\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
Tenth MSU Conference on Differential Equations and Computational
Simulations.\newline
\emph{Electronic Journal of Differential Equations},
Conference 23 (2016),  pp. 9--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document} \setcounter{page}{9}
\title[\hfilneg EJDE-2016/Conf/23 \hfil 
Second-order fully discretized projection method]
{Second-order fully discretized projection method for
incompressible Navier-Stokes equations}

\author[D. X. Guo \hfil EJDE-2015/conf/23 \hfilneg]
{Daniel X. Guo}  

\address{Daniel X. Guo \newline
 Department of Mathematics and Statistics \\
 University of North Carolina at Wilmington, NC 28403, USA}
\email{guod@uncw.edu}


\thanks{Published March 21, 2016.}
\subjclass[2000]{76D05, 74H15, 65C20, 65M12}
\keywords{Fully discretized; projection method; Navier-Stokes equations;
\hfill\break\indent stability; convergence}

\begin{abstract}
 A second-order fully discretized projection method for the
 incompressible Navier-Stokes equations is proposed.
 It is an explicit method for updating the pressure field.
 No extra conditions of immediate velocity fields are needed.
 The stability and convergence are investigated.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

The projection method (or fractional step method) for solving the incompressible
 Navier-Stokes equations was originally introduced and studied independently
by Chorin \cite{ch1, ch2} and Temam \cite{rt1, rt2}, it has had multiple
applications, see e.g., Guermond, Shen, and Yang \cite{gsy}, Guermond, Minev,
and Shen \cite{gms}, Kim and Moin \cite{km}, Temam \cite{rt3}, Yanenko \cite{ny},
and the references therein,
for the theoretical and numerical aspects. Despite many advantages and
extensive uses in the past by numerous researchers, the projection method
has a few major drawbacks for numerical computations. In general, the
original method is only first-order accurate in time.
It also needs the supplementary boundary conditions for the intermediate level
velocity
field and the pressure, which are not supplied in the original equations.

A fully discretized projection method was studied in Guo \cite{guo1, guo2}
on the staggered grid. The idea was originated in the block LU decomposition,
see Perot \cite{pt}. This leaded to a
whole class of methods (first-order, second-order and even higher order methods).
It depends on how the Navier-Stokes equations are discretized,
higher order methods can be possible.

In this article, we investigate the stability and convergence of a second-order
fully discretized projection method proposed in  Guo \cite{guo1, guo2}.
This article is organized as follows. In Section 2, we introduce the Navier-Stokes equations and the
boundary conditions. The space discretization is listed in Section 3. The full discretization of the Navier-Stokes equations is shown in Section 4. In
Section 5, we present fully discretized projection methods. The stability and convergence of second-order fully discretized projection method is studied in Section 6. The conclusion is in the last section.

\section{Incompressible Navier-Stokes equations}

We will consider the non-dimensionalized unsteady incompressible
Navier-Stokes equations in space dimension two and three on a given regular
domain $\Omega$, namely
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial  t}+(u\cdot \nabla )u-
\frac{1}{\mathrm{Re}}\Delta u+\nabla p=f, \\
\nabla \cdot u=0, \\
u|_{t=0}=u_{0},
\end{gathered} \label{e2.1}
\end{equation}
with the boundary conditions on $\partial \Omega$:
\[
\eta u+(1-\eta )\frac{\partial u}{\partial \overrightarrow{n} }=0.
\]
Here $\mathrm{Re}$ is the Reynolds number; the parameter $\eta $
has the limit values of $0$ for the
free-slip (no stress) condition (Neumann) and $1$ for the no-slip
condition (Dirichlet). In general, we will not specify $\eta$, but keep in
mind that $0 \le \eta \le 1$.

For the stability and convergence, what we need are the boundary
conditions that guarantee the existence of the solution for the original
equations. Actually, the boundary conditions are also important
in discretizing the equations near the boundary. We may need to use
different methods for the points near boundary. In this article, we
suppose that all discretizations have the same order on all of grid points.

\section{Space discretization}

Two families of finite-dimensional Hilbert spaces $X_h$ and $V_h$
are given, which
depend on a parameter $h\in \mathbb{R}_{+}^d$ ($d=2,3$). For finite
differences, $h$ is the mesh, i.e. $h=\{h_1,h_2\}=\{\Delta x,\Delta y\}$
in space dimension two and $h=\{h_1,h_2,h_3\}=\{\Delta x,\Delta y,\Delta z\}$
in space dimension three.

Two scalar products $((\cdot ,\cdot ))_h$ and $(\cdot ,\cdot )_h$ with
corresponding norms $\|\cdot \|_h$ and $|\cdot |_h$ are
defined on each $V_h$. Since $V_h$ is a finite-dimensional space the two
norms $\|\cdot \|_h$ and $|\cdot |_h$ are equivalent. We assume
that they are related as follows
\begin{gather}
|u_h|_h\leq c_1\|u_h\|_h, \label{e3.1} \\
\|u_h\|_h\leq S(h)|u_h|_h,\quad \forall u_h\in V_h.\label{e3.2}
\end{gather}
where $c_1$ is independent of $h$ and $S(h)$ depends on $h$.
We assume that
$S(h)\to \infty$  as $h \to 0$.

