text:
\begin{document}

\title[]
{A Relaxation/Finite Difference discretization\\
 of a 2D Semilinear Heat Equation\\
over a rectangular domain}

\author[]{Georgios E. Zouraris$^{\ddag}$}

\thanks
{$^{\ddag}$ Department of Mathematics and Applied Mathematics and
Applied Mathematics Laboratory (AML),
University of Crete, GR-700 13 Voutes Campus, Heraklion, Crete, Greece.
(e-mail: georgios.zouraris@uoc.gr)}

\subjclass{65M12, 65M60}

\keywords {Relaxation Scheme, semilinear heat equation,
finite differences, Dirichlet boundary conditions,
optimal order error estimates}

\begin{abstract}

We consider an initial and Dirichlet boundary value
problem for a semilinear, two dimensional heat equation over a
rectangular domain. The problem is discretized in time
by a version of the Relaxation Scheme proposed by C. Besse
(C. R. Acad. Sci. Paris S{\'e}r. I, vol. 326 (1998)) for the nonlinear
Schr{\"o}dinger equation and in space by a standard second order
finite difference method.

The proposed method is unconditionally well-posed and its
convergence is established by proving an optimal second order
error estimate allowing a mild mesh condition to hold.

\end{abstract}

\maketitle

\section{Introduction}

\subsection{Formulation of the problem}

Let $T>0$, $a_1,a_2,b_1,b_2\in{\mathbb R}$ with $a_2>a_1$ and
$b_2>b_1$, ${\mathcal D}:=[a_1,a_2]\times[b_1,b_2]$,
$Q:=[0,T]\times{\mathcal D}$ and $\uu\in C_{t,x_1,x_2}^{1,2,2}(Q,{\mathbb R})$
be the solution of the following initial and boundary value problem:

\begin{gather}

\uu_t=\varDelta\uu+g(\uu)+f
\quad\text{\rm on}\ \ Q,
\label{PSL_a}\\

\uu(t,x)=0\quad\forall\,(t,x)\in[0,T]\times\partial{\mathcal D},
\label{PSL_b}\\

\uu(0,x)=\uu_0(x)\quad\forall\,x\in{\mathcal D},
\label{PSL_c}

\end{gather}

where $g\in C({\mathbb R},{\mathbb R})$ with $g(0)=0$,
$f\in C(Q,{\mathbb R})$ and $\uu_0\in C({\mathcal D},{\mathbb R})$
with

\begin{equation}\label{PSL_d}
\uu_0\left|_{\ssy\partial{\mathcal D}}\right.=0.
\end{equation}

\subsection{Formulation of the numerical method}

Let $\nset$ be the set of all positive integers. For a given
$N\in\nset$, we define a uniform partition of the time interval $[0,T]$ with
time-step $\tau:=\tfrac{T}{N}$, nodes $t_n:=n\,\tau$ for $n=0,\dots,N$,
and intermediate nodes $t^{n+\half}=t_n+\tfrac{\tau}{2}$ for
$n=0,\dots,N-1$. Also, for given $J_1,J_2\in\nset$, we consider a uniform partition
of $[a_1,a_2]$ with mesh-width $h_1:=\tfrac{a_2-a_1}{J_1+1}$ and nodes
$x_{1,i}:=a_1+i\,h_1$ for $i=0,\dots,J_1+1$, and a uniform partition
of $[b_1,b_2]$ with mesh-width $h_2:=\tfrac{b_2-b_1}{J_2+1}$ and nodes
$x_{2,j}:=b_1+j\,h_2$ for $j=0,\dots,J_2+1$.

Also, we set
${\mathbb I}:=\{(i,j):\,\,i=0,\dots,J_1+1,\,\, j=0,\dots,J_2+1\}$,
${\mathbb I}^{\circ}:=\{(i,j):\,\,i=1,\dots,J_1,\,\, j=1,\dots,J_2\}$,
$\partial{\mathbb I}:={\mathbb I}\backslash{\mathbb I}^{\circ}$,
and introduce the discrete space

\begin{equation*}
{\dspace}:=\left\{\,V=(V_{i,j})_{(i,j)\in{\mathbb I}}\in\rset^{(J_1+2)\times(J_2+2)}:\,\,
V_{i,j}=0\,\,\,\forall(i,j)\in\,\partial{\mathbb I}\right\},
\end{equation*}

and a discrete Laplacian operator $\Deltah:\dspace\rightarrow\dspace$ by

\begin{equation*}
(\Deltah V)_{i,j}:=\tfrac{V_{i-1,j}-2\,V_{i,j}+V_{i+1,j}}{h_1^2}
+\tfrac{V_{i,j-1}-2\,V_{i,j}+V_{i,j+1}}{h_2^2},
\quad\forall\,(i,j)\in{\mathbb I}^{\circ}
\quad\forall\,V\in\dspace.
\end{equation*}

In addition, we introduce an o\-pe\-rator
${\sf I}_{\ssy\HH}^{\circ}:C({\mathcal D})\rightarrow\dspace$,
which, for given $z\in C({\mathcal D})$, is defined by
$({\sf I}^{\circ}_{\ssy\HH}[z])_{i,j}:=z(x_{1,i},x_{2,j})$
for all $(i,j)\in{\mathbb I}^{\circ}$.

Finally, for any $W\in\dspace$, we define $g(W)\in\dspace$ by
$(g(W))_{i,j}:=g(W_{i,j})$ for all $(i,j)\in{\mathbb I}^{\circ}$.

\par
The Relaxation Finite Difference (RFD) method uses a standard finite
difference scheme for space discretization along with a variant of the
Relaxation Scheme for time stepping (cf. \cite{Besse1}).

\par\noindent\vskip0.2truecm\par\noindent

{\tt Step 1}: First, set

\begin{equation}\label{BRS_1}
\UUU^0:={\sf I}_{\ssy\sf H}^{\circ}\left[\uu_0\right]\in\dspace
\end{equation}

and find $\UUU^{\half}\in\dspace$ such that

\begin{equation}\label{BRS_2}
\tfrac{\UUU^{\half}-\UUU^0}{(\tau/2)}=\Deltah\UUU^{\half}+g(\uu^0)
+\PP\left[f(t^{\half},\cdot)\right].
\end{equation}

\par\noindent\vskip0.2truecm\par\noindent

{\tt Step 2}: Set

\begin{equation}\label{BRS_13}
\Phi^{\half}:=g(\UUU^{\half})\in\dspace
\end{equation}

and find $\UUU^1\in\dspace$ such that

\begin{equation}\label{BRS_12}
\begin{split}
\tfrac{\UUU^1-\UUU^0}{\tau}=\Deltah\left(\tfrac{\UUU^1+\UUU^0}{2}\right)
+\Phi^{\half}+\PP\left[\tfrac{f(t_{1},\cdot)+f(t_0,\cdot)}{2}\right].
\end{split}
\end{equation}

\par\noindent\vskip0.2truecm\par\noindent

{\tt Step 3}: For $n=1,\dots,N-1$, first set

\begin{equation}\label{BRS_3}
\Phi^{n+\half}:=2\,g(\UUU^n)-\Phi^{n-\half}\in\dspace
\end{equation}

and then find $\UUU^{n+1}\in\dspace$ such that

\begin{equation}\label{BRS_4}
\tfrac{\UUU^{n+1}-\UUU^{n}}{\tau}
=\Deltah\left(\tfrac{\UUU^{n+1}+\UUU^{n}}{2}\right)
+\Phi^{n+\half}+\PP\left[\tfrac{f(t_{n+1},\cdot)+f(t_n,\cdot)}{2}\right].
\end{equation}

\subsection{References and main results}

The Relaxation Scheme (RS) has been introduced by C. Besse \cite{Besse1}
as a linearly implicit, conservative, time stepping method for the approximation
of the solution to the nonlinear Schr{\"o}dinger equation. Convergence results
has been obtained in \cite{Besse2} and \cite{Besse3} for the (RS) time-discrete
approximations of the Cauchy problem for the nonlinear Schr{\"o}dinger equation,
which can not be concluded in the fully-discrete case, because on the one hand they
are valid for small final time $T$, and on the other hand are based on the derivation
of $H^{s+2}$ a priori bounds  (with $s>\tfrac{d}{2}$) for the time discrete
approximations.

