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%\rightheadtext{NONEXISTENCE OF UNIFORMLY CONVERGENT METHOD FOR SEMILINEAR BVP}
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\begin{document}
\title[NONEXISTENCE OF UNIFORMLY CONVERGENT METHOD FOR SEMILINEAR BVP]
{On the non-existence of $\varepsilon$-uniform finite difference methods
on uniform meshes for semilinear two-point boundary value problems}
% author one information
\author{Paul A. Farrell}
\address{Department of Mathematics and Computer Science, Kent State University,
Kent, Ohio 44242, U.S.A.}
\email{farrell@mcs.kent.edu}
\thanks{Supported in part under NSF grant DMS-9627244.}
\thanks{The first author was supported in part by
The Research Council of Kent State University.}
% author two information
\author{John J. H. Miller}
\address{Department of Mathematics, Trinity College, Dublin 2, Ireland}
\email{jmiller@tcd.ie}
\author{Eugene O'Riordan}
\address{School of Mathematical Sciences, Dublin City University,
Glasnevin, Dublin 9, Ireland}
\email{oriordane@ccmail.dcu.ie}
\author{Grigorii I. Shishkin}
\address{Institute of Mathematics and Mechanics, Russian Academy of Sciences,
Ekaterinburg, Russia}
\email{grigorii@shishkin.ural.ru}
\thanks{The fourth author was supported in part by the Russian
Foundation for Basic Research under Grant N 95-01-00039.}
\subjclass{Primary 34B15, 65L12; Secondary 34L30, 65L10}
\keywords{Semilinear boundary value problem, singular
perturbation, finite-difference scheme, $\varepsilon$-uniform convergence,
uniform mesh, frozen fitting factor}
\date{October 18, 1996}
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\begin{abstract}
In this paper fitted finite difference methods
on a uniform mesh with internodal spacing $h$, are considered for a singularly
perturbed semilinear two point boundary value problem. It is proved that
a scheme of this type with a frozen fitting factor cannot converge
$\varepsilon$-uniformly in the maximum norm to the solution
of the differential equation as the
mesh spacing $h$ goes to zero. Numerical experiments are presented which show
that the same result is true for a
number of schemes with variable fitting factors.
\end{abstract}
\maketitle
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\section{Introduction}
In this paper fitted finite difference methods
on a uniform mesh are considered for a singularly
perturbed semilinear two point boundary value problem.
Singularly perturbed differential equations are all pervasive
in applications of mathematics to problems in the sciences and
engineering. Among these are the Navier-Stokes
equations of fluid flow at high Reynolds number,
the drift-diffusion equations of semiconductor device physics
\cite{vanroosbroeck50,markowich:ringhofer:selberherr:lentini:83},
the Michaelis-Menten theory for enzyme reactions \cite{jdmurray},
and mathematical models of liquid crystal materials and
of chemical reactions \cite{chem}.
The use of classical numerical methods for solving such problems may give rise
to difficulties when the singular perturbation parameter $ \ve $ is small.
In particular, methods based on centered differences or upwinded
differences on uniform meshes yield error bounds, in the maximum norm,
which depend on an inverse power of $\ve$.
Similarly Brandt and Yavneh \cite{brandt-yavneh} demonstrated that
anisotropic artificial viscosity in the first-order upwind
finite-difference scheme may result in inaccurate solutions, when
$\ve /h = O(1)$, where $h$ is the mesh width. Two alternative
approaches may be taken to the resolution of this problem.
Either additional information
about the solution may be used to produce accurate efficient methods,
which may involve {\em a priori} modification of the mesh or operator,
or an attempt may be made to produce {\em a postiori} adaptive methods
or black box methods.
The latter approach leads to codes that are designed to handle
a wider variety of problems than non-adaptive codes, usually at the expense
of greater execution time. Moreover, such methods are less suitable
than non-adaptive codes
to implementation in a parallel environment. This is because the adaption
process
inherent in {\em a posteriori\ } methods, introduces sequentiality to
the solution process, which is absent in the {\em a priori\ } case.
The {\em a priori} approach uses physical or mathematical knowledge
about the problem to enhance the solution strategy.
Such methods are widespread in
the literature. These include fitted finite difference methods
\cite{DMS},
finite element methods using special elements such as exponential
elements
\cite{RST},
and methods which use {\em a priori} refined or special meshes
\cite{MOS}.