When convergence be studied, we will be interested in the passage to the
limit $h \to 0$. The spaces $V_h$ with scalar product
$((\cdot,\cdot))_h$ will approximate in some sense the space $V$, while
the spaces $V_h$ with scalar product $(\cdot,\cdot)_h$ will approximate the
space $H$. The spaces $V$ and $H$ are defined as follows
\begin{gather*}
V = \text{ the closure of $\mathcal{V}$  in $H_0^1(\Omega)$}, \quad
 H = \text{ the closure of $\mathcal{V}$ in $L^2(\Omega)$},\\
\mathcal{V} = \{ u \in \mathcal{D} (\Omega), \operatorname{div} u = 0\}.
\end{gather*}


A trilinear operator $b_h$ is defined on $V_h\times V_h\times V_h$
as follows:
\[
b_h(u_h,v_h,w_h)=((u_h\cdot \nabla )v_h,w_h), \quad \forall
u_h,v_h,w_h\in V_h,
\]
and we have the properties:
\begin{equation}
|b_h(u_h,v_h,w_h)|\leq c_2|u_h|_h^{1/2}\|u_h\|_h^{1/2}
\|v_h\|_h|w_h|_h^{1/2}\|w_h\|_h^{1/2}, \quad
\forall u_h,v_h,w_h\in V_h, \label{e3.3}
\end{equation}
in space dimension two, and
\begin{equation}
|b_h(u_h,v_h,w_h)|\leq c_2|u_h|_h^{\frac{1}{4}}\|u_h\|_h^{3/4}
\|v_h\|_h|w_h|_h^{\frac{1}{4}}\|w_h\|_h^{3/4}, \quad
 \forall u_h,v_h,w_h\in V_h, \label{e3.4}
\end{equation}
in space dimension three. The constant $c_2$ in \eqref{e3.3}
and \eqref{e3.4} is independent of $h$.

We also assume the skewness property:
\begin{equation}
b_h(u_h,v_h,v_h)=0,\quad \forall u_h,v_h\in V_h,\label{e3.5}
\end{equation}
which implies
\[
b_h(u_h,v_h,w_h)=-b_h(u_h,w_h,v_h), \quad \forall
u_h,v_h,w_h\in V_h.
\]

We also define two operators $A_h$ and $B_h$ that are the discrete
analogs of $A$ and $B$ as follows:
\begin{gather*}
(A_hu_h,v_h)_h=((u_h,v_h))_h, \quad \forall u_h,v_h\in V_h, \\
(B_h(u_h,v_h),w_h)_h=b_h(u_h,v_h,w_h), \quad \forall
u_h,v_h,w_h\in V_h.
\end{gather*}

\section{Full discretization of Navier-Stokes equations}

Let $T>0$ be fixed, and let the time step be denoted by $\Delta t=T/N$
where $N$ is an integer. We construct recursively elements
$u_h^{n}\in V_h$, $n=0,1,2,\dots ,N$; $u_h^{0}$ is an approximation of $u_{0}$;
 we assume that $u_h^{1}$ is
constructed by a different scheme because our scheme has two levels in time.

Hence starting from $u_h^2$, we recursively define $u_h^{n}$, $
n=2,\dots ,N$, by applying the explicit
Adams-Bashforth scheme of order two
to the convective term and the Crank-Nicholson scheme of order two to the
viscous terms,
the pressure term and the right-hand side of equation \eqref{e2.1},
\begin{gather}
\begin{aligned}
&\frac{1}{\Delta} t (u_h^{n+1}-u_h^{n})+ \frac{3}{2} B_h(u_h^{n})-\frac{1}{2} B_h(u_h^{n-1})\\
&+\frac{1}{\mathrm{Re}}A_h(u_h^{n+1}+u_h^{n})
 +\frac{1}{2} G_h(p_h^{n+1}+p_h^{n})
=\frac{1}{2} (f_h^{n+1}+f_h^{n}),
\end{aligned} \label{e4.1} \\
D_hu_h^{n+1}=0.\label{e4.2}
\end{gather}
where $f_h^{n}$ is an approximation of $f^{n}$ in $V_h$; $G_h$ is
the discrete gradient operator (grad) mapping $X_h$ into $V_h$,
and $D_h$ is the discrete divergence operator (div).
In fact, they are related as follows,
\begin{equation}
(u_h, G_h p_h) = -(D_hu_h,p_h), \quad  \text{for }
u_h \in V_h, \; p_h \in X_h. \label{e4.3}
\end{equation}
One can easily show that \eqref{e4.1} and \eqref{e4.2} are a second-order
scheme in time for the Navier-Stokes equations \eqref{e2.1}. We will assume
that \eqref{e4.1}
and \eqref{e4.2} are second-order in space.
It is possible to get higher order discretizations, but other procedure
would be needed. Therefore, \eqref{e4.1} and \eqref{e4.2}
have the overall second-order accuracy. They are a fully implicit coupled system.
Generally speaking, it is hard to solve this system.

Notice that if we consider the boundary conditions in equation \eqref{e4.1} and
\eqref{e4.2}, we may need to add one or more terms to the right-hand side
of equations \eqref{e4.1} and \eqref{e4.2}. For the sake of simplicity,
we suppose that these terms are
combined in the terms of the right-hand side.
So, the idea is that during fully discretizing procedure in time and spaces, we
need all boundary conditions. And the final discretized equations are a complete
linear system.

For the rest of this article, we drop the subscript $h$.
Define $g^{n+1}$ as follows,
\begin{equation}
g^{n+1}=\frac{1}{2}(f^{n+1}+f^{n})-\frac{3}{2}B(u^{n})+\frac{1}{2}B(u^{n-1})
 \label{e4.4}
\end{equation}
then we rewrite \eqref{e4.1} and \eqref{e4.2} as
\begin{equation}
\frac{1}{\Delta t}(u^{n+1}-u^{n})+\frac{1}{2Re}A(u^{n+1}+u^{n})
+\frac{1}{2}G(p^{n+1}+p^{n})=g^{n+1}, \label{e4.5}
\end{equation}
and
\begin{equation}
D u^{n+1}=0.\label{e4.6}
\end{equation}

\section{Fully discretized projection method}

We then introduce the following method for the equations \eqref{e4.5} and \eqref{e4.6},
\begin{equation}
\begin{gathered}
\frac{1}{\Delta} t (u^{n+\frac{1}{2}}-u^{n})+\frac{1}{\mathrm{Re}}A(u^{n+
\frac{1}{2}}+u^{n})=g^{n+1}, \\
\frac{1}{\Delta} t (u^{n+1}-u^{n+\frac{1}{2}})
+\frac{1}{2} (I-\frac{\Delta t}{\mathrm{Re}}A)G(p^{n+1}+p^n)=0,
\\
Du^{n+1}=0.
\end{gathered} \label{e5.1}
\end{equation}

