non-empty intersection are either
identical, or share exactly one common vertex, or one common edge.
Additionally, every interior angle of any simplex is bounded from below by a
universal positive constant $\displaystyle\rho$. All the generic constants
hidden in the notation $\displaystyle\lesssim$ are solely determined by
$\displaystyle\rho>0$, thereby ensuring the triangulation is shape regular.
Next, we will establish the definitions for the jump and average of functions
that are scalar-valued and vector-valued. Consider an edge $\displaystyle
e\in\mathcal{E}_{h}^{i}$ that is shared by two adjacent triangles
$\displaystyle T_{+}$ and $\displaystyle T_{-}$, such that $\displaystyle
e=T_{+}\cap T_{-}$. Let $\displaystyle n_{+}$ be the unit normal of
$\displaystyle e$ pointing from $\displaystyle T_{+}$ to $\displaystyle
T_{-}$, and $\displaystyle n_{-}=-n_{+}$. In this context, we define the jumps
$\displaystyle[\hskip-1.5pt[\cdot]\hskip-1.5pt]$ and averages
$\displaystyle\\{\hskip-3.0pt\\{\cdot\\}\hskip-3.0pt\\}$ across the edge
$\displaystyle e$ as follows:

  * •

For a scalar valued function $\displaystyle w\in
H^{1}(\Omega,\mathcal{T}_{h})$, define

$\displaystyle\displaystyle[\hskip-1.5pt[w]\hskip-1.5pt]:=w_{+}\,n_{+}+w_{-}\,n_{-},\quad\quad\\{\hskip-3.0pt\\{w\\}\hskip-3.0pt\\}:=\frac{w_{+}+w_{-}}{2},$

where $\displaystyle w_{\pm}=w|_{T_{\pm}}$.

  * •

For a vector valued function $\displaystyle
v\in[H^{1}(\Omega,\mathcal{T}_{h})]^{2}$, define

$\displaystyle\displaystyle[\hskip-1.5pt[v]\hskip-1.5pt]:=v_{+}\cdot
n_{+}+v_{-}\cdot
n_{-},\quad\quad\\{\hskip-3.0pt\\{v\\}\hskip-3.0pt\\}:=\frac{v_{+}+v_{-}}{2},$

where $\displaystyle v_{\pm}=v|_{T_{\pm}}$.

To simplify the notation, we also introduce the concepts of jump and mean on
the boundary $\displaystyle\Gamma$. Consider any edge $\displaystyle
e\in\mathcal{E}_{h}^{b}$, where it is evident that there exists a triangle
$\displaystyle T\in\mathcal{T}_{h}$ such that $\displaystyle e=\partial
T\cap\Gamma$. Let $\displaystyle n_{e}$ be the unit normal of $\displaystyle
e$ pointing outward from $\displaystyle T$. For any $\displaystyle w\in
H^{1}(\Omega,\mathcal{T}_{h})$ and any $\displaystyle
v\in[H^{1}(\Omega,\mathcal{T}_{h})]^{2}$, we define the following on
$\displaystyle e\in\mathcal{E}_{h}^{b}$:

$\displaystyle\displaystyle[\hskip-1.5pt[w]\hskip-1.5pt]$
$\displaystyle\displaystyle:=w\,n_{e}\quad\text{and}\quad\\{\hskip-3.0pt\\{w\\}\hskip-3.0pt\\}=w,$
$\displaystyle\displaystyle[\hskip-1.5pt[v]\hskip-1.5pt]$
$\displaystyle\displaystyle:=v\cdot
n_{e}\quad\text{and}\quad\\{\hskip-3.0pt\\{v\\}\hskip-3.0pt\\}=v.$

Below, we define the Crouziex-Raviart finite element spaces.

$\displaystyle\displaystyle V_{h}$ $\displaystyle\displaystyle:=\\{v_{h}\in
L_{2}(\Omega):v_{h}|_{T}\in\mathbb{P}_{1}(T)~{}\forall~{}T\in\mathcal{T}_{h}~{}\text{and}~{}v_{h}~{}\text{is
continuous at}~{}m_{e}~{}\forall~{}e\in\mathcal{E}_{h}\\},$
$\displaystyle\displaystyle V_{h}^{0}$
$\displaystyle\displaystyle:=\\{v_{h}\in
V_{h}:v_{h}(m_{e})=0~{}\forall~{}e\in\mathcal{E}_{h}^{b}\\}.$

We need to modify the definition of $\displaystyle V_{h}$. The mean integral
of the function should be continuous i.e DOF should be modified to mean
integrals.
The discrete bilinear form $\displaystyle a_{pw}(\cdot,\cdot):V_{h}\times
V_{h}\rightarrow\mathbb{R}$ is defined by

$\displaystyle\displaystyle a_{pw}(v_{h},w_{h})$
$\displaystyle\displaystyle=\sum_{T\in\mathcal{T}_{h}}\int_{T}\nabla
v_{h}|_{T}\cdot\nabla w_{h}|_{T}\,dx\quad\forall~{}v_{h},w_{h}\in V_{h}.$

The mesh dependent bilinear form $\displaystyle a_{pw}(\cdot,\cdot)$ defines a
symmetric bilinear form which coincides with the continuous bilinear form
$\displaystyle a(\cdot,\cdot)$ on $\displaystyle H^{1}(\Omega)\times
H^{1}(\Omega)$. Now, we define the mesh dependent norm on $\displaystyle
V_{h}$. For any $\displaystyle v_{h}\in V_{h}$, define
$\displaystyle\|\cdot\|_{h}$ as follows:

$\displaystyle\displaystyle\|v_{h}\|_{h}:=\sqrt{a_{pw}(v_{h},v_{h})}.$

Note that, $\displaystyle\|\cdot\|_{h}$ defines only a semi-norm on
$\displaystyle V_{h}$ but a norm on $\displaystyle V_{h}^{0}$. Next, we define
a Crouziex-Raviart interpolation map $\displaystyle I_{CR}:V\cap
C(\bar{\Omega})\rightarrow V_{h}^{0}$ as follows:

$\displaystyle\displaystyle\int_{e}I_{CR}v\,ds=\int_{e}v\,ds\quad\forall~{}e\in\mathcal{E}_{h},~{}v\in
V\cap C(\bar{\Omega}).$ (3.1)

By density arguments, $\displaystyle I_{CR}$ can be extended continuously and
uniquely to $\displaystyle V$, and the extension is again denoted by
$\displaystyle I_{CR}$ for the ease of presentation. In the next lemma, we
state the approximation properties of $\displaystyle I_{CR}$ [15].

######  Lemma 3.1.

For $\displaystyle v\in V$, it holds that:

$\displaystyle\displaystyle\|v-I_{CR}v\|_{L_{2}(\Omega)}+h\|v-I_{CR}v\|_{h}$
$\displaystyle\displaystyle\lesssim h|v|_{H^{1}(\Omega)}.$

###  3.1 Raviart-Thomas Interpolation Operator

In this segment, Raviart-Thomas interpolation operator is defined which is
used further for the error analysis. Given a triangle $\displaystyle
T\in\mathcal{T}_{h}$, the lowest order Raviart-Thomas space is defined as:

$\displaystyle\displaystyle
RT_{0}(T):=\mathbb{P}_{0}(T)^{2}+x\,\mathbb{P}_{0}(T),$

where $\displaystyle x=(x_{1},x_{2})\in\mathbb{R}^{2}$.

######  Remark 3.2.

Let $\displaystyle T\in\mathcal{T}_{h}$ be any triangle. For any
$\displaystyle v\in RT_{0}(T)$, we have $\displaystyle v\cdot
n_{e}\in\mathbb{P}_{0}(e)$ for all $\displaystyle e\in\partial T$, where
$\displaystyle n_{e}$ is the unit outward normal vector to $\displaystyle e$.

For any $\displaystyle T\in\mathcal{T}_{h}$, define the local interpolation
operator $\displaystyle\Pi_{T}:H^{1}(T)^{2}\rightarrow RT_{0}(T)$ as follows:

$\displaystyle\displaystyle\int_{e}\Pi_{T}v\cdot n_{i}\,p\,ds$
$\displaystyle\displaystyle=\int_{e}v\cdot
n_{i}\,p\,ds\quad\forall~{}p\in\mathbb{P}_{0}(e),\quad\forall~{}e\in\partial
T.$

We state the approximation property of the interpolation operator
$\displaystyle\Pi_{T}$ in the following lemma.

######  Lemma 3.3.

Let $\displaystyle T\in\mathcal{T}_{h}$ be any fixed triangle. For any
$\displaystyle v\in H^{1}(T)^{2}$, the following holds:

$\displaystyle\displaystyle\|v-\Pi_{T}v\|_{L_{2}(T)}$
$\displaystyle\displaystyle\lesssim h_{T}|v|_{H^{1}(T)}.$

###### Proof.

We refer the readers to the article [1] for the proof. ∎

###  3.2 Enrichment Operator

The focus of this section is on introducing a new enrichment operator which
plays a crucial role in our upcoming analysis. It is well known that the
enrichment operators play a crucial role in the analysis for non-conforming
and discontinuous Galerkin finite element methods [4, 5, 6, 8, 9, 10, 22, 23].
These operators connect the non-conforming or discontinuous Galerkin spaces to
it’s conforming counterpart.
The novelty of the enrichment operator being introduced here is due to two
facts. The degrees of freedom for the finite element functions of it’s
conforming counterparts are taken in a non-standard manner which enables us to
prove an orthogonality result for the CR-finite element functions (Theorem
3.4). This result plays a crucial role in obtaining the optimal order energy
norm error estimate for the solution of the state equation.
To begin with we modify the definitions of degrees of freedoms of the
conforming counterpart $\displaystyle V_{c}$ of $\displaystyle V_{h}$.