Examples of these include methods for convection-diffusion problems
devised by the British Central Electricity Generating Board
\cite{morton},
fluid flow in aerodynamics
\cite{elmistikawy-werle},
semiconductor device physics
\cite{scharfetter:gummel:69,fargart2,acm2,miller-song},
chemical reactions \cite{vfl1}, and hydrologic models for the Nash
cascade model of flood routing \cite{szollosi-nagy1}.
It is of theoretical and practical interest to consider
numerical methods for such problems, which
exhibit $ \ve $-uniform convergence,
that is, numerical methods for which
there exists an $N_0$, independent of $\ve$, such that for all $N \ge N_0$,
where $N$ is the number of mesh elements,
the error constant and rate of convergence in the maximum norm are
independent of $\ve$.
Thus a numerical method is said to be $\ve$-uniform of order $p$
on the mesh $\Omega _N = \{ x_i, i = 0,1,\ldots ,N \}$ if
there exists an $N_0$ independent of $\ve$ such that for all $N \ge N_0$
\[
\sup_{0<\ve\le 1} \max_{\Omega _N} \vert u(x) - u_N(x) \vert
\leq CN^{-p} ,
\]
where $u$ is the solution of the differential equation,
$u_N$ is the numerical approximation to
$u$, $C$ and $p > 0$ are independent of $\ve$ and $N$.
Singularly perturbed boundary value problems
for linear elliptic equations, which reduce for $\ve = 0$
to zero-order equations, were examined
in \cite{4,5,6,7,8}. For such problems
$\ve$-uniform methods consisting of
exponentially fitted finite difference operators
on uniform meshes were thoroughly investigated and
applied successfully to ordinary
differential equations in \cite{DMS,3} and to
linear partial differential equations
in \cite{4,5,6,7}.
A sufficient condition for $\ve$-uniform convergence, for
linear ordinary differential equations,
is that the
scheme be fitted with the appropriate constant fitting factor in the
region of the boundary layer. This was shown for the non-selfadjoint case
in \cite{farrell-ntp}.
Schemes with constant exponential
fitting factors (a special case of the frozen fitting factor
schemes considered in this paper) for
the linear self-adjoint problem were
considered in \cite[Chap. 10]{DMS}, and shown there to be
$\ve$-uniform.
%$\ve$-uniformly convergent.
The semilinear problem considered in this paper
exhibits an exponential boundary layer, which is
asymptotically similar in behavior to the layers arising in
self-adjoint linear ordinary differential equations.
It has been an area of speculation in the community, which
considers $\ve$-uniformly convergent methods, whether results
of the type available widely in the literature for linear problems
could also be obtained for nonlinear equations using
fitted finite difference methods on uniform meshes.
Previous attempts in this direction include schemes which are
$\ve$-uniformly convergent in weaker norms, such as the $\ell_1$ and
$\ell_2$ norm (cf. Niijima \cite{niijima}).
The key issue in this paper is to show that, even in the case of
this very simple nonlinearity, $\ve$-uniform convergence
cannot be achieved in the $\ell_\infty$ norm using
fitted finite difference methods on uniform meshes.
%the results for linear ordinary differential equations
%do not hold.
%do not carry over to this
%nonlinear case.
In this paper, it is shown
that a general class of fitted finite difference
methods on a uniform mesh, which includes well known exponentially
fitted finite difference methods \cite{DMS}, are not
$\varepsilon$-uniform pointwise in the maximum norm
for a singularly perturbed semilinear two point boundary value problem.
To be precise, in section 2, we shall prove this result for schemes with
a {\em frozen} fitting factor. The fitting
factor is said to be frozen if, at points $x_{i}$ in a
neighborhood of the boundary layer at $x=0$, it is determined by the
quantities given at $x=0$ alone. In section 3 numerical results are
given, which indicate that this result holds not only for
schemes with a frozen fitting factor but also for some standard
fitted schemes from the literature, the fitting factors of which are
not frozen.
It should be noted however that the result does not indicate that
fitted methods on non-uniform meshes cannot be
$\ve$-uniform.
In fact, in \cite{fmors1b}, numerical methods, $\ve$-uniform
in the maximum norm,
are constructed for a class of
semilinear problems, using classical finite difference operators on
special piecewise-uniform meshes.
Thus $\ve$-uniform methods can be constructed on special piecewise uniform
meshes even though it is not possible on uniform meshes.