Observe that in \eqref{e5.1}, $A$ is a known operator, say a
square matrix; $G$ and $D$ also are two known operators, but they may be
two non-square matrices. However, they satisfy the equation below,
$$
(Du^n,p^n) = -(u^n, Gp^n), \quad \text{ for } u^n \in V_h, \, p^n \in X_h.
$$

Let us
check the difference between \eqref{e4.5} and \eqref{e5.1}.
First, we add the two equations
in \eqref{e5.1} and we obtain:
$$
\frac{1}{\Delta} t (u^{n+1}-u^{n})+\frac{1}{\mathrm{Re}}A(u^{n+1}+u^{n})
 =g^{n+1}-\frac{1}{2} G(p^{n+1}+p^{n})+R^{n+1};
$$
i.e.,
\begin{equation}
\begin{aligned}
&\frac{1}{\Delta} t (u^{n+1}-u^{n})+\frac{3}{2}B(u^{n})-\frac{1}{2} B(u^{n-1}) \\
&+\frac{1}{\mathrm{Re}}A(u^{n+1}+u^{n}) +\frac{1}{2} G(p^{n+1}+p^{n})
=\frac{1}{2} (f^{n+1}+f^{n})+R^{n+1},
\end{aligned} \label{e5.2}
\end{equation}
where
\[
R^{n+1}=\frac{1}{2}(\frac{\Delta t}{\mathrm{Re}})^2AAG(p^{n+1}+p^{n}).
\]
Since $R^{n+1}$ is of order $(\Delta t)^2$, this method is a
second-order method to the Navier-Stokes equations \eqref{e2.1}.
However,  to update $p^{n+1}$, we need to solve a linear system
with the same size as $A$ at each time step. In this paper,
we study the stability and convergence of the equations \eqref{e5.1}.

\section{Stability analysis and convergence}

Let
 \[
 \Phi^{n+1} = \frac{1}{2} (p^{n+1}+p^{n}).
\]
Then
\[
 R^{n+1}=  (\frac{\Delta t}{\mathrm{Re}})^2 AA G \Phi^{n+1}.
\]
Note that from the definition of $G$ and $D$ and the third equation of \eqref{e5.1},
it holds,
\[
(G\Phi^{n+1}, u^{n+1}) = -(\Phi^{n+1}, Du^{n+1}) =0.
\]

\subsection{Stability analysis}

It is assumed that
\begin{equation}
\begin{gathered}
|u^0| \leq K_1, \quad \Delta t \sum_{n=0}^{n=N}|f^n|^2\leq K_2,\\
|u^1|^2+|u^{1/2}|^2+\frac{\Delta t}{\text{$Re$}}
\{\|u^1\|^2+\|u^{1/2}\|^2\} \leq K_3,
\end{gathered} \label{e6.1}
\end{equation}
where $K_1$, $K_2$ and $K_3$ are independent of $h$ and $\Delta t$.
Let
$$
{\delta}^{n+1}=\frac{1}{2}(u^{n+1}-u^{n}),\quad
{\delta}^{n+\frac{1}{2}}=\frac{1}{2}(u^{n+\frac{1}{2}}-u^{n});
$$
then
\[
\frac{1}{2}(u^{n+1}+u^{n})=u^{n+1}-\delta ^{n+1},\quad
\frac{1}{2}(u^{n+\frac{1}{2}}+u^{n})=u^{n+\frac{1}{2}}-\delta ^{n+\frac{1}{2}}.
\]
Now rewriting the first equation in \eqref{e5.1} with $g^{n+1}$ defined
in \eqref{e4.4}, it reads
\begin{equation}
\begin{aligned}
&\frac{1}{\Delta t}(u^{n+\frac{1}{2}}-u^{n})
+\frac{1}{\text{$2 Re$}}A(u^{n+\frac{1}{2}}+u^n)  \\
&=\frac{1}{2} (f^{n+1}+f^{n})-\frac{3}{2}B(u^{n})+\frac{1}{2} B(u^{n-1}).
\end{aligned}  \label{e6.2}
\end{equation}
Solve the second equation in \eqref{e5.1} for $u^{n+\frac{1}{2}}$ as follows
$$
u^{n+\frac{1}{2}} = u^{n+1}+
\Delta t (I-\frac{\Delta t}{\mathrm{Re}}A)G \Phi^{n+1}.
$$
Substituting $u^{n+\frac{1}{2}}$  in \eqref{e6.2}, it results,
\begin{equation}
\begin{aligned}
&\frac{1}{\Delta t}(u^{n+1}-u^{n})+\frac{1}{\mathrm{Re}}A(u^{n+1}
-\delta^{n+1})  \\
&=\frac{1}{2} (f^{n+1}+f^{n})-\frac{3}{2}B(u^{n})+\frac{1}{2}
B(u^{n-1})-G\Phi^{n+1} +R^{n+1}.
\end{aligned} \label{e6.3}
\end{equation}
Multiply equation \eqref{e6.3} with $2\Delta t u^{n+1}$, we obtain
\begin{align*}
&2(u^{n+1}-u^{n},u^{n+1})+\frac{2 \Delta t}{\mathrm{Re}}((u^{n+1}-\delta
^{n+1},u^{n+1})) \\
&=\Delta t (f^{n+1}+f^{n},u^{n+1})-2\Delta t b(u^{n},u^{n},u^{n+1})
  \\
&\quad  -\Delta t [b(u^{n},u^{n},u^{n+1})-b(u^{n-1},u^{n-1},u^{n+1})]
+2\Delta t (R^{n+1},u^{n+1}),
\end{align*}
or
\begin{equation}
\begin{aligned}
&|u^{n+1}|^2-|u^{n}|^2+4|\delta ^{n+1}|^2+\frac{2 \Delta t}{\mathrm{Re}}
\|u^{n+1}\|^2\\
&=\frac{2 \Delta t}{\mathrm{Re}}((\delta ^{n+1},u^{n+1}))
+\Delta t (f^{n+1}+f^{n},u^{n+1})+4\Delta t b(u^{n},\delta ^{n+1},u^{n+1})  \\
&\quad  -2\Delta t [b(\delta^{n},u^{n},u^{n+1})+b(u^{n-1},
\delta ^{n},u^{n+1})]+2\Delta t (R^{n+1},u^{n+1}).
\end{aligned}\label{e6.4}
\end{equation}