$\displaystyle\displaystyle V_{c}=\\{$ $\displaystyle\displaystyle v_{h}\in
V:v_{h}|_{T}\in
P_{2}(T),\;v_{h}|_{+}(p)=v_{h}|_{-}(p)\;\text{for}\;\;p\in\nu\;\text{and}\;\int_{e}v_{h}|_{+}ds=\int_{e}v_{h}|_{-}ds\;$
$\displaystyle\displaystyle\text{where}\;v_{h}|_{+}=v_{h}|_{T_{+}},\;v_{h}|_{-}=v_{h}|_{T_{-}},\;e\in\mathcal{E}_{h}\\}.$
(3.2)

Note that this choice of $\displaystyle V_{c}$ enables the Lagrange
interpolation operator $\displaystyle I_{L}:H^{2}(\Omega)\rightarrow V_{c}$ to
have the following orthogonal property:

$\displaystyle\displaystyle\int_{e}(v-I_{L}v)ds=0,$ (3.3)

which plays a crucial role to prove Theorem 3.4.

Note that, $\displaystyle V_{c}$ denotes the Lagrange finite element space of
order $\displaystyle 2$, which is the conforming counterpart of $\displaystyle
V_{h}^{0}$. Let $\displaystyle I_{L}:V\cap C(\bar{\Omega})\rightarrow V_{c}$
denotes the Lagrange interpolation operator defined as follows: for
$\displaystyle v\in V\cap C(\bar{\Omega})$,

$\displaystyle\displaystyle(I_{L}v)(p)$
$\displaystyle\displaystyle=v(p)\quad\forall~{}p\in\mathcal{V}_{h},$
$\displaystyle\displaystyle\int_{e}I_{L}v\,ds$
$\displaystyle\displaystyle=\int_{e}v\,ds\quad\forall~{}e\in\mathcal{E}_{h}.$
(3.4)

Let $\displaystyle I_{c}:V_{h}^{0}\rightarrow V_{c}$ denotes the enriching map
defined by: for $\displaystyle v_{h}\in V_{h}^{0}$, define

$\displaystyle\displaystyle I_{c}v_{h}(p)$
$\displaystyle\displaystyle=\frac{1}{|\mathcal{T}_{p}|}\sum_{T\in\mathcal{T}_{p}}v_{h}|_{T}(p)\quad\forall~{}p\in\mathcal{V}_{h}^{i},$
$\displaystyle\displaystyle I_{c}v_{h}(p)$
$\displaystyle\displaystyle=0\quad\forall~{}p\in\mathcal{V}_{h}^{b},$ (3.5)
$\displaystyle\displaystyle\int_{e}I_{c}v_{h}\,ds$
$\displaystyle\displaystyle=\int_{e}v_{h}\,ds\quad\forall~{}e\in\mathcal{E}_{h},$
(3.6)

where $\displaystyle\mathcal{T}_{p}$ denotes the set of triangles sharing the
node $\displaystyle p\in\mathcal{V}_{h}$ and $\displaystyle|\mathcal{T}_{p}|$
refers to the cardinality of $\displaystyle\mathcal{T}_{p}$. Subsequently, we
prove an orthogonality result concerning $\displaystyle I_{c}$, which holds
significant importance in the subsequent analysis.

######  Theorem 3.4.

Let $\displaystyle p_{h}$ be a piecewise linear function (need not be in
$\displaystyle V_{h}$), it holds that:

$\displaystyle\displaystyle
a_{pw}(p_{h},v_{h}-I_{c}v_{h})=0\quad\forall~{}v_{h}\in V_{h}^{0}.$

###### Proof.

Let $\displaystyle v_{h}\in V_{h}^{0}$ be an arbitrary function. It follows
from triangle wise integration by parts that

$\displaystyle\displaystyle a_{pw}(p_{h},v_{h}-I_{c}v_{h})=$
$\displaystyle\displaystyle\sum_{e\in\mathcal{E}_{h}}\int_{e}\left\\{\hskip-5.0pt\left\\{\dfrac{\partial
p_{h}}{\partial
n}\right\\}\hskip-5.0pt\right\\}\left[\hskip-3.5pt\left[v_{h}-I_{c}v_{h})\right]\hskip-3.5pt\right]ds$
$\displaystyle\displaystyle+\sum_{e\in\mathcal{E}_{h}^{i}}\int_{e}\left[\hskip-3.5pt\left[\dfrac{\partial
p_{h}}{\partial
n}\right]\hskip-3.5pt\right]\left\\{\hskip-5.0pt\left\\{v_{h}-I_{c}v_{h}\right\\}\hskip-5.0pt\right\\}ds.$
(3.7)

Since, $\displaystyle p_{h}|_{T}\in\mathbb{P}_{1}(T)$ for all $\displaystyle
T\in\mathcal{T}_{h}$, the rest of the proof follows from using (3.5) and (3.6)
in (3.2). ∎

##  4 Crouzeix-Raviart Analysis of Various State Equations

This section considers the Crouzeix-Raviart (CR) finite element approximation
of the solutions of certain auxiliary state equations. While some of these
equations have a typical weak solution in $\displaystyle H^{1}(\Omega)$,
others do not. The optimal order $\displaystyle L_{2}$ norm error estimates
are derived for both of the cases. To begin with, we define the spaces as
follows:

$\displaystyle\displaystyle W_{h}$ $\displaystyle\displaystyle:=\\{v_{h}\in
H^{1}(\Omega):v_{h}|_{T}\in\mathbb{P}_{1}(T)~{}\forall~{}T\in\mathcal{T}_{h}\\},$
$\displaystyle\displaystyle U_{h}$ $\displaystyle\displaystyle:=\\{u_{h}\in
L_{2}(\Gamma):u_{h}|_{e}\in\mathbb{P}_{0}(e)~{}\forall~{}e\in\mathcal{E}_{h}^{b}\\},$
$\displaystyle\displaystyle V_{1}(\Gamma)$
$\displaystyle\displaystyle:=\\{v_{h}\in
C(\Gamma):v_{h}|_{e}\in\mathbb{P}_{1}(e)~{}\forall~{}e\in\mathcal{E}_{h}^{b}\\}.$

Firstly, we consider the equation which possesses standard weak solution in
$\displaystyle H^{1}(\Omega)$. Let $\displaystyle z_{h}\in V_{1}(\Gamma)$ be
an arbitrary function. Since, $\displaystyle V_{1}(\Gamma)\subset
H^{1/2}(\Gamma)$, thus $\displaystyle z_{h}\in H^{\frac{1}{2}}(\Gamma)$. Using
$\displaystyle z_{h}$, we define a function say $\displaystyle\tilde{z}_{h}$
in the following manner:

$\displaystyle\displaystyle\begin{cases}\tilde{z}_{h}(p)&=0\quad\forall~{}p\in\mathcal{V}_{h}^{i},\\\
\tilde{z}_{h}(p)&=z_{h}(p)\quad\forall~{}p\in\mathcal{V}_{h}^{b}.\end{cases}$
(4.1)

It is easy to see that $\displaystyle\tilde{z}_{h}\in W_{h}$. For any
$\displaystyle z_{h}\in V_{1}(\Gamma)$, define $\displaystyle y_{z_{h}}\in
H^{1}(\Omega)$ such that:

$\displaystyle\displaystyle y_{z_{h}}$
$\displaystyle\displaystyle=y_{0}+\tilde{z}_{h},~{}y_{0}\in V,$ (4.2)
$\displaystyle\displaystyle a(y_{0},v)$
$\displaystyle\displaystyle=-a(\tilde{z}_{h},v)\quad\forall~{}v\in V.$ (4.3)

Let $\displaystyle y_{z_{h},h}\in V_{h}$ be the CR-approximation of
$\displaystyle y_{z_{h}}\in H^{1}(\Omega)$, i.e. it satisfies the following
equation:

$\displaystyle\displaystyle y_{z_{h},h}$
$\displaystyle\displaystyle={y}_{0,h}+\tilde{z}_{h},~{}y_{0,h}\in V_{h}^{0},$
(4.4) $\displaystyle\displaystyle a_{pw}({y}_{0,h},v_{h})$
$\displaystyle\displaystyle=-a_{pw}(\tilde{z}_{h},v_{h})\quad\forall~{}v_{h}\in
V_{h}^{0}.$ (4.5)

By Lax-milgram lemma, auxiliary problems (4.2)-(4.3) and (4.4)-(4.5) are well
posed. The following Theorem yields an optimal order error estimate for
$\displaystyle y_{z_{h}}$.:

######  Theorem 4.1.

For $\displaystyle z_{h}\in V_{1}(\partial\Omega)$ let $\displaystyle
y_{z_{h}},\,y_{z_{h},h}$ satisfies (4.2), (4.2) and (4.4), (4.5) respectively.
Then $\displaystyle\|y_{z_{h}}-y_{z_{h},h}\|\lesssim
h[\|y_{z_{h}}-y_{z_{h},h}\|_{pw}+h^{1-\epsilon}\|z_{h}\|_{H^{\frac{3}{2}-\epsilon}(\partial\Omega)}]$
for any $\displaystyle\epsilon>0$.