Recently, in \cite{Georgios2}, the (RS) joined with a finite difference scheme
is proposed for the approximation of the solution to a semilinear heat equation
in the 1D case, and the corresponding error analysis arrived at optimal second order
error estimates. Here, we investigate the extension of the results obtained in
\cite{Georgios2} in the 2D case. We would like to stress that a fylly-discrete version
of the (RS) applied on a parabolic problem can be analyzed by using energy estimates,
which, however, are not efficient in the case of the nonlinear Schr{\"o}dinger equation
(see \cite{Georgios3}).

\par
To develop an error analysis for the (RFD) method, we introduce
the modified scheme (see Section~\ref{The_MBRFD}) that follows
from the (RFD) method after mollifying properly the terms with nonlinear
structure (cf. \cite{Akrivis1},  \cite{KarMak}, \cite{Georgios1}).

For the approximations obtained from the modified scheme, we provide an optimal,
second order error estimate in the discrete $H^1-$norm at the nodes and in the
discrete $L^2-$norm at the intermediate nodes (see Theorem~\ref{BR_CB_Conv}).
After applying an inverse inequality (see \eqref{SOBO}) and imposing a proper mesh condition
(see \eqref{BR_Xmesh}), the latter
convergence result implies that the discrete $L^{\infty}-$norm of the modified approximations
is uniformly bounded, and thus they are the same with those derived by the (RFD) method
and hence inherit their convergence properties o (see Theorem~\ref{DR_Final}),
i.e. that there exist constant $C>0$, independent of $\tau$, $h_1$, $h_2$, such that

\begin{equation*}
\max_{0\leq n\leq {\ssy N}}\left[\,
\big|\,\Phi^{n+\half}-{\sf I}_{\ssy\HH}^{\circ}[g(u(t^{n+\half},\cdot))]\,\big|_{0,\HH}
+\big|\,U^n-\PP[u(t_n,\cdot)]\,\big|_{1,\HH}\,\right]\leq\,C\,(\tau^2+h_1^2+h_2^2),
\end{equation*}

where $|\cdot|_{1,\HH}$ is a discrete $H^1-$norm
and $|\cdot|_{0,\HH}$ is a discrete $L^2-$norm.

\section{Preliminaries}\label{Section2}

\subsection{Discrete relations}

We provide $\dspace$ with the discrete inner product
$(\cdot,\cdot)_{0,\HH}$ given by

$$(V,Z)_{0,\HH}:=h_1\,h_2\,\sum_{(i,j)\in{\mathbb I}^{\circ}}V_{i,j}\,Z_{i,j}
\quad\forall\,V,Z\in\dspace$$

and we shall denote by $\|\cdot\|_{0,\HH}$ its induced norm, i.e.
$\|v\|_{0,\HH}:=\left[(V,V)_{0,\HH}\right]^{\ssy 1/2}$ for $V\in\dspace$.

Also, we equip $\dspace$ with a discrete $L^{\infty}$-norm $|\cdot|_{\infty,\HH}$
defined by $|W|_{\infty,\HH}:=\max_{(i,j)\in{\mathbb I}^{\circ}}|W_{i,j}|$ for
$W\in\dspace$, and with a discrete $H^1$-type norm $|\cdot|_{1,\HH}$
given by

$$|V|_{1,\HH}:=\left[h_1\,h_2\,\sum_{j=1}^{\ssy J_2}\sum_{i=0}^{\ssy J_1}
\left|\tfrac{V_{i+1,j}-V_{i,j}}{h_1}\right|^2
+h_1\,h_2\,\sum_{i=1}^{\ssy J_1}\sum_{j=0}^{\ssy J_2}
\left|\tfrac{V_{i,j+1}-V_{i,j}}{h_2}\right|^2\right]^{1/2}
\quad\forall\,V\in\dspace.$$

\par
In the convergence analysis of the method, we will make use of the following,
easy to verify, relation

\begin{equation}\label{SELF}
(\Deltah V,V)_{0,\HH}=-|V|_{1,\HH}^2\quad\forall\,V\in\dspace,
\end{equation}

of the discrete Poincar{\'e}-Friedrishs inequality

\begin{equation}\label{dpoincare}
\|V\|_{0,\HH}\leq\,\tfrac{1}{2}\,\min\{a_2-a_1,b_2-b_1\}
\,|V|_{1,\HH}\quad\forall\,V\in\dspace,
\end{equation}

of the inverse inequality

\begin{lem}\label{sobol}
For $V\in\dspace$ it holds that

\begin{equation}\label{SOBO}
|V|_{\infty,\HH}\leq \tfrac{{\sf L}}{\sqrt{h_1+h_2}}\,|V|_{1,\HH}
\end{equation}

with ${\sf L}:=\left[\tfrac{(a_2-a_1)\,(b_2-b_1)}{\min\{a_2-a_1,b_2-b_1\}}\right]^{1/2}$.
\end{lem}

\begin{proof}

Let $V\in\dspace$ and $(i_0,j_0)\in{\mathbb I}^{\circ}$ such that $|V|_{\infty,\HH}=|V_{i_0,j_0}|$.
Since $V_{0,j_0}=V_{i_0,0}=0$, we conclude (see \cite{Georgios2}) that

\begin{equation}\label{Sobo1}
\begin{split}
|V_{i_0,j_0}|^2\leq&\,(a_2-a_1)\,h_1\sum_{i=0}^{\ssy J_1}\left|
\tfrac{V_{i+1,j_0}-V_{i,j_0}}{h_1}\right|^2\\

\leq&\,\tfrac{a_2-a_1}{h_2}\,h_1\,h_2\sum_{j=1}^{\ssy J_2}
\sum_{i=0}^{\ssy J_1}\left| \tfrac{V_{i+1,j}-V_{i,j}}{h_1}\right|^2\\
\end{split}
\end{equation}

and

\begin{equation}\label{Sobo2}
\begin{split}
|V_{i_0,j_0}|^2\leq&\,(b_2-b_1)\,h_2\sum_{j=0}^{\ssy J_2}\left|
\tfrac{V_{i_0,j+1}-V_{i_0,j}}{h_2}\right|^2\\

\leq&\,\tfrac{b_2-b_1}{h_1}\,h_1\,h_2\sum_{i=1}^{\ssy J_1}
\sum_{j=0}^{\ssy J_2}\left| \tfrac{V_{i,j+1}-V_{i,j}}{h_2}\right|^2.\\
\end{split}
\end{equation}

From \eqref{Sobo1} and \eqref{Sobo2}, we conclude that

\begin{equation*}
|V|_{\infty,\HH}^2\leq\,\tfrac{(b_2-b_1)\,(a_2-a_1)}{(a_2-a_1)\,h_1+(b_2-b_1)\,h_2}
\,|V|_{1,\HH}^2
\end{equation*}

which, easily, yields \eqref{SOBO}.
\end{proof}

\noindent
and of the following Lipschitz-type inequality:

\begin{lem}\label{Lemma_D3}
Let ${\sf g}\in C^2(\rset;\rset)$ with $\sup_{\ssy\rset}(|{\sf  g}'|+|{\sf g}''|)<+\infty$.
Then, for $v^a,v^b,z^a,z^b\in\dspace$, it holds that

\begin{equation}\label{OPAP_1A}
\begin{split}
\|{\sf g}(v^a)-{\sf g}(v^b)-{\sf g}(z^a)+{\sf g}(z^b)\|_{0,\HH}
\leq&\,\sup_{\ssy\rset}|{\sf g}'|\,\,|z^a-z^b|_{\infty,\HH}\,\|v^b-z^b\|_{0,\HH}\\

&+\left(\sup_{\ssy\rset}|{\sf g}'|+\sup_{\ssy\rset}|{\sf g}''|
\,\,|z^a-z^b|_{\infty,\HH}\right)\,\|v^a-v^b-z^a+z^b\|_{0,\HH}.\\
\end{split}
\end{equation}

\end{lem}

\begin{proof}

It is similar to the proof of (2.9) in Lemma~2.3 in \cite{Georgios2}.