We now estimate each term in the right-hand side of \eqref{e6.4} in space dimension
two. The first one is
majorized, thanks to the Schwarz and Young inequalities, and \eqref{e3.2}
 (note, $S=S(h)$):
\begin{align*}
\frac{2 \Delta t}{\mathrm{Re}}((\delta ^{n+1},u^{n+1}))
& \leq \frac{2 \Delta t}{\mathrm{Re}}\|\delta ^{n+1}\|\|u^{n+1}\|\\
& \leq \frac{2 \Delta t}{\mathrm{Re}}S|\delta ^{n+1}\||u^{n+1}\| \\
& \leq \frac{1}{8}|\delta ^{n+1}|^2+\frac{8
\Delta t^2}{\mathrm{Re}^2}S^2\|u^{n+1}\|^2.
\end{align*}
For the second term we use the Schwarz and Young inequalities again
and thanks to \eqref{e3.1}:
\begin{align*}
\Delta t(f^{n+1}+f^{n},u^{n+1})
& \leq \Delta t (|f^{n+1}|+|f^{n}|)|u^{n+1}| \\
& \leq c_1\Delta t(|f^{n+1}|+|f^{n}|)\|u^{n+1}\| \\
& \leq \frac{\Delta t}{\text{$4Re$}}\|u^{n+1}\|^2+c_1^2\mathrm{Re}\Delta
t(|f^{n+1}|^2+|f^{n}|^2).
\end{align*}
For the third term we use \eqref{e3.1}, \eqref{e3.2} and \eqref{e3.3}
and majorize it by
\begin{equation}
\begin{aligned}
4\Delta t b(u^{n},\delta ^{n+1},u^{n+1})
& \leq 4c_2\Delta t|u^{n}|^{1/2}\|u^{n}\|^{1/2}|\delta ^{n+1}|^{1/2}
\|\delta ^{n+1}\|^{1/2}\|u^{n+1}\|  \\
& \leq 4c_2S\Delta t|u^{n}\|\delta ^{n+1}\||u^{n+1}\| \\
& \leq \frac{1}{8}|\delta ^{n+1}|^2+32c_2^2S^2\Delta
t^2|u^{n}|^2\|u^{n+1}\|^2.
\end{aligned} \label{e6.5}
\end{equation}
Thanks to \eqref{e3.3} the fourth term is approximated by
\begin{equation}
\begin{aligned}
&2\Delta t[b(\delta ^{n},u^{n},u^{n+1})+b(u^{n-1},\delta ^{n},u^{n+1})] \\
&\leq 2 c_2\Delta t|\delta ^{n}|^{1/2}\|\delta ^{n}\|^{\frac{1}{2}
}(|u^{n}|^{1/2}\|u^{n}\|^{1/2}+|u^{n-1}|^{\frac{1}{2}
}\|u^{n-1}\|^{1/2})\|u^{n+1}\| \\
&\leq 2c_2S\Delta t|\delta ^{n}|(|u^{n}|+|u^{n-1}|)\|u^{n+1}\| \\
&\leq \frac{1}{4}|\delta ^{n}|^2+4c_2^2S^2\Delta
t^2(|u^{n}|^2+|u^{n-1}|^2)\|u^{n+1}\|^2.
\end{aligned} \label{e6.6}
\end{equation}
Finally, by the definition of $R^{n+1}$ we have
\[
 2\Delta t(R^{n+1},u^{n+1}) = 2\Delta t (\frac{\Delta t}{\mathrm{Re}})^2
(AAG\Phi^{n+1},u^{n+1}).
\]
Rewriting the second equation of \eqref{e5.1}, it results
\begin{equation}
\begin{aligned}
G\Phi^{n+1}
&  = -\frac{1}{\Delta} t (I - \frac{\Delta t}{\mathrm{Re}}A)^{-1} (u^{n+1}-u^{n+\frac{1}{2}})  \\
& = -\frac{1}{\Delta} t [I + \frac{\Delta t}{\mathrm{Re}}A + (\frac{\Delta t}{\mathrm{Re}})^2AA
+ \dots] (u^{n+1}-u^{n+\frac{1}{2}}),
\end{aligned} \label{e6.7}
\end{equation}
if $\frac{\Delta t}{\mathrm{Re}}$ is small enough.

Then, substituting $G\Phi^{n+1}$ in $R^{n+1}$, we obtain
\begin{align*}
&2\Delta t(R^{n+1},u^{n+1}) \\
&= -2 (\frac{\Delta t}{\mathrm{Re}})^2(AA[I + \frac{\Delta t}{\mathrm{Re}}A
 + (\frac{\Delta t}{\mathrm{Re}})^2AA + \dots] u^{n+1}, u^{n+1}) \\
& \quad +2 (\frac{\Delta t}{\mathrm{Re}})^2(AA[I + \frac{\Delta t}{\mathrm{Re}}A
 + (\frac{\Delta t}{\mathrm{Re}})^2AA + \dots] u^{n+\frac{1}{2}}, u^{n+1}) \\
& = -2 (\frac{\Delta t}{\mathrm{Re}})^2[(AAu^{n+1}, u^{n+1})
 + \frac{\Delta t}{\mathrm{Re}}(AAAu^{n+1}, u^{n+1})+\dots] \\
& \quad + 2 (\frac{\Delta t}{\mathrm{Re}})^2[(AAu^{n+\frac{1}{2}}, u^{n+1})
 + \frac{\Delta t}{\mathrm{Re}}(AAAu^{n+\frac{1}{2}}, u^{n+1})+\dots] \\
& \le 2 (\frac{\Delta t}{\mathrm{Re}})^2(S^2\|u^{n+1}\|^2
 + \frac{\Delta t}{\mathrm{Re}}S^4\|u^{n+1}\|^2+\dots] \\
& \quad +2 (\frac{\Delta t}{\mathrm{Re}})^2[S^2(\frac{1}{4}\|u^{n+1}\|^2
 + \|u^{n+\frac{1}{2}}\|^2)
   +\frac{\Delta t}{\mathrm{Re}}S^4(\frac{1}{4}\|u^{n+1}\|^2
 +\|u^{n+\frac{1}{2}}\|^2)+\dots] \\
& \le 2 (\frac{\Delta t}{\mathrm{Re}})^2\frac{\text{5}}{4}(S^2
 +\frac{\Delta t}{\mathrm{Re}}S^4+\dots)\|u^{n+1}\|^2
  +2(\frac{\Delta t}{\mathrm{Re}})^2(S^2+\frac{\Delta t}{\mathrm{Re}}S^4
 +\dots)\|u^{n+\frac{1}{2}}\|^2\\
& \le 2(\frac{\Delta t}{\mathrm{Re}})^2\frac{S^2}{1-\frac{\Delta t}{\mathrm{Re}}S^2}
 (\frac{\text{5}}{4}\|u^{n+1}\|^2+\|u^{n+\frac{1}{2}}\|^2) \\
& \le \frac{\Delta t^2}{\mathrm{Re}^2}S^2 (\frac{\text{5}}{4}\|u^{n+1}\|^2
 +\|u^{n+\frac{1}{2}}\|^2),
\end{align*}
where we assume that $\frac{\Delta t}{\mathrm{Re}}S^2 < 1/2$.