###### Proof.

Next, we estimate $\displaystyle\|y_{z_{h}}-y_{z_{h},h}\|$. Again, by duality,
we find that

$\displaystyle\displaystyle\|y_{z_{h}}-y_{z_{h},h}\|$
$\displaystyle\displaystyle=\sup_{g\in L_{2}(\Omega),g\neq
0}\frac{(y_{z_{h}}-y_{z_{h},h},g)}{\|g\|}.$ (4.6)

Using (4.2) and (4.4), we have

$\displaystyle\displaystyle y_{z_{h}}-y_{z_{h},h}$
$\displaystyle\displaystyle=y_{0}-\bar{y}_{0,h}.$ (4.7)

For any $\displaystyle g\in L_{2}(\Omega)$, define $\displaystyle\phi_{g}\in
H^{1}_{0}(\Omega)$ such that

$\displaystyle\displaystyle a(\phi_{g},v)$
$\displaystyle\displaystyle=(g,v)\quad\forall~{}v\in H^{1}_{0}(\Omega).$ (4.8)

By elliptic regularity theory, we have $\displaystyle\phi_{g}\in
H^{2}(\Omega)$ such that $\displaystyle\|\phi_{g}\|_{2,\Omega}\lesssim\|g\|$.
Let $\displaystyle\phi_{g,h}\in V_{h}^{0}$ be the standard non-conforming
finite element approximation of $\displaystyle\phi_{g}$. Thus, it satisfies
the following equation:

$\displaystyle\displaystyle a_{pw}(\phi_{g,h},v_{h})$
$\displaystyle\displaystyle=(g,v_{h})\quad\forall~{}v_{h}\in V_{h}^{0}.$ (4.9)

Since $\displaystyle y_{0}\in H^{1}_{0}(\Omega)$ and
$\displaystyle\bar{y}_{0,h}\in V_{h}$, it follows from (4.3),(4.5), (4.8) and
(4.9) that

$\displaystyle\displaystyle(g,y_{0}-\bar{y}_{0,h})$
$\displaystyle\displaystyle=a(\phi_{g},y_{0})-a_{pw}(\phi_{g,h},\bar{y}_{0,h})$
$\displaystyle\displaystyle=a_{pw}(\phi_{g}-\phi_{g,h},y_{0}-\bar{y}_{0,h})+a_{pw}(\phi_{g}-\phi_{g,h},\bar{y}_{0,h})+a_{pw}(\phi_{g,h},y_{0}-\bar{y}_{0,h})$
$\displaystyle\displaystyle=a_{pw}(\phi_{g}-\phi_{g,h},y_{0}-\bar{y}_{0,h})+a_{pw}(\phi_{g},\bar{y}_{0,h})-(g,\bar{y}_{0,h})$
$\displaystyle\displaystyle\quad+a_{pw}(\phi_{g,h},y_{0})+a_{pw}(\tilde{z}_{h},\phi_{g,h})$
$\displaystyle\displaystyle=a_{pw}(\phi_{g}-\phi_{g,h},y_{0}-\bar{y}_{0,h})+a_{pw}(\phi_{g},\bar{y}_{0,h}-y_{0})-(g,\bar{y}_{0,h}-y_{0})$
$\displaystyle\displaystyle\quad+a_{pw}(y_{0},\phi_{g,h}-\phi_{g})+a_{pw}(\tilde{z}_{h},\phi_{g,h}-\phi_{g})$
$\displaystyle\displaystyle=I+II+III+IV+V.$ (4.10)

We estimate each term on the right hand side of (4). First, consider the term
$\displaystyle IV+V$.

$\displaystyle\displaystyle IV+V$
$\displaystyle\displaystyle=a_{pw}(y_{\tilde{z}_{h}},\phi_{g,h}-\phi_{g})$

Note that since $\displaystyle z_{h}$ is a piecewise linear and globally
continuous on each line segment $\displaystyle\Gamma_{j}$ ($\displaystyle
1\leq j\leq n$) consisting $\displaystyle\partial\Omega$. Therefore
$\displaystyle z_{h}|_{\Gamma_{j}}\in H^{\frac{3}{2}-\epsilon}(\Gamma_{j})$
for any $\displaystyle\epsilon>0$ [3]. Since $\displaystyle z_{h}$ satisfies
the compatibility conditions that $\displaystyle z_{h}$ is continuous at the
vertices of $\displaystyle\Omega$ and as $\displaystyle\Omega$ is convex
therefore an application of result from real interpolation spaces [7] yields
that $\displaystyle y_{z_{h}}\in H^{2-\epsilon}(\Omega)$ [18]Theorem 1.8.
Consider the term $\displaystyle
a_{pw}(y_{\tilde{z}_{h}},\phi_{g,h}-\phi_{g})$. A use of integration by parts
yields

$\displaystyle\displaystyle a_{pw}(y_{\tilde{z}_{h}},\phi_{g,h}-\phi_{g})$
$\displaystyle\displaystyle=\sum_{T\in\mathcal{T}_{h}}\int_{T}\nabla
y_{\tilde{z}_{h}}\cdot\nabla(\phi_{g,h}-\phi_{g})\,dx$
$\displaystyle\displaystyle=-\sum_{T\in\mathcal{T}_{h}}\int_{T}\Delta
y_{\tilde{z}_{h}}\,(\phi_{g,h}-\phi_{g})\,dx+\sum_{T\in\mathcal{T}_{h}}\int_{\partial
T}\nabla y_{\tilde{z}_{h}}\cdot n\,(\phi_{g,h}-\phi_{g})\,ds$
$\displaystyle\displaystyle=-\sum_{T\in\mathcal{T}_{h}}\int_{T}\Delta
y_{\tilde{z}_{h}}\,(\phi_{g,h}-\phi_{g})\,dx$
$\displaystyle\displaystyle\quad+\sum_{T\in\mathcal{T}_{h}}\int_{\partial
T}\frac{\partial{y}_{\tilde{z}_{h}}}{\partial n}\,(\phi_{g,h}-\phi_{g})\,ds,$
(4.11) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\sum_{T\in\mathcal{T}_{h}}\int_{T}\Delta
y_{\tilde{z}_{h}}\,(\phi_{g,h}-\phi_{g})\,dx$
$\displaystyle\displaystyle+\sum_{T\in\mathcal{T}_{h}}\int_{\partial
T}\frac{\partial{y}_{\tilde{z}_{h}}}{\partial
n}\,((\phi_{g,h}-\phi_{g})-I_{CR}((\phi_{g,h}-\phi_{g}))\,ds+$
$\displaystyle\displaystyle\sum_{T\in\mathcal{T}_{h}}\int_{\partial
T}\frac{\partial{y}_{\tilde{z}_{h}}}{\partial
n}\,(I_{CR}(\phi_{g,h}-\phi_{g})-I_{c}I_{CR}((\phi_{g,h}-\phi_{g}))ds+\sum_{T\in\mathcal{T}_{h}}\int_{\partial
T}\frac{\partial{y}_{\tilde{z}_{h}}}{\partial
n}\,I_{c}I_{CR}((\phi_{g,h}-\phi_{g}))ds$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\sum_{T\in\mathcal{T}_{h}}\int_{T}\Delta
y_{\tilde{z}_{h}}\,(\phi_{g,h}-\phi_{g})\,dx$
$\displaystyle\displaystyle+\sum_{T\in\mathcal{T}_{h}}\int_{\partial
T}(\nabla{y}_{\tilde{z}_{h}}-\Pi_{T}\nabla{y}_{\tilde{z}_{h}})\cdot
n\,((\phi_{g,h}-\phi_{g})-I_{CR}((\phi_{g,h}-\phi_{g}))\,ds+$
$\displaystyle\displaystyle\sum_{T\in\mathcal{T}_{h}}\int_{\partial
T}(\nabla{y}_{\tilde{z}_{h}}-\Pi_{T}\nabla{y}_{\tilde{z}_{h}})\cdot
n\,(I_{CR}(\phi_{g,h}-\phi_{g})-I_{c}I_{CR}((\phi_{g,h}-\phi_{g}))ds+$
$\displaystyle\displaystyle\sum_{e\in\mathcal{E}_{h}}\int_{e}\frac{\partial{y}_{\tilde{z}_{h}}}{\partial
n}\,\left[\hskip-3.5pt\left[I_{c}I_{CR}((\phi_{g,h}-\phi_{g}))\right]\hskip-3.5pt\right]ds.$
(4.12)

Where $\displaystyle\Pi_{T}$ is lowest order Raviart-Thomas interpolation
operator as in Lemma 3.3. Note that $\displaystyle\Delta y_{\tilde{z}}=0$ and
$\displaystyle\int_{e}\frac{\partial{y}_{\tilde{z}_{h}}}{\partial
n}\,\left[\hskip-3.5pt\left[I_{c}I_{CR}((\phi_{g,h}-\phi_{g}))\right]\hskip-3.5pt\right]ds=0$.
Therefore application of Cauchy Schwarz inequality, discrete trace inequality,
Crauzeix Raviart and Raviart Thomas interpolation error estimates in (4)
yields

$\displaystyle\displaystyle a_{pw}(y_{\tilde{z}_{h}},\phi_{g,h}-\phi_{g})$
$\displaystyle\displaystyle=\lesssim
h^{1-\epsilon}|y_{\tilde{z}_{h}}|_{2-\epsilon,\Omega}\|\phi_{g,h}-\phi_{g}\|_{pw}.$
(4.13)

Now, consider the terms $\displaystyle II+III$.