\end{proof}

\subsection{Consistency Errors}\label{Section3}

To simplify the notation, we set
$\uu^{n}:={\sf I}_{\ssy\HH}^{\circ}[\uu(t_{n},\cdot)]$
for $n=0,\dots,N$, and $\uu^{n+\half}:={\sf I}_{\ssy\HH}^{\circ}[\uu(t^{n+\half},\cdot)]$
for $n=0,\dots,N-1$.

\subsubsection{Consistency error in time}

For $n=1,\dots,N-1$, let ${\sf r}^n\in\dspace$ be defined by

\begin{equation}\label{NORAD_22}
\tfrac{g(\uu^{n+\half})+g(\uu^{n-\half})}{2}=g(\uu^n)+{\sf r}^n.
\end{equation}

Then, setting $\zeta(t,x):=g(u(t,x))$ and expanding, by using the Taylor formula
around $t=t_n$, we obtain:

\begin{equation*}\label{USNORTH_21}
{\sf r}^n=\tfrac{\tau^2}{2}\,\,{\sf I}_{\ssy\HH}^{\circ}\left[\int_{0}^{\half}
\left[\,(\tfrac{1}{2}-s)\,\zeta_{tt}(t_n+s\,\tau,\cdot)
+s\,\zeta_{tt}(t^{n-\half}+s\,\tau,\cdot)\right]\;ds\right],
\quad n=1,\dots,N-1,
\end{equation*}

which, easily, yields

\begin{equation}\label{SHELL_21}
\tau\,\max_{1\leq{n}\leq{\ssy N-1}}|{\sf r}^n|_{\infty,{\HH}}
+\max_{2\leq{n}\leq{\ssy N-1}}|{\sf r}^n-{\sf r}^{n-1}|_{\infty,\HH}
\leq\,{\sf C}_{\ssy\tt I}\,\tau^3\,\max_{\ssy Q}\left(|\zeta_{tt}|+|\zeta_{ttt}|\right).
\end{equation}

\par
Let ${\sf r}^{\frac{1}{4}}\in\dspace$ be defined by

\begin{equation}\label{NORAD_000}
\tfrac{\uu^{\half}-\uu^0}{(\tau/2)}={\sf I}_{\ssy\HH}^{\circ}\left[\uu_{xx}(t^{\half},\cdot)\right]
+g(\uu^0)+{\sf I}_{\ssy\HH}^{\circ}\left[f(t^{\half},\cdot)\right]+{\sf r}^{\frac{1}{4}}
\end{equation}

and, for $n=0,\dots,N-1$, let ${\sf r}^{n+\half}\in\dspace$ be specified by

\begin{equation}\label{NORAD_02}
\tfrac{\uu^{n+1}-\uu^n}{\tau}={\sf I}_{\ssy\HH}^{\circ}
\left[\tfrac{\uu_{xx}(t_{n+1},\cdot)+\uu_{xx}(t_n,\cdot)}{2}\right]
+g(\uu^{n+\half})+{\sf I}_{\ssy\HH}^{\circ}\left[\,\tfrac{f(t_{n+1},\cdot)+f(t_n,\cdot)}{2}
\,\right]+{\sf r}^{n+\half}.
\end{equation}

Using \eqref{PSL_a},  from \eqref{NORAD_000} and \eqref{NORAD_02},
we obtain

\begin{equation}\label{USNORTH_00B}
{\sf r}^{\frac{1}{4}}={\sf r}_{\ssy A}^{\frac{1}{4}}+{\sf r}_{\ssy B }^{\frac{1}{4}},
\quad\text{\rm and}\quad
{\sf r}^{n+\half}:={\sf r}_{\ssy A}^{n+\half}+{\sf r}_{\ssy B}^{n+\half},
\quad n=0,\dots,N-1,
\end{equation}

where

\begin{equation*}
{\sf r}^{\frac{1}{4}}_{\ssy A}=\tfrac{u^{\half}-u^0}{(\tau/2)}
-{\sf I}_{\ssy\HH}^{\circ}\left[u_t(t^{\half},\cdot)\right],
\quad
{\sf r}_{\ssy B}^{\frac{1}{4}}:=g(u^{\half})-g(u^0)
\end{equation*}

and

\begin{equation*}
\begin{split}
{\sf r}^{n+\half}_{\ssy A}:=&\,\tfrac{\uu^{n+1}-\uu^n}{\tau}
-{\sf I}_{\ssy\HH}^{\circ}\left[u_{t}(t^{n+\half},\cdot)\right]

-{\sf I}_{\ssy\HH}^{\circ}\left[\tfrac{\uu_t(t_{n+1},\cdot)
+\uu_t(t_{n},\cdot)}{2}-u_{t}(t^{n+\half},\cdot)\right],\\

{\sf r}^{n+\half}_{\ssy B}:=&\,\tfrac{g(u^{n+1})+g(u^n)}{2}
-g(u^{n+\half}).\\
\end{split}
\end{equation*}

Applying the Taylor formula we obtain

\begin{equation}\label{USNORTH_04A}
{\sf r}_{\ssy A}^{\quarter}=-2\,\tau\,{\sf I}^{\circ}_{\ssy\HH}\left[
\int_0^{\half}s\,u_{tt}(s\,\tau,\cdot)\;ds\right],\quad

{\sf r}^{\quarter}_{\ssy B}={\sf I}_{\ssy\HH}^{\circ}
\left[\int_{t_0}^{t_1}\zeta_t(s,\cdot)\;ds\right]
\end{equation}

and

\begin{equation}\label{USNORTH_02A}
\begin{split}
{\sf r}_{\ssy A}^{n+\half}=&\,\tfrac{\tau^2}{2}\,{\sf I}_{\ssy\HH}^{\circ}
\left[\int_0^{\half}\left[\,s^2\,\uu_{ttt}(t_n+s\,\tau,\cdot)
+(\tfrac{1}{2}-s)^2\,\uu_{ttt}(t^{n+\half}+s\,\tau,\cdot)\,\right]\;ds\right]\\

&\quad-\tfrac{\tau^2}{2}\,{\sf I}_{\ssy\HH}^{\circ}\left[
\int_0^{\frac{1}{2}}\left[\,s\,u_{ttt}(t_n+\tau\,s,\cdot)
+(\tfrac{1}{2}-s)\,u_{ttt}(t^{n+\half}+\tau\,s,\cdot)\,\right]\;ds\right],\\

{\sf r}_{\ssy B}^{n+\half}=&\,\tfrac{\tau^2}{2}\,{\sf I}_{\ssy\HH}^{\circ}\left[
\int_0^{\frac{1}{2}}\left[\,s\,\zeta_{tt}(t_n+\tau\,s,\cdot)
+(\tfrac{1}{2}-s)\,\zeta_{tt}(t^{n+\half}+\tau\,s,\cdot)\,\right]\;ds\right],
\quad n=0,\dots,N-1.\\
\end{split}
\end{equation}