Gathering all these estimates, we deduce from \eqref{e6.4} that
\begin{equation}
\begin{aligned}
&|u^{n+1}|^2-|u^{n}|^2+\frac{15}{4}|\delta ^{n+1}|^2-
\frac{1}{4}|\delta^{n}|^2 +\frac{7}{4}
\frac{\Delta t}{\mathrm{Re}}\|u^{n+1}\|^2 \\
&\leq c_1^2\mathrm{Re}\Delta t(|f^{n+1}|^2+|f^{n}|^2)
 +\frac{\Delta t^2}{\mathrm{Re}^2}S^2\|u^{n+\frac{1}{2}}\|^2 \\
&\quad  +\frac{\text{37}}{4}\frac{\Delta t^2}{\mathrm{Re}^2}S^2\|u^{n+1}\|^2
 +c_3S^2\Delta t^2(|u^{n}|^2+|u^{n-1}|^2)\|u^{n+1}\|^2,
\end{aligned} \label{e6.8}
\end{equation}
where $c_3 = 32 c_2^2$.

We multiply equation \eqref{e6.2} by $2\Delta t u^{n+\frac{1}{2}}$ to obtain
\begin{equation}
\begin{aligned}
&2(u^{n+\frac{1}{2}}-u^{n},u^{n+\frac{1}{2}})+\frac{2\Delta t}{\mathrm{Re}}
((u^{n+\frac{1}{2}}-\delta ^{n+\frac{1}{2}},u^{n+\frac{1}{2}})) \\
&=\Delta t (f^{n+1}+f^{n},u^{n+\frac{1}{2}})
      -2\Delta t b(u^{n},u^{n},u^{n+\frac{1}{2}}) \\
&\quad  -\Delta t [b(u^{n},u^{n},u^{n+\frac{
1}{2}})-b(u^{n-1},u^{n-1},u^{n+\frac{1}{2}})].
\end{aligned} \label{e6.9}
\end{equation}
Thanks to \eqref{e3.5}, the above equation can be rewritten as
\begin{equation}
\begin{aligned}
&|u^{n+\frac{1}{2}}|^2-|u^{n}|^2+4|\delta ^{n+\frac{1}{2}}|^2
+\frac{2 \Delta t}{\mathrm{Re}}\|u^{n+\frac{1}{2}}\|^2\\
&=\frac{2 \Delta t}{\mathrm{Re}}
((\delta ^{n+\frac{1}{2}},u^{n+\frac{1}{2}}))
+\Delta t(f^{n+1}+f^{n},u^{n+
\frac{1}{2}}) +4\Delta t b(u^{n},\delta ^{n+\frac{1}{2}},u^{n+\frac{1}{2}})
\\
&\quad  -2\Delta t [b(\delta ^{n},u^{n},u^{n+\frac{1}{2}})+
b(u^{n-1},\delta ^{n},u^{n+\frac{1}{2}})].
\end{aligned} \label{e6.10}
\end{equation}