$\displaystyle\displaystyle II+III=$ $\displaystyle\displaystyle
a_{pw}(\phi_{g},\bar{y}_{0,h}-y_{0})-(g,\bar{y}_{0,h}-y_{0})$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
a_{pw}(\phi_{g},(\bar{y}_{0,h}-y_{0})-I_{CR}(\bar{y}_{0,h}-y_{0}))+a_{pw}(\phi_{g},I_{CR}(\bar{y}_{0,h}-y_{0})-I_{c}(I_{CR}(\bar{y}_{0,h}-y_{0})))+$
$\displaystyle\displaystyle
a_{pw}(\phi_{g},I_{c}(I_{CR}(\bar{y}_{0,h}-y_{0})))-\quad(g,(\bar{y}_{0,h}-y_{0})-I_{CR}(\bar{y}_{0,h}-y_{0}))-$
$\displaystyle\displaystyle(g,I_{CR}(\bar{y}_{0,h}-y_{0})-I_{c}(I_{CR}(\bar{y}_{0,h}-y_{0})))-(g,I_{c}(I_{CR}(\bar{y}_{0,h}-y_{0}))).$
(4.14)

From (4) consider the term $\displaystyle
a_{pw}(\phi_{g},(\bar{y}_{0,h}-y_{0})-I_{CR}(\bar{y}_{0,h}-y_{0}))$.

$\displaystyle\displaystyle
a_{pw}(\phi_{g},(\bar{y}_{0,h}-y_{0})-I_{CR}(\bar{y}_{0,h}-y_{0}))$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\sum_{T\in\mathcal{T}_{h}}\int_{T}\Delta\phi_{g}((\bar{y}_{0,h}-y_{0})-I_{CR}(\bar{y}_{0,h}-y_{0}))\,dx+\sum_{T\in\mathcal{T}_{h}}\int_{\partial
T}\dfrac{\partial\phi_{g}}{\partial
n}((\bar{y}_{0,h}-y_{0})-I_{CR}(\bar{y}_{0,h}-y_{0}))\,ds$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{T\in\mathcal{T}_{h}}\int_{T}{g}((\bar{y}_{0,h}-y_{0})-I_{CR}(\bar{y}_{0,h}-y_{0}))\,dx+\sum_{T\in\mathcal{T}_{h}}\int_{\partial
T}(\nabla\phi_{g}-(\Pi_{T}\nabla\phi_{g}))\cdot n_{e}$
$\displaystyle\displaystyle((\bar{y}_{0,h}-y_{0})-I_{CR}(\bar{y}_{0,h}-y_{0}))\,ds$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\|g\|\|(\bar{y}_{0,h}-y_{0})-I_{CR}(\bar{y}_{0,h}-y_{0})\|+\sum_{e\in\mathcal{E}_{h}}\|(\nabla\phi_{g}-(\Pi_{T}\nabla\phi_{g}))\cdot
n_{e}\|_{\partial
T_{e}}\|(\bar{y}_{0,h}-y_{0})-I_{CR}(\bar{y}_{0,h}-y_{0})\|_{\partial T_{e}}$
(4.15)

An application of the discrete trace-inequality and the CR interpolation error
estimation in (4) yields,

$\displaystyle\displaystyle
a_{pw}(\phi_{g},(\bar{y}_{0,h}-y_{0})-I_{CR}(\bar{y}_{0,h}-y_{0}))\lesssim
h\|g\|\|\bar{y}_{0,h}-y_{0}\|_{pw}.$ (4.16)

Arguments in the similar line yields

$\displaystyle\displaystyle
a_{pw}(\phi_{g},I_{CR}(\bar{y}_{0,h}-y_{0})-I_{c}(I_{CR}(\bar{y}_{0,h}-y_{0})))\lesssim
h\|g\|\|\bar{y}_{0,h}-y_{0}\|_{pw}.$ (4.17)

Combining (4.8), (4), (4), (4.16), (4.17) and standard $\displaystyle L_{2}$
norm error-estimates for $\displaystyle I_{CR}$ interpolation operator yields,

$\displaystyle\displaystyle II+III$
$\displaystyle\displaystyle\lesssim\|g\|\|\bar{y}_{0,h}-y_{0}\|_{pw}.$ (4.18)

Next we aim to consider $\displaystyle\|y_{0,h}-y_{0}\|_{pw}$.
From (4.2), (4.3), (4.4), Theorem 3.4 and (4.5) we obtain

$\displaystyle\displaystyle\|y_{0,h}-I_{CR}y_{0}\|^{2}_{pw}$
$\displaystyle\displaystyle\leq
a_{pw}(I_{CR}y_{0}-y_{0,h},I_{CR}y_{0}-y_{0,h})$
$\displaystyle\displaystyle=a_{pw}(I_{CR}y_{0},I_{CR}y_{0}-y_{0,h})-a_{pw}(y_{0,h},I_{CR}y_{0}-y_{0,h})$
$\displaystyle\displaystyle=a_{pw}(I_{CR}y_{0}-y_{0},I_{CR}y_{0}-y_{0,h})+a_{pw}(y_{0},I_{CR}y_{0}-y_{0,h})+$
$\displaystyle\displaystyle\quad\,\,a_{pw}(\tilde{P_{1}}(\bar{u}_{h}),I_{CR}y_{0}-y_{0,h}-I_{c}(I_{CR}y_{0}-y_{0,h}))+a_{pw}(\tilde{P_{1}}(\bar{u}_{h}),I_{c}(I_{CR}y_{0}-y_{0,h}))$
$\displaystyle\displaystyle=a_{pw}(I_{CR}y_{0}-y_{0},I_{CR}y_{0}-y_{0,h})+a_{pw}(y_{0},I_{CR}y_{0}-y_{0,h})+a_{pw}(\tilde{P_{1}}(\bar{u}_{h}),I_{c}(I_{CR}y_{0}-y_{0,h}))$
$\displaystyle\displaystyle=a_{pw}(I_{CR}y_{0}-y_{0},I_{CR}y_{0}-y_{0,h})+a_{pw}(y_{0},I_{CR}y_{0}-y_{0,h}-I_{c}(I_{CR}y_{0}-y_{0,h}))+$
$\displaystyle\displaystyle\quad\,\,a_{pw}(y_{0},I_{c}(I_{CR}y_{0}-y_{0,h}))+a_{pw}(\tilde{P_{1}}(\bar{u}_{h}),I_{c}(I_{CR}y_{0}-y_{0,h}))$
$\displaystyle\displaystyle=a_{pw}(I_{CR}y_{0}-y_{0},I_{CR}y_{0}-y_{0,h})+a_{pw}(y_{0},I_{CR}y_{0}-y_{0,h}-I_{c}(I_{CR}y_{0}-y_{0,h}))$
$\displaystyle\displaystyle=a_{pw}(I_{CR}y_{0}-y_{0},I_{CR}y_{0}-y_{0,h})+a_{pw}(y_{0}-I_{CR}y_{0},I_{CR}y_{0}-y_{0,h}-I_{c}(I_{CR}y_{0}-y_{0,h}))$
(4.19)

Then applying the similar arguments as in (4), (4) and Young’s inequality in
the right hand side of (4.19) we obtain

$\displaystyle\displaystyle\|y_{0,h}-I_{CR}y_{0}\|_{pw}\lesssim\|y_{0}-I_{CR}y_{0}\|_{pw}.$
(4.20)

Finally combining (4.6), (4.7), (4), (4.13), (4.18) and (4.20) the result is
obtained. ∎

Corollary: For $\displaystyle z_{h}\in V_{1}(\partial\Omega)$ let
$\displaystyle y_{z_{h}},\,y_{z_{h},h}$ satisfies (4.2)-(4.3) and (4.4)-(4.5)
respectively. Then $\displaystyle\|y_{z_{h}}-y_{z_{h},h}\|\lesssim
h[\|y_{z_{h}}-y_{z_{h},h}\|_{pw}+\|z_{h}\|_{H^{\frac{1}{2}}(\partial\Omega)}]$.