Thus, from \eqref{USNORTH_00B}, \eqref{USNORTH_02A} and \eqref{USNORTH_04A},
we arrive at

\begin{equation}\label{SHELL_01}
\tau\,|{\sf r}^{\quarter}|_{\infty,\HH}
+\max_{0\leq{n}\leq{\ssy N-1}}|{\sf r}^{n+\half}|_{\infty,\HH}
\leq\,{\sf C}_{\ssy\tt I\!I}\,\tau^2\,\max_{\ssy Q}\left(|\zeta_t|+|\zeta_{tt}|+|u_{tt}|+|u_{ttt}|\right).
\end{equation}

\subsubsection{Consistency error in space}

Also, let ${\sf s}^{\quarter}\in\dspace$ be defined by

\begin{equation}\label{NORAD_011}
\tfrac{\uu^1-\uu^0}{(\tau/2)}=\Deltah\uu^1
+g(\uu^0)+\PP\left[f(t^{\half},\cdot)\right]+{\sf s}^{\quarter}
\end{equation}

and, for $n=0,\dots,N-1$, let ${\sf s}^{n+\half}\in\dspace$ be given by

\begin{equation}\label{NORAD_12}
\tfrac{\uu^{n+1}-\uu^n}{\tau}=\Deltah\left(\tfrac{\uu^{n+1}+\uu^n}{2}\right)
+g(\uu^{n+\half})+\PP\left[\tfrac{f(t_{n+1},\cdot)+f(t_n,\cdot)}{2}\right]
+{\sf s}^{n+\half}.
\end{equation}

Subtracting \eqref{NORAD_011} from \eqref{NORAD_000} and \eqref{NORAD_12} from
\eqref{NORAD_02}, we obtain

\begin{equation}\label{USNORTH_11}
\begin{split}
{\sf r}^{\frac{1}{4}}-{\sf s}^{\frac{1}{4}}=&\,\PP\left[\uu_{xx}(t^{\half},\cdot)\right]
-\Deltah\uu^{\half},\\

{\sf r}^{n+\half}-{\sf s}^{n+\half}=&\,\PP\left[\tfrac{\uu_{xx}(t_{n+1},\cdot)+\uu_{xx}(t_n,\cdot)}{2}\right]
-\Deltah\left(\tfrac{\uu^{n+1}+\uu^n}{2}\right),\quad n=0,\dots,N-1.
\end{split}
\end{equation}

For $t\in[0,T]$, we use of the Taylor formula with respect to the space variables around $x=(x_{1,i},x_{2,j})$
to get

\begin{equation*}\label{USARCTIC}
\begin{split}
\PP\left[\uu_{xx}(t,\cdot)\right]
-\Deltah\left({\sf I}_{\ssy\HH}[u(t,\cdot)]\right)_{i,j}
=&\,\tfrac{h_1^2}{6}\,\int_0^1(1-z)^3\,\partial_{x_1}^4\uu(t,x_{1,i}+h_1\,z,x_{2,j})\;dz\\

&\,+\tfrac{h_1^2}{6}\,\int_0^1z^3\,\partial_{x_1}^4\uu(t,x_{1,i-1}+h_1\,z,x_{2,j})\;dz\\

&\,+\tfrac{h_2^2}{6}\,\int_0^1(1-z)^3\,\partial_{x_2}^4\uu(t,x_{1,i},x_{2,j}+h_2\,z)\;dz\\

&\,+\tfrac{h_2^2}{6}\,\int_0^1z^3\,\partial_{x_2}^4\uu(t,x_{1,i},x_{2,j-1}+h_2\,z)\;dz
\quad\forall\,(i,j)\in{\mathbb I}^{\circ},\\
\end{split}
\end{equation*}

which, along with \eqref{USNORTH_11}, yields

\begin{equation}\label{SHELL_11}
|{\sf s}^{\quarter}-{\sf r}^{\quarter}|_{\infty,\HH}
+\max_{0\leq{n}\leq{\ssy N-1}}|{\sf s}^{n+\half}-{\sf r}^{n+\half}|_{\infty,\HH}
\leq\,{\sf C}_{\ssy\tt I\!I\!I}\,(h_1^2+h_2^2)\,
\max_{\ssy Q}\left(|\partial_{x_1}^4u|+|\partial_{x_2}^4u| \right).
\end{equation}

\section{Convergence Analysis}\label{Section4}

\subsection{A mollifier}

For $\delta>0$, let $\gff_{\delta}\in C^3(\rset;\rset)$ (cf. \cite{Akrivis1},
\cite{KarMak}, \cite{Georgios1}) be an odd fuction defined by

\begin{equation}\label{ni_defin}
\gff_{\delta}(x):=\left\{
\begin{aligned}
&x,\hskip2.40truecm
\mbox{if}\ \ x\in[0,\delta],\\

&p_{\delta}(x),\hskip1.75truecm
\mbox{if}\ \ x\in (\delta,2\delta],\\

&2\,\delta,\hskip2.22truecm\mbox{if}\ \ x> 2\delta,\\
\end{aligned}
\right.\quad\forall\,x\ge 0,
\end{equation}

where $p_{\delta}$ is the unique polynomial of ${\mathbb P}^7[\delta,2\delta]$
that satisfies the following conditions:

\begin{equation*}
p_{\delta}(\delta)=\delta,\,\,\,p_{\delta}'(\delta)=1,
\,\,\,p_{\delta}''(\delta)=p_{\delta}'''(\delta)=0,
\,\,\,p_{\delta}(2\,\delta)=2\,\delta,
\,\,\,p_{\delta}'(2\,\delta)=p_{\delta}''(2\,\delta)=p_{\delta}'''(2\,\delta)=0.
\end{equation*}

\subsection{The modified scheme}\label{The_MBRFD}

For given $\delta>0$, the modified version of the (RFD) method (cf. \cite{Akrivis1}, \cite{KarMak},
\cite{Georgios1}), derives approximations
$(V^n_{\delta})_{n=0}^{\ssy N}\subset\dspace$ of the solution
$\uu$ as follows:

\par\vskip0.2truecm\par\noindent

{\sf Step M1}: First, we set

\begin{equation}\label{Mstep1}
V_{\delta}^0:=U^0\quad\text{and}\quad V_{\delta}^{\half}:=U^{\half}.
\end{equation}

\par\vskip0.2truecm\par\noindent

{\sf Step M2}: Set

\begin{equation}\label{Mstep2}
\Phi^{\frac{1}{2}}_{\delta}:=g\big(\gff_{\delta}\big(V_{\delta}^{\half}\big)\big)
\end{equation}

and find $V^{1}_{\delta}\in\dspace$ such that

\begin{equation}\label{BR_CB6}
\tfrac{V^{1}_{\delta}-V^0_{\delta}}{\tau}=
\Deltah\left(\tfrac{V^{1}_{\delta}+V^0_{\delta}}{2}\right)

+\Phi_{\delta}^{\half}
+\PP\left[\tfrac{f(t_1,\cdot)+f(t_0,\cdot)}{2}\right].
\end{equation}

\par\vskip0.2truecm\par\noindent

{\sf Step M3}: For $n=1,\dots,N-1$, first define
$\Phi_{\delta}^{n+\half}\in\dspace$ by

\begin{equation}\label{BR_CB5}
\Phi_{\delta}^{n+\half}:=2\,g\big(\gff_{\delta}\big(V_{\delta}^n\big)\big)
-\Phi_{\delta}^{n-\half}
\end{equation}

and, then, find $V^{n+1}_{\delta}\in\dspace$ such
that

\begin{equation}\label{BR_CB6}
\tfrac{V^{n+1}_{\delta}-V^n_{\delta}}{\tau}=
\Deltah\left(\tfrac{V^{n+1}_{\delta}+V^n_{\delta}}{2}\right)

+\Phi_{\delta}^{n+\half}
+\PP\left[\tfrac{f(t_{n+1},\cdot)+f(t_n,\cdot)}{2}\right].
\end{equation}

\subsection{Convergence of the modified scheme}\label{Section_Conv_MBRFD}

\begin{thm}\label{BR_CB_Conv}
Let $u_{\ssy\max}:=\max\limits_{\ssy Q}|\uu|$ and
$\ddelta\ge u_{\ssy\max}$.