Note that \eqref{e6.10} is similar to \eqref{e6.4} with $u^{n+1}$ and
$\delta^{n+1}$ replaced by $u^{n+1/2}$ and $\delta^{n+1/2}$ respectively.
 So, we can repeat the estimation procedures as for \eqref{e6.4}.
In space dimension two we have the following estimates:
\begin{gather}
\frac{2 \Delta t}{\mathrm{Re}}((\delta ^{n+\frac{1}{2}},u^{n+\frac{1}{2}}))\leq
\frac{1}{8}|\delta ^{n+\frac{1}{2}}|^2+\frac{8\Delta t^2}{\mathrm{Re}^2}
S^2\|u^{n+\frac{1}{2}}\|^2.
\nonumber\\
\Delta t (f^{n+1}+f^{n},u^{n+\frac{1}{2}})\leq \frac{\Delta t}{4\mathrm{Re}}
\|u^{n+\frac{1}{2}}\|^2+c_1^2\mathrm{Re}\Delta t (|f^{n+1}|^2+|f^{n}|^2)
\nonumber \\
4\Delta t b(u^{n},\delta ^{n+\frac{1}{2}},u^{n+\frac{1}{2}})\leq \frac{1}{8}
|\delta ^{n+\frac{1}{2}}|^2+32c_2^2S^2\Delta t^2|u^{n}|^2\|u^{n+\frac{1}{2}
}\|^2 \label{e6.11}
\\
\begin{aligned}
&2\Delta t [b(\delta ^{n},u^{n},u^{n+\frac{1}{2}})+b(u^{n-1},\delta ^{n},u^{n+
\frac{1}{2}})]  \\
& \leq \frac{1}{4}|\delta ^{n}|^2+4c_2^2S^2\Delta
t^2(|u^{n}|^2+|u^{n-1}|^2)\|u^{n+\frac{1}{2}}\|^2.
\end{aligned} \label{e6.12}
\end{gather}
We gather these inequalities with \eqref{e6.10}, and obtain
\begin{equation}
\begin{aligned}
&|u^{n+\frac{1}{2}}|^2-|u^{n}|^2+\frac{15}{4}|\delta ^{n+\frac{1}{2}}|^2-
\frac{1}{4}|\delta^{n}|^2
 +\frac{7}{4} \frac{\Delta t}{\mathrm{Re}}\|u^{n+\frac{1}{2}}\|^2 \\
& \leq c_1^2\mathrm{Re}\Delta t(|f^{n+1}|^2+|f^{n}|^2)
  +\frac{\text{8}\Delta t^2}{\mathrm{Re}^2}S^2\|u^{n+\frac{1}{2}}\|^2\\
&\quad +c_3S^2\Delta t^2(|u^{n}|^2+|u^{n-1}|^2)\|u^{n+\frac{1}{2}}\|^2.
\end{aligned} \label{e6.13}
\end{equation}
Adding \eqref{e6.13} to \eqref{e6.8}, we obtain
\begin{equation}
\begin{aligned}
&|u^{n+1}|^2+|u^{n+\frac{1}{2}}|^2-2|u^{n}|^2+
\frac{15}{4}|\delta^{n+1}|^2+\frac{15}{4}
|\delta ^{n+\frac{1}{2}}|^2-\frac{1}{2} |\delta^{n}|^2 \\
&+\frac{7}{4} \frac{\Delta t}{\mathrm{Re}}[1-\frac{37\Delta t}{7\mathrm{Re} }S^2
-c_3\mathrm{Re} \Delta t S^2(|u^{n}|^2+|u^{n-1}|^2)](\|u^{n+1}\|^2
+\|u^{n+\frac{1}{2}}\|^2) \\
& \leq c_1^2\mathrm{Re}\Delta t(|f^{n+1}|^2+|f^{n}|^2),
\end{aligned} \label{e6.14}
\end{equation}
where $c_3$ is the same as in \eqref{e6.8}.

We now prove the following result.

\begin{theorem} \label{thm1}
In dimension two we assume that \eqref{e6.1} holds. Then there exists $K_4$
independent of $h$ and $\Delta t$ such that if
\begin{equation}
{\Delta t} < 16 c_1^2\mathrm{Re},\quad
{\Delta t}S(h)^2\leq \max\{\frac{\mathrm{Re}}{148},\frac{c}{\mathrm{Re}K_4}\}
  \label{e6.15}
\end{equation}
where $c_1$ is defined in \eqref{e3.1} and
 $c = \frac{1}{56 c_3}$, $c_3$
the constant is defined in \eqref{e6.8}. Then
\begin{subequations}
\begin{gather}
  |u^{n}|^2\leq K_4,\quad n=0,\dots ,N  \label{e6.16a}\\
 |u^{n+\frac{1}{2}}|^2\leq 14K_4,\quad n=0,\dots ,N \label{e6.16b}\\
 \sum_{n=1}^{N}|u^{n}-u^{n-1}|^2\leq 3K_4,  \label{e6.16c}\\
 \sum_{n=1}^{N}|u^{n}-u^{n-\frac{1}{2}}|^2\leq 6K_4,\label{e6.16d}\\
 \sum_{n=1}^{N}|u^{n-\frac{1}{2}}-u^{n-1}|^2\leq 3K_4,\label{e6.16e}\\
 \frac{\Delta t}{\mathrm{Re}}\sum_{n=1}^{N}\|u^{n}\|^2\leq 17K_4, \label{e6.16f}\\
\frac{\Delta t}{\mathrm{Re}}\sum_{n=1}^{N}\|u^{n-\frac{1}{2}}\|^2 \leq 17K_4,
 \label{e6.16g}
\end{gather}
\end{subequations}
\end{theorem}

\begin{proof}
Taking the inner product of the equation \eqref{e6.5} with
$u^{n+1}$ and in light of \eqref{e3.3}, we obtain
$$
([I + \frac{\Delta t}{\mathrm{Re}}A + (\frac{\Delta t}{\mathrm{Re}})^2AA
+ \dots](u^{n+1}-u^{n+\frac{1}{2}}),u^{n+1}) = 0;
$$
i.e.,
$$
(u^{n+1}-u^{n+\frac{1}{2}},u^{n+1}) = -\frac{\Delta t}{\mathrm{Re}}([A + (\frac{\Delta t}{\mathrm{Re}})AA + \dots](u^{n+1}-u^{n+\frac{1}{2}}),u^{n+1}).
$$
Thanks to \eqref{e3.1}, we have
$$
(u^{n+1}-u^{n+\frac{1}{2}},u^{n+1}) \le \frac{\text{5}}{4}
(\frac{\Delta t}{\mathrm{Re}})\frac{1}{1
-\frac{\Delta t}{\mathrm{Re}}S^2} (\|u^{n+1}\|^2+\|u^{n+\frac{1}{2}}\|^2).
$$
Since  $ \Delta t S^2 \le \mathrm{Re}/148$ from \eqref{e6.15}, then
 $\frac{\Delta t}{\mathrm{Re}} S^2 \le 1/6$ and it results
$$
\frac{1}{2}(|u^{n+1}|^2 - |u^{n+\frac{1}{2}}|^2+|u^{n+1} - u^{n+\frac{1}{2}}|^2)
\le \frac{\text{3}}{\text{4}} \frac{\Delta t}{\mathrm{Re}}
(\|u^{n+1}\|^2+\|u^{n+\frac{1}{2}}\|^2)
$$
or
\begin{equation}
|u^{n+1}|^2 - |u^{n+\frac{1}{2}}|^2+|u^{n+1} - u^{n+\frac{1}{2}}|^2
\le \frac{\text{3}}{\text{2}} \frac{\Delta t}{\mathrm{Re}}
(\|u^{n+1}\|^2+\|u^{n+\frac{1}{2}}\|^2). \label{e6.17}
\end{equation}