We note that since we are considering triangulations
$\displaystyle\mathcal{T}_{h}$ such that it induces an odd number of edges on
$\displaystyle\partial\Omega$ and $\displaystyle
dim~{}V_{1}(\partial\Omega)=dim~{}U_{h}$ therefore $\displaystyle
P_{0}:V_{1}(\partial\Omega)\rightarrow U_{h}$ is bijective. Define
$\displaystyle\tilde{P}_{1}:U_{h}\rightarrow V_{1}(\partial\Omega)$ by

$\displaystyle\displaystyle\tilde{P}_{1}=P_{0}^{-1}.$ (4.21)

Note that the matrix representation of $\displaystyle
P_{0}:V_{1}(\partial\Omega)\rightarrow U_{h}$ is

$\displaystyle\begin{bmatrix}1&0&0&\cdot&\cdot&0\\\ 1&1&0&\cdot&\cdot&0\\\
0&1&1&\cdot&\cdot&0\\\ \cdot&\cdot&\cdot&\cdot&\cdot&0\\\
\cdot&\cdot&\cdot&1&1&0\\\ 0&0&\cdot&0&1&1\end{bmatrix}$

Therefore if
$\displaystyle\|\tilde{P}_{1}\|_{\mathcal{L}(U_{h},V_{1}(\partial\Omega))}$
denotes the operator norm of $\displaystyle\tilde{P}_{1}$ subordinate to
$\displaystyle L_{2}$ norm then it is uniformly bounded with respect to the
mesh refinement parameter $\displaystyle h>0$ [21]. $\displaystyle i.\;e.$

$\displaystyle\displaystyle\|\tilde{P}_{1}\|_{\mathcal{L}(U_{h},V_{1}(\partial\Omega))}\leq
C,$ (4.22)

where $\displaystyle C>0$ is independent of $\displaystyle h>0$. This operator
plays a central role in associating the discrete control variable with the
discrete state variable, which is discussed in the next section.

######  Theorem 4.2.

For a piecewise constant function $\displaystyle{u}_{h}$ defined on
$\displaystyle\partial\Omega$, the following holds:

$\displaystyle\displaystyle\|y_{{u}_{h}}-{y}_{\tilde{P}_{1}(u_{h})}\|\lesssim{h}\|\tilde{P}_{1}{u}_{h}\|_{H^{\frac{1}{2}}(\partial\Omega)}..$
$\displaystyle\displaystyle\|y_{{u}_{h}}-{y}_{P_{1}(u_{h})}\|\lesssim h.$

###### Proof.

Since $\displaystyle{u}_{h}\in L_{2}(\Gamma)$, $\displaystyle y_{{u}_{h}}\in
L_{2}(\Omega)$ is the solution of the following ultra weak formulation:

$\displaystyle\displaystyle(y_{u_{h}},\Delta\phi)$
$\displaystyle\displaystyle=-\langle{u}_{h},\partial\phi/\partial
n\rangle\quad\forall~{}\phi\in X.$ (4.23)

Note that $\displaystyle\tilde{P}_{1}({u}_{h})\in H^{1/2}(\Gamma)$ and
$\displaystyle y_{\tilde{P}_{1}(\bar{u}_{h})}\in L_{2}(\Omega)$ also satisfies

$\displaystyle\displaystyle(y_{\tilde{P}_{1}({u}_{h})},\Delta\phi)$
$\displaystyle\displaystyle=-\langle\tilde{P}_{1}({u}_{h}),\partial\phi/\partial
n\rangle\quad\forall~{}\phi\in X.$ (4.24)

By duality, we have

$\displaystyle\displaystyle\|y_{{u}_{h}}-y_{\tilde{P}_{1}({u}_{h})}\|$
$\displaystyle\displaystyle=\sup_{g\in L_{2}(\Omega),g\neq
0}\frac{(g,y_{{u}_{h}}-y_{\tilde{P}_{1}({u}_{h})})}{\|g\|}.$ (4.25)

For any $\displaystyle\phi\in X$, it follows from (4.23), (4.24) that

$\displaystyle\displaystyle(y_{\tilde{P}_{1}({u}_{h})}-y_{{u}_{h}},\Delta\phi)$
$\displaystyle\displaystyle=\langle{u}_{h}-\tilde{P}_{1}({u}_{h}),\partial\phi/\partial
n\rangle$ $\displaystyle\displaystyle=\langle
P_{0}(\tilde{P}_{1}{u}_{h})-\tilde{P}_{1}({u}_{h}),\partial\phi/\partial
n-P_{0}(\partial\phi/\partial n)\rangle.$ (4.26)

We know that for any $\displaystyle g\in L_{2}(\Omega)$, there exists
$\displaystyle\phi\in X$ such that
$\displaystyle-\Delta\phi=g~{}\text{in}~{}\Omega$ and since
$\displaystyle\Omega$ is convex from elliptic regularity theory
$\displaystyle\|\phi\|_{H^{2}(\Omega)}\lesssim\|g\|$. Thus, it follows from
(4.26) and Cauchy-Schwarz inequality that,

$\displaystyle\displaystyle(y_{{u}_{h}}-y_{\tilde{P}_{1}({u}_{h})},g)$
$\displaystyle\displaystyle=\langle{u}_{h}-\tilde{P}_{1}({u}_{h}),\partial\phi/\partial
n-P_{0}(\partial\phi/\partial n)\rangle$ $\displaystyle\displaystyle=\langle
P_{0}(\tilde{P}_{1}{u}_{h})-\tilde{P}_{1}({u}_{h}),\partial\phi/\partial
n-P_{0}(\partial\phi/\partial n)\rangle$
$\displaystyle\displaystyle\lesssim\|P_{0}(\tilde{P}_{1}{u}_{h})-\tilde{P}_{1}{u}_{h}\|_{0,\Gamma}\|\partial\phi/\partial
n-P_{0}(\partial\phi/\partial n)\|_{0,\Gamma}.$ (4.27)

Then combining (4.25), (4.27) and the elliptic regularity estimate for
$\displaystyle\phi$ we obtain

$\displaystyle\displaystyle\|y_{{u}_{h}}-y_{P_{1}({u}_{h})}\|$
$\displaystyle\displaystyle\lesssim{h}\|\tilde{P}_{1}{u}_{h}\|_{H^{\frac{1}{2}}(\partial\Omega)}.$
(4.28)

∎

######  Theorem 4.3.

For a piecewise constant function $\displaystyle u_{h}$ defined on
$\displaystyle\partial\Omega$ it’s extension $\displaystyle y_{u_{h}}$ and
it’s CR approximation $\displaystyle y_{u_{h},h}$ satisfies the following
estimate:

$\displaystyle\displaystyle\|y_{u_{h}}-y_{u_{h},h}\|\lesssim h.$

###### Proof.

Applying Theorems 4.1, 4.2 the proof follows . ∎

######  Theorem 4.4.

The following estimate holds:

$\displaystyle\displaystyle\|y_{P_{0}\bar{u}}-y_{\tilde{P}_{1}P_{0}\bar{u}}\|\lesssim
h\|\bar{u}\|_{H^{\frac{1}{2}}(\partial\Omega)}.$

###### Proof.

The main ingredients of the proof consists of orthogonality properties of the
$\displaystyle L_{2}$ projection operator and bijection of
$\displaystyle\tilde{P}_{1}$. Note that since we consider the triangulation
such that it induces an odd numbers of edges on $\displaystyle\partial\Omega$
therefore given $\displaystyle{P}_{0}\bar{u}\in U_{h}$ there exists an unique
$\displaystyle v_{1}\in V_{1}(\partial\Omega)$ such that

$\displaystyle\displaystyle{P}_{0}\bar{u}=P_{0}v_{1}.$ (4.29)

Note that

$\displaystyle\displaystyle\|y_{P_{0}\bar{u}}-y_{\tilde{P}_{1}P_{0}\bar{u}}\|=\sup_{g\in
L_{2}(\Omega),g\neq
0}\dfrac{(y_{P_{0}\bar{u}}-y_{\tilde{P}_{1}P_{0}\bar{u}},g)}{\|g\|}.$ (4.30)

Then consider the auxiliary problem:

$\displaystyle\displaystyle-\Delta\phi$
$\displaystyle\displaystyle=g~{}\text{in}~{}\Omega,$ (4.31)
$\displaystyle\displaystyle\phi$
$\displaystyle\displaystyle=0~{}\text{on}~{}\partial\Omega.$

Considering $\displaystyle(y_{P_{0}\bar{u}}-y_{\tilde{P}_{1}P_{0}\bar{u}},g)$,
(4.31), $\displaystyle P_{0}^{2}=P_{0}$ along with integration by parts
implies

$\displaystyle\displaystyle(y_{\tilde{P}_{1}P_{0}\bar{u}}-y_{P_{0}\bar{u}},g)$
$\displaystyle\displaystyle=(\tilde{P}_{1}P_{0}\bar{u}-P_{0}\bar{u},\dfrac{\partial\phi}{\partial
n})$
$\displaystyle\displaystyle=(\tilde{P}_{1}P_{0}v_{1}-P_{0}v_{1},\dfrac{\partial\phi}{\partial
n})$
$\displaystyle\displaystyle=(v_{1}-P_{0}v_{1},\dfrac{\partial\phi}{\partial
n})$
$\displaystyle\displaystyle=(v_{1}-P_{0}v_{1},\dfrac{\partial\phi}{\partial
n}-P_{0}\dfrac{\partial\phi}{\partial n}).$ (4.32)

Therefore combination of (4.30), (4.29) and (4) implies that

$\displaystyle\displaystyle\|y_{P_{0}\bar{u}}-y_{\tilde{P}_{1}P_{0}\bar{u}}\|\lesssim
h\|\tilde{P}_{1}P_{0}\bar{u}\|_{H^{\frac{1}{2}}(\partial\Omega)}.$ (4.33)