Then, there exist positive constants ${\sf C}^{\ssy{\sf BCV},1}_{\ddelta}$,
${\sf C}^{\ssy{\sf BCV},2}_{\ddelta}$ and
${\sf C}^{\ssy{\sf BCV},3}_{\ddelta}$, independent of $\tau$ and $h$, such that:
if $\tau\,{\sf C}^{\ssy{\sf BCV},1}_{\ddelta}\leq\tfrac{1}{2}$, then

\begin{equation}\label{BR_CB_cnv_2}
|\uu^{\half}-V_{\ddelta}^{\half}|_{1,\HH}
\leq\,{\sf C}_{\ddelta}^{\ssy{\sf BCV},2}\,\big(\tau^{\frac{3}{2}}
+\tau^{\half}\,h_1^2+\tau^{\half}\,h_2^2\big)
\end{equation}

and

\begin{equation}\label{BR_CB_cnv_1}
\max_{0\leq{m}\leq{\ssy N-1}}\|g(\uu^{m+\half})-\Phi_{\ddelta}^{m+\half}\|_{0,\HH}
+\max_{0\leq{m}\leq{\ssy N}}|\uu^m-V_{\ddelta}^m|_{1,\HH}
\leq\,{\sf C}_{\ddelta}^{\ssy{\sf BCV},3}\,\big(\tau^2+h_1^2+h_2^2\big)
\end{equation}

\end{thm}

\begin{proof}

To simplify the notation, we set
$\ee^m:=\uu^m-V_{\ddelta}^m$ for $m=0,\dots,N$, and
$\btheta^m:=g(u^{m+\half})-\Phi_{\ddelta}^{m+\half}$
for $m=0,\dots,N-1$.

In the sequel,  we will use the symbol $C$ to denote a generic constant that is
independent of $\tau$, $h$ and $\ddelta$, and may changes value from one line to the other.
Also, we will use the symbol $C_{\ddelta}$ to denote a generic constant that depends on
$\ddelta$ but is independent of $\tau$, $h$, and may changes value from one line to the other.

\par\noindent\vskip0.4truecm\par\noindent

$\boxed{{\tt Part\,\,\,1}:}$ Since $\ee^0=0$, after subtracting
\eqref{BRS_2} and \eqref{BRS_12} from \eqref{NORAD_011} and
\eqref{NORAD_12} (with $n=0$), respectively, we obtain

\begin{gather}
\ee^{\half}=\tfrac{\tau}{2}\,\Deltah\ee^{\half}+\tfrac{\tau}{2}\,{\sf s}^{\frac{1}{4}},
\label{BR_EE_a}\\

\ee^{1}=\tfrac{\tau}{2}\,\Deltah\ee^{1}+\tau\,\btheta^0+\tau\,{\sf s}^{\half}.
\label{BR_EE_b}
\end{gather}

Taking the $(\cdot,\cdot)_{0,\HH}-$inner product of
\eqref{BR_EE_a} with $\ee^{\half}$, and then using \eqref{SELF},
the Cauchy-Schwarz inequality, \eqref{SHELL_01},
\eqref{SHELL_11} and the arithmetic mean inequality,
we get

\begin{equation*}
\begin{split}
\|\ee^{\half}\|_{0,\HH}^2+\tfrac{\tau}{2}\,|\ee^{\half}|_{1,\HH}^2
=&\,\tfrac{\tau}{2}\,({\sf s}^{\frac{1}{4}},\ee^{\half})_{0,\HH}\\

\leq&\,\tfrac{\tau}{2}\,\left(\,\|{\sf s}^{\frac{1}{4}}-{\sf r}^{\frac{1}{4}}\|_{0,\HH}
+\|{\sf r}^{\frac{1}{4}}\|_{0,\HH}\,\right)\,\|\ee^{\half}\|_{0,\HH}\\

\leq&\,C\,\big(\tau^2+\tau\,h_1^2+\tau\,h_2^2\big)\,\|\ee^{\half}\|_{0,\HH}\\

\leq&\,C\,\big(\tau^2+\tau\,h_1^2+\tau\,h_2^2\big)^2+\tfrac{1}{2}\,\|\ee^{\half}\|_{0,\HH}^2,\\
\end{split}
\end{equation*}

which, obviously, yields

\begin{equation}\label{BR_reco_1}
\|\ee^{\half}\|_{0,\HH}^2+\tau\,|\ee^{\half}|_{1,\HH}^2
\leq\,C\,\big(\tau^2+\tau\,h_1^2+\tau\,h_2^2\big)^2.
\end{equation}

Since $\ddelta\ge u_{\ssy\max}$, using \eqref{ni_defin}
and \eqref{BR_reco_1}, we have

\begin{equation}\label{BR_reco_6}
\begin{split}
\|\btheta^0\|_{0,h}^2=&\,\big\|g(\gff_{\ddelta}(u^{\half}))
-g\big(\gff_{\ddelta}\big(V^{\half}_{\ddelta}\big)\big)\big\|_{0,h}^2\\

\leq&\,\sup_{\ssy\rset}|(g\circ\gff_{\ssy\ddelta})'|^2\,\|\ee^{\half}\|_{0,h}^2\\

\leq&\,C_{\ddelta}\,(\tau^2+\tau\,h_1^2+\tau\,h_2^2)^2.\\
\end{split}
\end{equation}

\par
Taking the $(\cdot,\cdot)_{0,\HH}-$inner product of
\eqref{BR_EE_b} with $\ee^{1}$, and then using \eqref{SELF},
the Cauchy-Schwarz inequality, \eqref{SHELL_11},
\eqref{SHELL_01},\eqref{BR_reco_6}
and the arithmetic mean inequality, we obtain

\begin{equation*}
\begin{split}
\|\ee^{1}\|_{0,\HH}^2+\tfrac{\tau}{2}\,|\ee^{1}|_{1,\HH}^2
=&\,\tau\,({\sf s}^{\half},\ee^1)_{0,\HH}+\tau\,(\btheta^0,\ee^1)_{0,\HH}\\

\leq&\,\tau\,\left(\,\|{\sf s}^{\half}-{\sf r}^{\half}\|_{0,\HH}
+\|{\sf r}^{\half}\|_{0,\HH}+\|\btheta\|_{0,\HH}\,\right)\,\|\ee^1\|_{0,\HH}\\

\leq&\,C\,\tau\,\big(\tau^2+h_1^2+h_2^2\big)\,\|\ee^1\|_{0,\HH}\\

\leq&\,C\,\tau^2\,\big(\tau^2+h_1^2+h_2^2\big)^2
+\tfrac{1}{2}\,\|\ee^1\|_{0,\HH}^2,\\
\end{split}
\end{equation*}

which, obviously, yields

\begin{equation}\label{REL1_covid}
\|\ee^1\|_{0,\HH}^2+\tau\,|\ee^1|_{1,\HH}^2
\leq\,C\,\tau^2\,\big(\tau^2+h_1^2+h_2^2\big)^2.
\end{equation}

Taking the $(\cdot,\cdot)_{0,\HH}-$inner product of
\eqref{BR_EE_b} with $\Deltah\ee^1$,
and then using \eqref{SELF}, we obtain

\begin{equation*}\label{BR_reco_2}
|\ee^1|_{1,\HH}^2+\tfrac{\tau}{2}\,\|\Deltah\ee^1\|_{0,\HH}^2
=-\tau\,({\sf s}^{\half},\Deltah\ee^1)_{0,\HH}-\tau\,(\btheta^0,\Deltah\ee^1)_{0,\HH}.
\end{equation*}