We first show \eqref{e6.16a} by induction. Assuming that \eqref{e6.16a} holds for
$n=0,\dots ,m,$ we infer from \eqref{e6.12}, \eqref{e6.13} that
\begin{equation}
\begin{aligned}
&|u^{n+1}|^2 + |u^{n+\frac{1}{2}}|^2-2|u^n|^2+
\frac{15}{4}|\delta^{n+1}|^2\\
&+\frac{15}{4}|\delta^{n+\frac{1}{2}}|^2-\frac{1}{2} |\delta^{n}|^2
+\frac{\text{13}\Delta t}{8\mathrm{Re}}\{\|u^{n+1}\|^2+\|u^{n+\frac{1}{2}}\|^2\}\\
&\leq c_1^2Re {\Delta t}\{|f^{n+1}|^2+|f^{n}|^2\}.
\end{aligned} \label{e6.18}
\end{equation}
Adding \eqref{e6.18} to \eqref{e6.17}, we obtain
\begin{align*}
&2|u^{n+1}|^2 -2|u^n|^2+|u^{n+1}-u^{n+\frac{1}{2}}|^2+
\frac{15}{4}|\delta^{n+1}|^2\\
&+\frac{15}{4}|\delta^{n+\frac{1}{2}}|^2-\frac{1}{2} |\delta^{n}|^2
 +\frac{\Delta t}{\text{$8Re$}}\{\|u^{n+1}\|^2+\|u^{n+\frac{1}{2}}\|^2\}\\
&\leq c_1^2Re {\Delta t}\{|f^{n+1}|^2+|f^{n}|^2\};
\end{align*}
i.e.,
\begin{equation}
\begin{aligned}
|u^{n+1}|^2 -|u^n|^2+\frac{1}{2} |u^{n+1}-u^{n+\frac{1}{2}}|^2+
\frac{15}{8}|\delta^{n+1}|^2
+\frac{15}{8}|\delta^{n+\frac{1}{2}}|^2-\frac{1}{4}|\delta^{n}|^2
  \\
\quad +\frac{\Delta t}{16}\{\|u^{n+1}\|^2+\|u^{n+\frac{1}{2}}\|^2\}
\leq c_1^2Re {\Delta t}\{|f^{n+1}|^2+|f^{n}|^2\}.
\end{aligned} \label{e6.19}
\end{equation}
Thanks to \eqref{e3.1}, we obtain
\begin{equation}
\begin{aligned}
&|u^{n+1}|^2 -|u^n|^2+\frac{15}{8}|\delta^{n+1}|^2
-\frac{1}{4}|\delta^{n}|^2
+\frac{\Delta t}{16c_1^2\mathrm{Re}}|u^{n+1}|^2  \\
& \leq c_1^2Re {\Delta t}\{|f^{n+1}|^2+|f^{n}|^2\}.
\end{aligned} \label{e6.20}
\end{equation}
Therefore, if we define
$$
\xi^n = |u^n|^2 + \frac{1}{4}|\delta^n|^2,  \quad
\gamma^n = c_1^2\mathrm{Re}\Delta t \{|f^{n+1}|^2 + |f^n|^2\},
$$
then, from \eqref{e6.20} the following inequality holds
$$
(1+\frac{\Delta t}{16c_1^2Re}) \xi^{n+1} \leq \xi^{n} + \gamma^n.
$$
By applying  Lemma \ref{lem1} below with $\beta= 0$ and
$\alpha=-\frac{\Delta t}{\text{$16c_1^2Re$}}$ (from \eqref{e6.15}, $\alpha < 1$),
we find, for $n=1,2,\dots,m$, that
$$
\xi^{n+1} \leq (1+\frac{\Delta t}{16c_1^2Re})^{-n}
\{\xi^1+c_1^2Re{\Delta t} \sum^n_{j=1}(|f^{n+1}|^2+|f^n|^2)\};
$$
i.e.,
\begin{align*}
|u^{n+1}|^2 + \frac{1}{16}|u^{n+1}-u^n|^2
& \leq |u^1|^2+\frac{1}{16}|u^1-u^0|^2+
c_1^2\mathrm{Re}\Delta t \sum_{j=1}^{N}|f^j|^2,  \\
& \leq \frac{9}{8}|u^1|^2+\frac{1}{8}|u^0|^2+
c_1^2\mathrm{Re}\Delta t \sum_{j=1}^{N}|f^j|^2,  \\
& \leq \frac{9}{8}K_3 + \frac{1}{8}K_1^2+c_1^2\mathrm{Re}K_2.
\end{align*}
We choose
$$
K_4 = \frac{3}{2}K_3 + K_1^2 + 2c_1^2\mathrm{Re} K_2,
$$
and conclude that \eqref{e6.16a} is proved for $n=0, 1, \dots,m+1$, hence
for all $n$. Then \eqref{e6.17} is valid for all $n$ and summing these relations
for $n=1,2,\dots,N-1$, we find
\begin{align*}
&|u^N|^2+\frac{1}{2} \sum_{j=2}^{N}|u^j-u^{j-\frac{1}{2}}|^2+
\sum_{j=2}^{N}|\delta^j|^2+\frac{15}{8}
\sum_{j=2}^{N}|\delta^{j-\frac{1}{2}}|^2  \\
&+\frac{\Delta t}{16}\sum_{j=2}^{N}\{\|u^j\|^2
+\|u^{j-\frac{1}{2}}\|^2\}  \\
& \leq |u^1|^2+\frac{1}{4}|\delta^1|^2
+ 2c_1^2\mathrm{Re}\Delta t \sum_{j=1}^{N}|f^j|^2  \\
& \leq |u^1|^2+\frac{1}{16}|u^1-u^0|^2+2c_1^2\mathrm{Re}\Delta t
\sum_{j=1}^{N}|f^j|^2
\leq K_4.
\end{align*}
Hence
\begin{gather*}
\sum_{j=2}^{N}|u^j-u^{j-\frac{1}{2}}|^2 \leq 2K_4, \quad
\sum_{j=2}^{N}|u^j-u^{j-1}|^2 \leq K_4, \\
\sum_{j=2}^{N}|u^{j-\frac{1}{2}}-u^{j-1}|^2 \leq K_4, \quad
\frac{\Delta t}{Re}\sum_{j=2}^{N}\{\|u^j\|^2+\|u^{j-\frac{1}{2}}\|^2\}
\leq 16K_4.
\end{gather*}
Thanks to \eqref{e6.1}, we know that
\begin{gather*}
|u^1-u^{1/2}|^2 \leq 2|u^1|^2+2|u^{1/2}|^2 \leq 4K_3\leq 4K_4, \\
|u^1-u^0|^2 \leq 2|u^1|^2+2|u^0|^2 \leq 2K_3+2K_1^2\leq 2K_4, \\
|u^{1/2}-u^0|^2 \leq 2|u^{1/2}|^2+2|u^0|^2 \leq 2K_3+2K_1^2 \leq 2K_4, \\
\frac{\Delta t}{Re}(\|u^1\|^2+\|u^{1/2}\|^2) \leq K_3.
\end{gather*}
Then \eqref{e6.16c}--\eqref{e6.16g} are proved.