Next we show that
$\displaystyle\|\tilde{P}_{1}P_{0}\bar{u}\|_{H^{\frac{1}{2}}(\partial\Omega)}\lesssim\|\bar{u}\|_{H^{\frac{1}{2}}(\partial\Omega)}$.
Consider

$\displaystyle\displaystyle\|\tilde{P}_{1}P_{0}\bar{u}\|_{H^{\frac{1}{2}}(\partial\Omega)}$
$\displaystyle\displaystyle\leq\|\tilde{P}_{1}P_{0}(\bar{u}-\pi_{1}\bar{u})\|_{H^{\frac{1}{2}}(\partial\Omega)}+\|\tilde{P}_{1}P_{0}(\pi_{1}\bar{u})\|_{H^{\frac{1}{2}}(\partial\Omega)}$
$\displaystyle\displaystyle\leq
Ch^{-\frac{1}{2}}\|\tilde{P}_{1}P_{0}(\bar{u}-\pi_{1}\bar{u})\|+\|\pi_{1}\bar{u}\|_{H^{\frac{1}{2}}(\partial\Omega)}$
$\displaystyle\displaystyle\leq
Ch^{-\frac{1}{2}}\|P_{0}(\bar{u}-\pi_{1}\bar{u})\|+\|\pi_{1}\bar{u}\|_{H^{\frac{1}{2}}(\partial\Omega)}$
$\displaystyle\displaystyle\leq
Ch^{-\frac{1}{2}}\|\bar{u}-\pi_{1}\bar{u}\|+\|\bar{u}\|_{H^{\frac{1}{2}}(\partial\Omega)}$
$\displaystyle\displaystyle\leq
C\|\bar{u}\|_{H^{\frac{1}{2}}(\partial\Omega)}.$ (4.34)

Therefore combining (4.30), (4), (4.33) and (4) the result is obtained. ∎

######  Theorem 4.5.

Let $\displaystyle z\in H^{\frac{1}{2}}(\partial\Omega)$, then the following
holds:

$\displaystyle\displaystyle\|y_{z}-y_{P_{0}(z),h}\|\lesssim
h\|z\|_{H^{1/2}(\Gamma)}.$

###### Proof.

By triangle inequality, we have

$\displaystyle\displaystyle\|y_{z}-y_{P_{0}(z),h}\|\leq\|y_{z}-y_{\tilde{P}_{1}(P_{0}(z))}\|+\|y_{\tilde{P}_{1}(P_{0}(z))}-y_{P_{0}(z),h}\|.$
(4.35)

Consider $\displaystyle\|y_{z}-y_{\tilde{P}_{1}(P_{0}z)}\|$.

$\displaystyle\displaystyle\|y_{z}-y_{\tilde{P}_{1}(P_{0}z)}\|\leq\|y_{z}-y_{P_{0}z}\|+\|y_{P_{0}z}-y_{\tilde{P}_{1}(P_{0}z)}\|.$
(4.36)

Consider $\displaystyle\|y_{P_{0}z}-y_{\tilde{P}_{1}(P_{0}z)}\|$. We write the
$\displaystyle L_{2}$ norm in duality form

$\displaystyle\displaystyle\|y_{P_{0}z}-y_{\tilde{P}_{1}(P_{0}z)}\|=\sup_{g\in
L_{2}(\Omega),g\neq 0}\dfrac{(y_{P_{0}z}-y_{\tilde{P}_{1}(P_{0}z)},g)}{\|g\|}$
(4.37)

Consider the auxiliary problem

$\displaystyle\displaystyle-\Delta\phi$
$\displaystyle\displaystyle=g~{}\text{in}~{}\Omega,$ (4.38)
$\displaystyle\displaystyle\phi$
$\displaystyle\displaystyle=0~{}\text{on}~{}\partial\Omega.$

Consider $\displaystyle(y_{P_{0}z}-y_{\tilde{P}_{1}(P_{0}z)},g)$. Combining
(4.37), (4.38) and integration by parts we obtain

$\displaystyle\displaystyle(y_{P_{0}z}-y_{\tilde{P}_{1}(P_{0}z)},g)$
$\displaystyle\displaystyle=(y_{\tilde{P}_{1}(P_{0}z)}-y_{P_{0}z},\Delta\phi)$
$\displaystyle\displaystyle=(P_{0}z-\tilde{P}_{1}(P_{0}z),\dfrac{\partial\phi}{\partial
n})$
$\displaystyle\displaystyle=(P_{0}\tilde{P}_{1}(P_{0}z))-\tilde{P}_{1}(P_{0}z),\dfrac{\partial\phi}{\partial
n})$
$\displaystyle\displaystyle=(P_{0}(\tilde{P}_{1}P_{0}z)-\tilde{P}_{1}P_{0}z,\dfrac{\partial\phi}{\partial
n}-P_{0}\dfrac{\partial\phi}{\partial n})$ $\displaystyle\displaystyle\lesssim
h\|\tilde{P}_{1}P_{0}z\|_{H^{\frac{1}{2}}(\partial\Omega)}\|g\|$ (4.39)

Therefore combining (4.37), (4) we obtain

$\displaystyle\displaystyle\|y_{P_{0}z}-y_{\tilde{P}_{1}(P_{0}z)}\|\lesssim
h\|\tilde{P}_{1}P_{0}z\|_{H^{\frac{1}{2}}(\partial\Omega)}.$ (4.40)

Next consider $\displaystyle\|y_{z}-y_{P_{0}z}\|$. Applying duality arguments
as above we obtain

$\displaystyle\displaystyle\|y_{z}-y_{P_{0}z}\|\lesssim
h\|z\|_{H^{\frac{1}{2}}(\partial\Omega)}.$ (4.41)

Applying Corollary 1 we obtain

$\displaystyle\displaystyle\|y_{\tilde{P}_{1}(P_{0}(z))}-y_{P_{0}(z),h}\|\lesssim
h\|\tilde{P}_{1}P_{0}(z)\|_{H^{\frac{1}{2}}(\partial\Omega)}.$ (4.42)

Combining (4), (4.35), (4.36), (4), (4.40), (4.41), (4.42) we obtain the
result. ∎

##  5 Discrete Control Problem and Error Estimate

In this section the fully discrete control problem is introduced. The control
space is discretized by piecewise constant functions. Define the discrete
control problem by:

$\displaystyle\displaystyle
J_{h}(y_{h},u_{h})=\dfrac{1}{2}\|y_{h}-y_{d}\|^{2}+\dfrac{\alpha}{2}\|\tilde{P}_{1}(u_{h})\|^{2},$
(5.1)

subject to,

$\displaystyle\displaystyle{y}_{h}={y}_{f,h}+{y}_{{u}_{h},h},\quad{y}_{f,h}\in
V_{h}^{0},~{}{y}_{{u}_{h},h}={y}_{0,h}+\tilde{P_{1}}({u}_{h})\in V_{h},$ (5.2)
$\displaystyle\displaystyle
a_{pw}({y}_{f,h},v_{h})=(f,v_{h})\quad\forall~{}v_{h}\in V_{h}^{0},$ (5.3)
$\displaystyle\displaystyle
a_{pw}({y}_{0,h},v_{h})=-a_{pw}(\widetilde{\tilde{P_{1}}({u}_{h})},v_{h})\quad\forall~{}v_{h}\in
V_{h}^{0}.$ (5.4)

The existence and uniqueness of the solution of the problem follows from
finite dimensional and strict convexity arguments. The optimal variables
$\displaystyle(\bar{u}_{h},\bar{y}_{h})\in U_{h,ad}\times V_{h}$ satisfies the
following system

######  Theorem 5.1.

There exists a unique solution $\displaystyle(\bar{y}_{h},\bar{u}_{h})\in
V_{h}\times U_{h,ad}$ of the discrete optimal control problem. Furthermore, it
satisfy the following system of equations:

$\displaystyle\displaystyle\bar{y}_{h}=\bar{y}_{f,h}+\bar{y}_{\bar{u}_{h},h},\quad\bar{y}_{f,h}\in
V_{h}^{0},~{}\bar{y}_{\bar{u}_{h},h}=\bar{y}_{0,h}+\tilde{P_{1}}(\bar{u}_{h})\in
V_{h},$ (5.5) $\displaystyle\displaystyle
a_{pw}(\bar{y}_{f,h},v_{h})=(f,v_{h})\quad\forall~{}v_{h}\in V_{h}^{0},$ (5.6)
$\displaystyle\displaystyle
a_{pw}(\bar{y}_{0,h},v_{h})=-a_{pw}(\widetilde{\tilde{P_{1}}(\bar{u}_{h})},v_{h})\quad\forall~{}v_{h}\in
V_{h}^{0},$ (5.7)
$\displaystyle\displaystyle(\bar{y}_{h}-y_{d},y_{p_{h},h}-\bar{y}_{\bar{u}_{h},h})+\alpha\langle\tilde{P}_{1}(\bar{u}_{h}),\tilde{P}_{1}(p_{h})-\tilde{P}_{1}(\bar{u}_{h})\rangle\geq
0\quad\forall~{}p_{h}\in U_{h,ad}.$ (5.8)

Next we prove the following auxiliary estimate which is important to obtain
the optimal order error estimate for the optimal control.