Then, we use the Cauchy-Schwarz inequality, \eqref{SHELL_11},
\eqref{SHELL_01}, \eqref{BR_reco_6} and the arithmetic mean inequality, to have

\begin{equation*}
\begin{split}
|\ee^1|_{1,\HH}^2+\tfrac{\tau}{2}\,|\Deltah\ee^{1}|_{0,\HH}^2
\leq&\,\tau\,\left(\,\|{\sf s}^{\half}-{\sf r}^{\half}\|_{0,\HH}
+\|{\sf r}^{\half}\|_{0,\HH}+\|\btheta\|_{0,\HH}\,\right)\,\|\Deltah\ee^1\|_{0,\HH}\\

\leq&\,\tau\,\left(\,h_1^2+h_2^2+\tau^2+\|\btheta\|_{0,\HH}\,\right)\,\|\Deltah\ee^1\|_{0,\HH}\\

\leq&\,C\,\tau\,\big(\tau^2+h_1^2+h_2^2\big)\,\|\Deltah\ee^1\|_{0,\HH}\\

\leq&\,C\,\tau\,\big(\tau^2+h_1^2+h_2^2\big)^2
+\tfrac{\tau}{4}\,\|\Deltah\ee^1\|_{1,\HH}^2,\\
\end{split}
\end{equation*}

which, obviously, yields

\begin{equation}\label{REL2_covid}
|\ee^1|_{1,\HH}^2+\tau\,\|\Deltah\ee^1\|_{0,\HH}^2
\leq\,C\,\tau\,\big(\tau^2+h_1^2+h_2^2\big)^2.
\end{equation}

\par\noindent\vskip0.4truecm\par\noindent

$\boxed{{\tt Part\,\,\,2}}$:

Subtracting \eqref{BR_CB5} from \eqref{NORAD_22}, and then
using \eqref{ni_defin} and the assumption $\ddelta\ge u_{\ssy\max}$,
we obtain:

\begin{equation}\label{covid_2.1}
\btheta^{n}+\btheta^{n-1}
=2\,\left[g(\gff_{\ddelta}(u^n))-g(\gff_{\ddelta}(V_{\ddelta}^n))\right]
+2\,{\sf r}^{n},\quad n=1,\dots,N-1,
\end{equation}

which, easily, yields that

\begin{equation}\label{covid_2.2}
\btheta^{n}-\btheta^{n-2}
=2\,{\sf R}^n+2\,({\sf r}^n-{\sf r}^{n-1}),\quad n=2,\dots,N-1,
\end{equation}

where

\begin{equation}\label{covid_2.3}
{\sf R}^n:=g(\gff_{\ddelta}(V_{\ddelta}^{n-1}))-g(\gff_{\ddelta}(V_{\ddelta}^n))
-g(\gff_{\ddelta}(u^{n-1}))+g(\gff_{\ddelta}(u^n))\in\dspace.
\end{equation}

Then, we use \eqref{OPAP_1A} (with ${\sf g}=g\circ\gff_{\ddelta}$),
\eqref{ni_defin} and the mean value theorem, to get

\begin{equation}\label{covid_2.4}
\begin{split}
\|{\sf R}^n\|_{0,\HH}\leq&\,\sup_{\ssy\rset}|(g\circ\gff_{\ssy\ddelta})'|
\,\|\ee^n-\ee^{n-1}\|_{0,\HH}\\

&\quad+\sup_{\ssy\rset}|(g\circ\gff_{\ssy\ddelta})''|\,|\uu^{n-1}-\uu^{n}|_{\infty,\HH}
\,\left(\|\ee^n-\ee^{n-1}\|_{0,\HH}+\|\ee^{n}\|_{0,\HH}\right)\\

\leq&\,C_{\ddelta}\,\left(\|\ee^n-\ee^{n-1}\|_{0,\HH}
+\tau\,\|\ee^{n}\|_{0,\HH}\right),
\quad n=2,\dots,N-1.
\end{split}
\end{equation}

Taking the $(\cdot,\cdot)_{0,\HH}-$inner product of both sides of \eqref{covid_2.2}
with $\tau\big(\btheta^{n}+\btheta^{n-2}\big)$, and then using the
Cauchy-Schwarz inequality, \eqref{covid_2.4} and \eqref{SHELL_21}, it follows that

\begin{equation*}
\begin{split}
\tau\,\|\btheta^{n}\|_{0,\HH}^2-\tau\,\|\btheta^{n-2}\|^2_{0,\HH}\leq&\,
\left(2\,\tau\,\|{\sf R}^n\|_{0,\HH}
+2\,\tau\,\|{\sf r}^n-{\sf r}^{n-1}\|_{0,\HH}\right)
\,\|\btheta^{n}+\btheta^{n-2}\|_{0,\HH}\\

\leq&\,C_{\ddelta}\,\left(\,\tau\,\|\ee^n-\ee^{n-1}\|_{0,\HH}
+\tau^2\,\|\ee^{n}\|_{0,\HH}+\tau^4\right)\,\|\btheta^{n}+\btheta^{n-2}\|_{0,\HH}\\

\leq&\,C_{\ddelta}\,\tau\,\|\ee^n-\ee^{n-1}\|_{0,\HH}
\,\left(\|\btheta^n\|_{0,\HH}+\|\btheta^{n-2}\|_{0,\HH}\right)\\

&\quad+C_{\ddelta}\,\tau^2\,\|\ee^{n}\|_{0,\HH}
\,\left(\|\btheta^{n}\|_{0,\HH}+\|\btheta^{n-2}\|_{0,\HH}\right)\\

&\quad+C\,\tau^4\,\left(\|\btheta^{n}\|_{0,\HH}+\|\btheta^{n-2}\|_{0,\HH}\right),
\quad n=2,\dots,N-1,\\
\end{split}
\end{equation*}

which, along with the application of the arithmetic mean inequality, yields

\begin{equation}\label{covid_2.5}
\begin{split}
\|\btheta^{n}\|_{0,\HH}^2+\|\btheta^{n-1}\|_{0,\HH}^2

\leq&\,\|\btheta^{n-1}\|_{0,\HH}^2+\|\btheta^{n-2}\|^2_{0,\HH}\\

&\quad+\tau^{-1}\,\|\ee^n-\ee^{n-1}\|_{0,\HH}^2+C\,\tau^5\\

&\quad+C_{\ddelta}\,\tau\,\left(\|\ee^{n}\|_{0,\HH}^2
+\|\btheta^{n}\|_{0,\HH}^2+\|\btheta^{n-2}\|_{0,\HH}^2\right),
\quad n=2,\dots,N-1.\\
\end{split}
\end{equation}

\par\noindent\vskip0.4truecm\par\noindent

$\boxed{{\tt Part\,\,\,3}}$:
We subtract \eqref{BR_CB6} from \eqref{NORAD_12},
to obtain

\begin{equation}\label{BR_CB_EQ}
2\,(\ee^{n+1}-\ee^n)=\tau\,\Deltah\left(\ee^{n+1}+\ee^n\right)
+2\,\tau\,{\sf s}^{n+\frac{1}{2}}+2\,\tau\,\btheta^n,
\quad n=1,\dots,N-1.
\end{equation}

Now, we take the $(\cdot,\cdot)_{0,{\sf H}}-$inner product
of \eqref{BR_CB_EQ} with $(\ee^{n+1}-\ee^{n})$,
and then, use \eqref{SELF}, to have

\begin{equation}\label{BR_CB_Gat2}
\begin{split}
2\,\|\ee^{n+1}-\ee^{n}\|_{0,{\sf H}}^2
+\tau\,|\ee^{n+1}|_{1,{\sf H}}^2
-\tau\,|\ee^n|_{1,{\sf H}}^2
=&\,2\,\tau\,({\sf s}^{n+\frac{1}{2}},\ee^{n+1}-\ee^n)_{0,{\sf H}}\\