Finally, from the inequality
$$
|u^{n+\frac{1}{2}}|^2 \le 2|u^{n+1}|^2 + 2|u^{n+1}-u^{n+\frac{1}{2}}|^2,
$$
then \eqref{e6.16b} holds following \eqref{e6.16a} and \eqref{e6.16d}.
\end{proof}

We state the Lemma used in the proof of Theorem \ref{thm1}.

\begin{lemma} \label{lem1}
Consider two sequences of numbers $\xi ^{n}$, $\gamma _{n}$ $\geq 0$ such
that
$$
(1-\alpha )\xi ^{n}\leq (1+\beta )\xi ^{n-1}+\gamma _{n}, \label{e6.21}
$$
for all $n\geq 1$ and for some $\alpha <1$ and $\beta >-1$. Then for all $n$:
\begin{equation}
\xi ^{n}\leq (\frac{1+\beta }{1-\alpha })^{n}\xi ^{0}+\frac{(1+\beta )^{n-1}
}{(1-\alpha )^{n}}\sum_{j=1}^{n}\gamma _{j}.\label{e6.22}
\end{equation}
If $\gamma _{j}\leq \gamma$, for all  $j$, we also have
\begin{equation}
\xi ^{n}\leq (\frac{1+\beta }{1-\alpha })^{n}(\xi ^{0}+\frac{\gamma }{\beta
+\alpha }). \label{e6.23}
\end{equation}
\end{lemma}

\begin{proof} For $m=0,\dots ,n-1$, we write
\[
\xi ^{n-m}\leq \frac{1+\beta }{1-\alpha }\xi ^{n-m-1}+\frac{1}{1-\alpha }
\gamma _{n-m}.
\]
We multiply this relation by $((1+\beta )/(1-\alpha ))^{m}$ and add the
corresponding inequalities for $m=0,\dots,n-1$; \eqref{e6.22} and \eqref{e6.23}
follow easily.
\end{proof}


\subsection{Convergence}

We introduce the approximate functions $u_{1k}$, $u_{2k}$, $\tilde{u}_k$
($k=\Delta t$) defined as follows:
\begin{gather*}
u_{1k}= u^{n+\frac{1}{2}} \quad \text{for } t \in [n\Delta t, (n+1)\Delta t),\;
n = 0,\dots,N-1,  \\
u_{2k}= u^{n+1} \quad\text{for } t \in [n\Delta t, (n+1)\Delta t),\;
n = 0,\dots,N-1,  \\
\parbox{9cm}{$\tilde{u}_k$ is continuous from $[0,T]$ to $H$,
 linear on each interval $((n-1)\Delta t, n\Delta t)$ and equal to $u^n$ at $n\Delta t$,
$n = 0,\dots,N$.}
\end{gather*}

Then \eqref{e6.14} yields the following result.

\begin{theorem} \label{thm2}
As $k=\Delta t \to 0$, the functions $u_{1k}$, $u_{2k}$ and $\tilde{u}_k$ remain
bounded in $L^{\infty}(0,T;L^2(\Omega)^d)$;  $u_{1k}$ and $u_{2k}$ remain
bounded in $L^{\infty}(0,T;H^1(\Omega)^d)$, and the same is true for
$\tilde{u}_k$ if $u_0 \in V$.

Furthermore $u_{1k}-u_{2k}$ and $u_{2k}-\tilde{u}_{k}$ converge to $0$ in
$L^{\infty}(0,T;L^2(\Omega)^d)$ as $k=\Delta t \to 0$, their norm
being bounded.
\end{theorem}

The above theorem follows directly  from Theorem \ref{thm1}.

Because of Theorem \ref{thm2} there exists a subsequence $k' \to 0$ such that
\begin{gather*}
\text{$u_{1k'} \to u_1$ in $L^{\infty}(0,T;H)$ weak-star and
$L^2(0,T;H^1(\Omega)^d)$ weakly},  \\
\text{$u_{2k'} \to u_2$  in $L^{\infty}(0,T;H)$ weak-star and
$L^2(0,T;H^1(\Omega)^d)$  weakly},  \\
\text{$\tilde{u}_{k'} \to u$  in $L^{\infty}(0,T;H)$ weak-star and
$L^2(0,T;H^1(\Omega)^d)$ weakly}.
\end{gather*}
Theorem \ref{thm2} also implies that $u_1=u_2=u$ and thus
$$
u \in L^{\infty}(0,T;H) \cap L^2(0,T;V).
$$

\subsection*{Conclusions}
In this article, we proved the stability and convergence of a second
order fully discretized projection method for the incompressible Navier-Stokes
equations. We did not specify the boundary conditions, but all necessary
boundary conditions should be supplied in order to have a solution.
It is possible to construct higher order methods, but the numerical analysis
is much more complicated.



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\end{document}