######  Theorem 5.2.

For the discrete optimal control $\displaystyle\bar{u}_{h}$ the following
error estimate holds:

$\displaystyle\displaystyle\|y_{\bar{u}_{h}}-y_{\bar{u}_{h},h}\|\lesssim h$

###### Proof.

From Theorem 4.2 we obtain

$\displaystyle\displaystyle\|y_{\bar{u}_{h}}-y_{\tilde{P}_{1}(\bar{u}_{h})}\|$
$\displaystyle\displaystyle\lesssim
h\|\tilde{P}_{1}(\bar{u}_{h})\|_{H^{\frac{1}{2}}(\partial\Omega)}.$ (5.9)

Next it remains to find an uniform bound for
$\displaystyle\|\tilde{P}_{1}(\bar{u}_{h})\|_{H^{\frac{1}{2}}(\partial\Omega)}$.
Let $\displaystyle\xi_{h}\in V_{h}$ satisfies

$\displaystyle\displaystyle
a_{pw}(\xi_{h},v_{h})=(\bar{y}_{h}-y_{d},v_{h}),\forall v_{h}\in V_{h}$ (5.10)

Then note that (5.8) is equivalent to

$\displaystyle\displaystyle(\bar{\theta}_{h}+\alpha\tilde{P}_{1}\bar{u}_{h},\tilde{P}_{1}p_{h}-\tilde{P}_{1}\bar{u}_{h})\geq
0,\forall p_{h}\in U_{ad},$ (5.11)

where for any $\displaystyle r_{h}\in V_{1}(\partial\Omega)$,
$\displaystyle\bar{\theta}_{h}\in V_{1}(\partial\Omega)$ is defined by

$\displaystyle\displaystyle(\bar{\theta}_{h},r_{h})=a_{pw}(\bar{\xi}_{h},\tilde{r}_{h})-(\bar{y}_{h}-y_{d},\tilde{r}_{h}),$
(5.12)

where $\displaystyle\tilde{r}_{h}\in V_{h}$ is a piecewise linear, globally
continuous extension of $\displaystyle r_{h}$ in $\displaystyle\Omega$. Next
we prove

$\displaystyle\displaystyle\|\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n}-\bar{\theta}_{h}\|\lesssim\sqrt{h}.$ (5.13)

Where $\displaystyle\theta_{y_{\bar{u}_{h}}}\in H^{1}_{0}(\Omega)$ satisfies

$\displaystyle\displaystyle-\Delta\theta_{y_{\bar{u}_{h}}}$
$\displaystyle\displaystyle=y_{\bar{u}_{h}}-y_{d}\;\text{in}\;\Omega,$ (5.14)
$\displaystyle\displaystyle\theta_{y_{\bar{u}_{h}}}$
$\displaystyle\displaystyle=0\;\text{on}\;\partial\Omega.$

Note that

$\displaystyle\displaystyle\|\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n}-\bar{\theta}_{h}\|^{2}\lesssim\|\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n}-\tilde{P}_{1}P_{0}(\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n})\|^{2}+\|\tilde{P}_{1}P_{0}(\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n})-\bar{\theta}_{h}\|^{2}$ (5.15)

Also

$\displaystyle\displaystyle\|\tilde{P}_{1}P_{0}(\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n})-\bar{\theta}_{h}\|^{2}\lesssim\|\tilde{P}_{1}P_{0}(\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n})-\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n}\|^{2}+\|\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n}-P_{1}\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n}\|^{2}+\|P_{1}\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n}-\bar{\theta}_{h}\|^{2}.$ (5.16)

Consider,

$\displaystyle\displaystyle\|P_{1}(\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n})-\bar{\theta}_{h}\|^{2}=(\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n}-\bar{\theta}_{h},P_{1}(\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n})-\bar{\theta}_{h})$ (5.17)

To estimate $\displaystyle\|\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n}-\bar{\theta}_{h}\|$ consider the following auxiliary problem: Find
$\displaystyle z_{h}\in\tilde{V}_{h}$ such that

$\displaystyle\displaystyle a_{pw}(z_{h},v_{h})$
$\displaystyle\displaystyle=0,\forall v_{h}\in V_{h}$ (5.18)
$\displaystyle\displaystyle z_{h}$
$\displaystyle\displaystyle=P_{1}(\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n})-\bar{\theta}_{h}\;\text{on}\;\partial\Omega.$

Then from (5.12), (5.18) and (5.14) we obtain,

$\displaystyle\displaystyle\int_{\partial\Omega}\left(\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n}-\bar{\theta}_{h}\right)\left(P_{1}\left(\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n}\right)-\bar{\theta}_{h}\right)$
$\displaystyle\displaystyle=a_{pw}(\theta_{y_{\bar{u}_{h}}}-\xi_{h},z_{h})$
$\displaystyle\displaystyle=a_{pw}(\theta_{y_{\bar{u}_{h}}}-I_{CR}\theta_{y_{\bar{u}_{h}}},z_{h})$
$\displaystyle\displaystyle\leq\|\theta_{y_{\bar{u}_{h}}}-I_{CR}\theta_{y_{\bar{u}_{h}}}\|_{pw}\|z_{h}\|_{pw}$
$\displaystyle\displaystyle\lesssim h\|y_{\bar{u}_{h}}-y_{d}\|]\|z_{h}\|_{pw}$
$\displaystyle\displaystyle\lesssim
h\|\bar{u}_{h}\|_{H^{-\frac{1}{2}}(\partial\Omega)}\|P_{1}(\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n})-\bar{\theta}_{h}\|_{H^{\frac{1}{2}}(\partial\Omega)}$
$\displaystyle\displaystyle\lesssim\sqrt{h}\|\bar{u}_{h}\|\|P_{1}(\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n})-\bar{\theta}_{h}\|$ (5.19)

Combining (5.17) and (5) and applying the fact that
$\displaystyle\|\bar{u}_{h}\|$ is uniformly bounded with respect to
$\displaystyle h>0$ we obtain

$\displaystyle\displaystyle\|\dfrac{\partial\theta_{y_{\bar{u}_{h}}}}{\partial
n}-\bar{\theta}_{h}\|\lesssim\sqrt{h}.$ (5.20)

Hence application of triangle inequality, interpolation error estimates and
inverse inequality implies that

$\displaystyle\displaystyle\|\bar{\theta}_{h}\|_{H^{\frac{1}{2}}(\partial\Omega)}\leq
C.$ (5.21)

Note that (5.11) implies

$\displaystyle\displaystyle\tilde{P}_{1}\bar{u}_{h}=P_{\tilde{P}_{1}(U_{h,ad})}(\dfrac{1}{\alpha}\bar{\theta}_{h}).$
(5.22)

Also from (4.22) it is evident that $\displaystyle\tilde{P}_{1}(U_{h,ad})$ is
an uniformly bounded convex subset of $\displaystyle V_{1}(\partial\Omega)$.
Therefore by the stability of $\displaystyle P_{\tilde{P}_{1}(U_{h,ad})}$ with
respect to $\displaystyle H^{\frac{1}{2}}(\partial\Omega)$ norm [12] we obtain
that

$\displaystyle\displaystyle\|\tilde{P}_{1}\bar{u}_{h}\|_{H^{\frac{1}{2}}(\partial\Omega)}\leq
C.$ (5.23)

Combining (5.9) and (5.23) the result follows. ∎

######  Theorem 5.3.

Suppose $\displaystyle\bar{u}\in U_{ad}$, $\displaystyle\bar{u}_{h}\in
U_{h,ad}$ be the optimal control and discrete optimal control respectively.
The following holds:

$\displaystyle\displaystyle\|\bar{u}-\bar{u}_{h}\|_{L_{2}(\Gamma)}\lesssim
h^{1/2}.$

###### Proof.

We begin by choosing $\displaystyle p_{h}=P_{0}\bar{u}$ in (5.8) to obtain,

$\displaystyle\displaystyle\alpha\|\tilde{P}_{1}\bar{u}_{h}-\tilde{P}_{1}P_{0}\bar{u}\|^{2}\leq$
$\displaystyle\displaystyle\alpha(\tilde{P}_{1}P_{0}\bar{u},\tilde{P}_{1}P_{0}\bar{u}-\tilde{P}_{1}\bar{u}_{h})+(\bar{y}_{h}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\alpha(\tilde{P}_{1}P_{0}\bar{u}-P_{0}\bar{u},\tilde{P}_{1}P_{0}\bar{u}-\tilde{P}_{1}\bar{u}_{h})+(P_{0}\bar{u},\tilde{P}_{1}P_{0}\bar{u}-\tilde{P}_{1}\bar{u}_{h})+(\bar{y}_{h}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\alpha(\tilde{P}_{1}P_{0}\bar{u}-P_{0}(\tilde{P}_{1}P_{0}\bar{u}),\tilde{P}_{1}P_{0}\bar{u}-\tilde{P}_{1}\bar{u}_{h})+\alpha(P_{0}\bar{u},\tilde{P}_{1}P_{0}\bar{u}-\tilde{P}_{1}\bar{u}_{h})+$
$\displaystyle\displaystyle(\bar{y}_{h}-\bar{y},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})$
(5.24)