&\quad+2\,\tau\,(\btheta^n,\ee^{n+1}-\ee^n)_{0,{\sf H}},
\quad n=1,\dots,N-1.
\end{split}
\end{equation}

Using the Cauchy-Schwarz ine\-quality,
the arithmetic mean ine\-quality, \eqref{SHELL_01}
and \eqref{SHELL_11}, we have

\begin{equation}\label{BR_CB_Gat3}
\begin{split}
2\,\tau\,({\sf s}^{n+\frac{1}{2}},\ee^{n+1}-\ee^n)_{0,{\sf H}}
\leq&\,2\,\tau\,\left(\,\|{\sf s}^{n+\half}-{\sf r}^{n+\half}\|_{0,{\sf H}}
+\|{\sf r}^{n+\half}\|_{0,{\sf H}}\,\right)\,
\|\ee^{n+1}-\ee^{n}\|_{0,{\sf H}}\\

\leq&\,2\,\tau\,\big(\tau^2+h_1^2+h_2^2\big)\,
\|\ee^{n+1}-\ee^{n}\|_{0,{\sf H}}\\

\leq&\,C\,\tau^2\,\big(\tau^2+h_1^2+h_2^2\big)^2\,
+\tfrac{1}{6}\,\|\ee^{n+1}-\ee^{n}\|_{0,{\sf H}}^2,
\quad n=1,\dots,N-1.
\end{split}
\end{equation}

Next, we use the Cauchy-Schwarz inequality,  \eqref{dpoincare},
\eqref{ni_defin} and the arithmetic mean inequality, to get

\begin{equation}\label{BR_CB_Gat4}
\begin{split}
2\,\tau\,(\btheta^n,\ee^{n+1}-\ee^n)_{0,\HH}\leq&\,2\,\tau
\,\|\btheta^n\|_{0,\HH}\,\|\ee^{n+1}-\ee^{n}\|_{0,\HH}\\

\leq&\,6\,\tau^2\,\|\btheta^{n}\|_{0,\HH}^2
+\tfrac{1}{6}\,\|\ee^{n+1}-\ee^{n}\|_{0,\HH}^2,
\quad n=1,\dots,N-1.\\
\end{split}
\end{equation}

From \eqref{BR_CB_Gat2}, \eqref{BR_CB_Gat3} and
\eqref{BR_CB_Gat4}, we conclude that

\begin{equation}\label{BR_CB_Gat5.1}
\begin{split}
\tau^{-1}\,\|\ee^{n+1}-\ee^{n}\|_{0,\HH}^2
+|\ee^{n+1}|_{1,{\sf H}}^2\leq&\,|\ee^n|_{1,\HH}^2
+6\,\tau\,\|\btheta^{n}\|_{0,\HH}^2

+C\,\tau\,\big(\tau^2+h_1^2+h_2^2\big)^2,
\quad n=1,\dots,N-1.\\
\end{split}
\end{equation}

\par\noindent\vskip0.4truecm\par\noindent

$\boxed{{\tt Part\,\,\,4}}$:
Let us, first, introduce the error quantities:

\begin{equation}\label{BR_Zerror}
{\sf E}^m:=\tau^{-1}\,\|\ee^{m}-\ee^{m-1}\|_{0,\HH}^2
+|\ee^m|_{1,\HH}^2
+\|\btheta^{m-1}\|_{0,\HH}^2
+\|\btheta^{m-2}\|_{0,\HH}^2,\quad m=2,\dots,N.
\end{equation}

Then, from \eqref{BR_CB_Gat5.1}, \eqref{covid_2.5} and \eqref{dpoincare}, we
conclude that there exists a constant ${\sf C}_{\ddelta}^{\ssy{\sf BR, I}}>0$
such that:

\begin{equation}\label{BR_CB_Gat9}
(1-{\sf C}_{\ddelta}^{\ssy{\sf BR, I}}\,\tau)\,{\sf E}^{n+1}
\leq\,(1+{\sf C}_{\ddelta}^{\ssy{\sf BR, I}}\,\tau)\,{\sf E}^n
+C\,\tau\,\big(\tau^2+h_1^2+h_2^2\big)^2,\quad n=2,\dots,N-1.
\end{equation}

Assuming that $\tau\,{\sf C}^{\ssy{\sf BR, I}}_{\ddelta}\leq\frac{1}{2}$,
a standard discrete Gronwall argument based on \eqref{BR_CB_Gat9} yields

\begin{equation}\label{BR_CB_Gat10}
\begin{split}
\max_{2\leq{m}\leq{\ssy N}}{\sf E}^m
\leq&\,C_{\ddelta}\,\left[\,{\sf E}^2+\big(\tau^2+h_1^2+h_2^2\big)^2\,\right]\\

\leq&\,C_{\ddelta}\,\left[
\tau^{-1}\,\|\ee^{2}-\ee^{1}\|_{0,\HH}^2
+|\ee^2|_{1,\HH}^2
+\|\btheta^{1}\|_{0,\HH}^2

+\|\btheta^{0}\|_{0,\HH}^2
+\big(\tau^2+h_1^2+h_2^2\big)^2\,\right].\\
\end{split}
\end{equation}

Setting $n=1$ in \eqref{covid_2.1} and then using \eqref{SHELL_21},
\eqref{BR_reco_6}, \eqref{REL1_covid}, we get

\begin{equation}\label{BR_Papagalini}
\begin{split}
\|\btheta^1\|_{0,\HH}\leq&\,\|\btheta^0\|_{0,\HH}
+2\,\sup_{\ssy\rset}|(g\circ\gff_{\ddelta})'|\,\|\ee^1\|_{0,\HH}
+2\,\|{\sf r}^1\|_{0,\HH}\\

\leq&\,C_{\ddelta}\,\big(\tau^2+\tau\,h_1^2+\tau\,h_2^2\big).\\
\end{split}
\end{equation}

Also, setting $n=1$ in \eqref{BR_CB_Gat5.1}, and then
using \eqref{BR_Papagalini} and \eqref{REL2_covid},
we have

\begin{equation}\label{BR_reco_8}
\begin{split}
\tau^{-1}\,\|\ee^2-\ee^1\|_{0,\HH}^2+|\ee^{2}|_{1,\HH}^2
\leq&\,C\,\tau\,\big(\tau^2+h_1^2+h_2^2\big)^2
+|\ee^{1}|_{1,\HH}^2+6\,\tau\,\|\btheta^1\|_{0,\HH}^2\\

\leq&\,C_{\ddelta}\,\tau\,\big(\tau^2+h_1^2+h_2^2\big)^2.\\
\end{split}
\end{equation}

Thus, \eqref{BR_CB_Gat10}, \eqref{BR_reco_8},
\eqref{BR_Papagalini} and \eqref{BR_reco_6}
yield

\begin{equation}\label{BR_main_res_2}
\max_{2\leq{m}\leq{\ssy N}}{\sf E}^m\leq\,C_{\ddelta}\,\big(\tau^2+h_1^2+h_2^2\big)^2.
\end{equation}

\par
Thus, \eqref{BR_CB_cnv_2} follows from \eqref{BR_reco_1}. Since $\ee^0=0$,
\eqref{BR_CB_cnv_1} is established, easily,
from \eqref{BR_main_res_2}, \eqref{BR_Zerror}  and \eqref{REL2_covid}.