Application of Young’s inequality to (5) and the fact that $\displaystyle
P_{0}\tilde{P}_{1}=I_{U_{h}}$ yields,

$\displaystyle\displaystyle(\alpha-\epsilon)\|\tilde{P}_{1}\bar{u}_{h}-\tilde{P}_{1}P_{0}\bar{u}\|^{2}\leq$
$\displaystyle\displaystyle\alpha(P_{0}\bar{u},\tilde{P}_{1}P_{0}\bar{u}-\tilde{P}_{1}\bar{u}_{h})+(\bar{y}_{h}-\bar{y},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+$
$\displaystyle\displaystyle(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+\|\tilde{P}_{1}P_{0}\bar{u}-P_{0}(\tilde{P}_{1}P_{0}\bar{u})\|^{2}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\alpha(P_{0}\bar{u},(\tilde{P}_{1}P_{0}\bar{u}-\tilde{P}_{1}\bar{u}_{h})-P_{0}(\tilde{P}_{1}P_{0}\bar{u}-\tilde{P}_{1}\bar{u}_{h}))+(\bar{y}_{h}-\bar{y},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+$
$\displaystyle\displaystyle(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+\|\tilde{P}_{1}P_{0}\bar{u}-P_{0}(\tilde{P}_{1}P_{0}\bar{u})\|^{2}+\alpha(P_{0}\bar{u},P_{0}\bar{u}-\bar{u}_{h})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle(\bar{y}_{h}-\bar{y},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+$
$\displaystyle\displaystyle\frac{\alpha}{\epsilon}\|\tilde{P}_{1}P_{0}\bar{u}-P_{0}(\tilde{P}_{1}P_{0}\bar{u})\|^{2}+\alpha(P_{0}\bar{u},P_{0}\bar{u}-\bar{u}_{h})$
(5.25)

From (5) consider
$\displaystyle(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+\alpha(P_{0}\bar{u},P_{0}\bar{u}-\bar{u}_{h})$.

$\displaystyle\displaystyle(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+\alpha(P_{0}\bar{u},P_{0}\bar{u}-\bar{u}_{h})=$
$\displaystyle\displaystyle(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+$
$\displaystyle\displaystyle\alpha(P_{0}\bar{u}-\bar{u},P_{0}\bar{u}-\bar{u}_{h})+\alpha(\bar{u},P_{0}\bar{u}-\bar{u}_{h})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+\alpha(\bar{u},P_{0}\bar{u}-\bar{u}_{h})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+\alpha(\bar{u},P_{0}\bar{u}-\bar{u})+\alpha(\bar{u},\bar{u}-\bar{u}_{h})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\alpha\|\bar{u}-P_{0}\bar{u}\|^{2}+\alpha(P_{0}\bar{u},P_{0}\bar{u}-\bar{u})+$
$\displaystyle\displaystyle(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+\alpha(\bar{u},\bar{u}-\bar{u}_{h})$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+\alpha(\bar{u},\bar{u}-\bar{u}_{h})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\alpha(\bar{u},\bar{u}-\bar{u}_{h})+(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{P_{0}\bar{u}}+$
$\displaystyle\displaystyle
y_{P_{0}\bar{u}}-y_{\bar{u}}+y_{\bar{u}}-y_{\bar{u}_{h}}+y_{\bar{u}_{h}}-y_{\bar{u}_{h},h}).$
(5.26)

Combining (2.11) and (5) we obtain

$\displaystyle\displaystyle(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+\alpha(P_{0}\bar{u},P_{0}\bar{u}-\bar{u}_{h})\leq$
$\displaystyle\displaystyle(\bar{y}-y_{d},y_{P_{0}\bar{u},h}-y_{P_{0}\bar{u}})+$
$\displaystyle\displaystyle(\bar{y}-y_{d},y_{P_{0}\bar{u}}-y_{\bar{u}})+(\bar{y}-y_{d},y_{\bar{u}_{h}}-y_{\bar{u}_{h},h}).$
(5.27)

Consider
$\displaystyle(\bar{y}_{h}-\bar{y},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})$ from
(5).

$\displaystyle\displaystyle(\bar{y}_{h}-\bar{y},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})$
$\displaystyle\displaystyle=(y_{f,h}-y_{f},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+(y_{\bar{u}_{h},h}-y_{\bar{u}},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})$
$\displaystyle\displaystyle\leq(y_{f,h}-y_{f},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+(y_{P_{0}\bar{u},h}-y_{\bar{u}},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h}).$
(5.28)

From (5) consider $\displaystyle\|y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h}\|$.

$\displaystyle\displaystyle\|y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h}\|\leq\|y_{P_{0}\bar{u},h}-y_{P_{0}\bar{u}}\|+\|y_{P_{0}\bar{u}}-y_{\bar{u}_{h}}\|+\|y_{\bar{u}_{h}}-y_{\bar{u}_{h},h}\|$
(5.29)

From (5.29) consider $\displaystyle\|y_{P_{0}\bar{u},h}-y_{P_{0}\bar{u}}\|.$

$\displaystyle\displaystyle\|y_{P_{0}\bar{u},h}-y_{P_{0}\bar{u}}\|\leq\|y_{P_{0}\bar{u}}-y_{P_{1}(P_{0}\bar{u})}\|+\|y_{P_{1}(P_{0}\bar{u})}-y_{P_{0}\bar{u},h}\|.$
(5.30)

Combining (5.30), Theorem 4.4, Theorem 4.5 we obtain

$\displaystyle\displaystyle\|y_{P_{0}\bar{u},h}-y_{P_{0}\bar{u}}\|\lesssim
h\|\bar{u}\|_{H^{\frac{1}{2}}(\partial\Omega)}.$ (5.31)

Theorem 5.2 yields,

$\displaystyle\displaystyle\|y_{\bar{u}_{h},h}-y_{\bar{u}_{h}}\|\lesssim h.$
(5.32)

An application of ultra weak formulation yields that

$\displaystyle\displaystyle\|y_{P_{0}\bar{u}}-y_{\bar{u}_{h}}\|\lesssim\|P_{0}\bar{u}-\bar{u}_{h}\|\lesssim
C,$ (5.33)

where $\displaystyle C>0$ is depends only upon
$\displaystyle|u_{a}|,~{}|u_{b}|$, $\displaystyle|\Omega|$ and shape-
regularity of the triangulation. Combining (5.29), (5.30), (5.31), (5.32) and
(5.33) we obtain

$\displaystyle\displaystyle\|y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h}\|\lesssim C$
(5.34)

Therefore combining (5) and (5.34) one obtains

$\displaystyle\displaystyle(\bar{y}_{h}-\bar{y},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})$
$\displaystyle\displaystyle\leq(y_{f,h}-y_{f},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})+(y_{\bar{u}_{h},h}-y_{\bar{u}},y_{P_{0}\bar{u},h}-y_{\bar{u}_{h},h})\lesssim
h,$ (5.35)

where the generic constant is independent of the discretization parameter
$\displaystyle h>0$ and depends only upon
$\displaystyle\|\bar{u}\|_{H^{\frac{1}{2}}(\partial\Omega)}$,
$\displaystyle\|f\|$, shape-regularity of the triangulation,
$\displaystyle\Omega$. Applying ultraweak formulation (1.4) for $\displaystyle
y_{\bar{u}-P_{0}\bar{u}}$ and duality argument we obtain

$\displaystyle\displaystyle\|y_{\bar{u}}-y_{P_{0}\bar{u}}\|\lesssim
h\|\bar{u}\|_{H^{\frac{1}{2}}(\partial\Omega)}.$ (5.36)

Hence combination of (5.31), (5.36) and triangle inequality yields

$\displaystyle\displaystyle\|y_{\bar{u}}-y_{P_{0}\bar{u},h}\|\lesssim
h\|\bar{u}\|_{H^{\frac{1}{2}}(\partial\Omega)}.$ (5.37)

Combining (5), (5), (5), (5), (5.29), (5.30), (5.31), (5.32), (5.36), (5.37)
we obtain

$\displaystyle\displaystyle\|\tilde{P}_{1}\bar{u}_{h}-\tilde{P}_{1}P_{0}\bar{u}\|\lesssim
h^{1/2}$ (5.38)

Note that

$\displaystyle\displaystyle\|\bar{u}-\bar{u}_{h}\|\leq\|\bar{u}-P_{0}\bar{u}\|+\|P_{0}\bar{u}-\tilde{P}_{1}P_{0}\bar{u}\|+\|\tilde{P}_{1}\bar{u}_{h}-\tilde{P}_{1}P_{0}\bar{u}\|+\|\bar{u}_{h}-\tilde{P}_{1}\bar{u}_{h}\|.$
(5.39)

Note that

$\displaystyle\displaystyle\|\bar{u}_{h}-\tilde{P}_{1}\bar{u}_{h}\|=\|P_{0}(\tilde{P}_{1}\bar{u}_{h})-\tilde{P}_{1}\bar{u}_{h}\|\lesssim
h^{1/2}\|\tilde{P}_{1}\bar{u}_{h}\|.$ (5.40)

We conclude our proof with the help of (5.39), (5.38), (5.40) (4) and (5.23).
∎

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