\end{proof}

\subsection{Convergence of the (RFD) method}

\begin{thm}\label{DR_Final}

Let $u_{\ssy\max}:=\max\limits_{\ssy Q}|\uu|$,
$\ddelta\ge\,2\,u_{\ssy\max}$,
${\sf L}$ be the constant in the inequality in Lemma~\ref{sobol},
and ${\sf C}^{\ssy{\sf BCV},1}_{\ddelta}$
${\sf C}^{\ssy{\sf BCV},2}_{\ddelta}$,
${\sf C}^{\ssy{\sf BCV},3}_{\ddelta}$ be the positive constants
specified in Theorem~\ref{BR_CB_Conv}. If

\begin{equation}\label{BR_Xmesh}
\tau\,{\sf C}^{\ssy{\sf BCV},1}_{\ddelta}\leq\tfrac{1}{2},
\quad
{\sf C}_{\ddelta}^{\ssy{\sf BCV},2}\,{\sf L}\,\,\,\left[
\tfrac{\tau^2+h_1^2+h_2^2}{\sqrt{h_1+h_2}}\right]
\leq\,\tfrac{\ddelta}{2},
\quad
{\sf C}_{\ddelta}^{\ssy{\sf BCV},3}\,{\sf L}\,\,\,\left[
\tfrac{\tau^{\frac{3}{2}}+\tau^{\half}\,h_1^2+\tau^{\half}\,h_2^2}{\sqrt{h_1+h_2}}\right]
\leq\,\tfrac{\ddelta}{2},
\end{equation}

then

\begin{equation}\label{BR_true_1}
\max_{0\leq{m}\leq{\ssy N-1}}|g(\uu^{m+\half})-\Phi^{m+\half}|_{0,\HH}
+\max_{0\leq{m}\leq{\ssy N}}|\uu^m-\UUU^m|_{1,\HH}
\leq\,{\sf C}_{\ddelta}^{\ssy{\sf BCV},3}\,(\tau^2+h_1^2+h_2^2).
\end{equation}

\end{thm}

\begin{proof}

Since $\ddelta\ge 2\,u_{\ssy\max}$, the convergence estimates
\eqref{BR_CB_cnv_2} and \eqref{BR_CB_cnv_1},
the discrete Sobolev inequality \eqref{SOBO}
and the mesh size conditions \eqref{BR_Xmesh} imply that

\begin{equation*}
\begin{split}
\big|V_{\ddelta}^{\half}\big|_{\infty,\HH}\leq&\,\big|\uu^{\half}
-V^{\half}_{\ddelta}\big|_{\infty,\HH}+\big|\uu^{\half}|_{\infty,\HH}\\

\leq&\,{\sf L}\,(h_1+h_2)^{-\half}\,\big|\uu^{\half}
-V^{\half}_{\ddelta}\big|_{1,\HH}+u_{\ssy\max}\\

\leq&\,{\sf C}_{\ddelta}^{\ssy{\sf BCV},2}\,{\sf L}\,(h_1+h_2)^{-\half}
\,(\tau^{\frac{3}{2}}+\tau^{\half}\,h_1^{2}+\tau^{\half}\,h_2^2)+\tfrac{\ddelta}{2}\\

\leq&\,\ddelta,
\end{split}
\end{equation*}

and

\begin{equation*}
\begin{split}
\big|V_{\ddelta}^{n}\big|_{\infty,\HH}\leq&\,\big|\uu^{n}
-V^{n}_{\ddelta}\big|_{\infty,\HH}+\big|\uu^{n}|_{\infty,\HH}\\

\leq&\,{\sf L}\,(h_1+h_2)^{-\half}\,\big|\uu^{n}
-V^{n}_{\ddelta}\big|_{1,\HH}+u_{\ssy\max}\\

\leq&\,{\sf C}_{\ddelta}^{\ssy{\sf BCV},3}\,{\sf L}
\,(\tau^2+h_1^2+h_2^2)\,(h_1+h_2)^{-\half}+\tfrac{\ddelta}{2}\\

\leq&\,\ddelta,\quad n=1,\dots,N,
\end{split}
\end{equation*}

which, along with \eqref{ni_defin} and \eqref{Mstep1}, yield

\begin{equation}\label{BR_crucial_1}
\gff_{\ddelta}(V^{\half}_{\ddelta})=V^{\half}_{\ddelta},\quad

\gff_{\ddelta}\big(V_{\ddelta}^{n}\big)
=V_{\ddelta}^{n},\quad n=1,\dots,N.
\end{equation}

Thus, for $\delta=\ddelta$, the modified (RFD) approximations defined
in Section~\ref{The_MBRFD} are the (RFD) approximations defined by
\eqref{BRS_1}-\eqref{BRS_4},

and the error estimate \eqref{BR_true_1} follow as a natural outcome
of \eqref{BR_CB_cnv_1}.

\end{proof}


TITLE: Do Rational Functions Act $n$-Transitively on $\mathbb{C}$?
QUESTION [1 upvotes]: Let $p_1, ..., p_n$ be $n$ distinct points in $\mathbb{C}$, and let $q_1, ..., q_n$ be $n$ distinct points in $\mathbb{C}$. Is there a rational function $f \in \mathbb{C} (x)$ such that $f(p_i) = q_i$ for each $1 \leq i \leq n$? Thinking of $\mathbb{C}(x)$ as a monoid under composition, this question asks whether $\mathbb{C}(x)$ acts $n$-transitively on $\mathbb{C}$.
More importantly, if this is false, then is there a nice description of the set of pairs $((a_1, ..., a_n), (b_1, ..., b_n)) \in \mathbb{C}^n \times \mathbb{C}^n$ for which there is a rational function $f \in \mathbb{C}(x)$ such that $f(a_i) = b_i$?
A rational function is determined by its preimages counting multiplicities at 3 distinct points, so I suspect this is false.

REPLY [3 votes]: By Lagrange interpolation, you don't even need rational functions — polynomials suffice.


text:
\begin{definition}[Definition:Independent Variable/Real Function]
Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Let $\map f x = y$.
Then $x$ is referred to as an '''independent variable'''.
\end{definition}

config: wiki


TITLE: Proof of the product rule in probability theory for causal independence
QUESTION [0 upvotes]: There is a common attitude in the text books on probability that the so-called product rule is an obvious property, when events are independent, i.e., $P(A\cap B)=P(A)P(B)$ when $A$ and $B$ are independent events. Yet, this is NOT an axiom that a probability must satisfy, nor it is a property that follows from the axioms. That is because there is no definition of the concept of independent events in the axioms.
Of course, the product rule is the actual definition of the (stochastic) independence of two events, if one wants to evade the concept of conditional probability. So you, clearly, see the chicken-and-the-egg conundrum.
Still, we continue to use the rule because we intuitively feel the independence property is true, which seems natural in those cases when our random experiment is obtained as a succession of two (simpler) random experiments which are causal independent.
And now my question: how is this idea (of causal independence) formalized in the context of the axiomatic approach to probabilities such that we can prove from here the product rule?

REPLY [0 votes]: $(|)=(),(|)=\frac{(∩)}{()}→(∩)=(|)()→(∩)=()()$


TITLE: How many ways to seat $8$ of $31$ people on a boat with restrictions
QUESTION [0 upvotes]: Task condition:
On each side of the boat should be seating for $4$ persons. How many ways you can select for this boat, if there are $31$ candidates, with ten people who want to sit on the left side of the boat, twelve who want to sit on the right side, and nine indifferent as to where they sit?
I have no thoughts

REPLY [1 votes]: There is nothing too difficult about it, only it is tedious. Fill up $0$ through $4$ "specialists" on their favoured sides and fill the balance from the 9 others, e.g. if you choose 2 specialists for left and 3 for right,
$${10\choose2}{12\choose3}{9\choose2}{7\choose1}$$ which generalises to
$$\sum_{l=0}^4 \sum_{r=0}^4{10\choose l}{12\choose r}{9\choose 4-l}{5+l\choose 4-r}$$