\section{Theoretical Result for frozen fitting factors}
In this section the class ${\mathcal C}$ of semilinear two-point boundary value
problems
on $\Omega=(0,1)$ of the form
\[
(P) \ \ \ \left\{ \begin{array}{c}
\varepsilon^{2} u''(x) - c(u(x)) u (x) = 0, ~~ x \in \Omega ,\\
u(0) = 1, ~~~ u (1) = 0
\end{array}\right.
\]
are considered. Here $c$ is a smooth function satisfying
\renewcommand{\theequation}{2.1}
\begin{equation}
c(u(x)) \geq \alpha > 0, ~ x \in \Omega \label{2.1}
\end{equation}
and the singular perturbation parameter $\varepsilon$ satisfies
$\varepsilon > 0$.
When $\varepsilon << 1$ the solutions of such problems exhibit boundary
layers in small neighborhoods of the boundary point $x = 0$.
These boundary layers are the cause of significant numerical
difficulties, some consequences of which are given in the theorem below.
A finite difference method is considered on a uniform
mesh $\Omega_{N} = \{ x_{i} \}_{1}^{N-1}$, where
$x_{i} = ih, ~ 0 \leq i \leq N$ and $Nh = 1$. On this mesh the standard
second order central difference
operator $\delta_{x}^{2}$ is used to approximate the second order
derivative, where $\delta_{x}^{2}$ is defined by:
\[
\delta_{x}^{2} w(x_{i}) =
\frac{w (x_{i+1}) - 2w(x_{i}) + w(x_{i-1})}{h^{2}},
\]
for any mesh function $w$.
The discrete problem corresponding to
continuous problem ($P$) is then
\[
(P_h)\ \ \ \left\{ \begin{array}{l}
\varepsilon^{2} \gamma_{i} \delta_x^{2} z(x_{i}) - c(z (x_{i})) z(x_{i})
= 0, ~~~ x_{i} \in \Omega_{N} ,\\
~~~~~~ \\
z(0) = 1, ~~~z(1) = 0,
\end{array}\right.
\]
where $\gamma_{i}$ is the fitting factor. In general, the
fitting factor $\gamma_{i}$ is determined at each
point $x_{i} \in \Omega_{N}$ by the quantities $\varepsilon, h,
c(z (x_{i-1})), c(z (x_{i}))$ and $c(z (x_{i+1}))$. The fitting
factor is said to be frozen if, at points $x_{i}$ in a
neighborhood of the boundary layer at $x=0$, it is determined by the
quantities $\varepsilon, h, c(z(0))$ alone.
The main theoretical result of this paper states that
there is no fitted central finite difference method $(P_h)$
with a frozen fitting
factor on a uniform mesh, whose solutions converge $\varepsilon$-uniformly
to the solution of problem ($P$). \\
\begin{theorem}
Let $u$ be the solution of any problem ($P$) in
the class ${\mathcal C}$
and $z$ the solution of the corresponding discrete problem ($P_h$) on
the uniform mesh $\Omega_{N}$. Assume that the fitting factor $\gamma$
depends continuously on its arguments and that it is frozen so that
for all $x \in [0,1/4), \gamma (x) = \gamma (\varepsilon, h, c(0))$.
Then, there is no choice of the fitting factor $\gamma$ for which
the solutions of ($P_h$) converge $\varepsilon$-uniformly to the
solution $u$ of ($P$), as $N \rightarrow \infty$ for all
problems ($P$) in ${\mathcal C}$. \\
\end{theorem}
\begin{proof}
The theorem is proved by assuming that it is false and
then deriving a contradiction. Thus, it is assumed that for
all problems ($P$) in ${\mathcal C}$, there is an $\varepsilon$-uniform fitted
finite difference method ($P_h$) on the
uniform mesh $\Omega_{N}$, with a frozen fitting factor such that
$\gamma (x) = \gamma (\varepsilon, h, c(0))$ for
all $x \in [0,1/4)$, with $\gamma$ depending continuously on its
arguments. That is, there exists
$\mu = \mu (h)$, independent of $\varepsilon$, such that
$|u(x_{i}) - z(x_{i})| \leq
\mu (h)$ where $\mu (h) \rightarrow 0$ as $h \rightarrow 0$.
Under these assumptions it will be shown that for any choice of the fitting
factor the error at the point $x_{1}$, namely $u(x_{1}) - z (x_{1})$, does not
converge to zero as $N\rightarrow \infty$ for a sequence of
problems in ${\mathcal C}$ for which $\varepsilon N$ is held constant.
This provides the required contradiction.
It suffices to consider problems in ${\mathcal C}$ corresponding to the following
two choices of the coefficient $c$,
\renewcommand{\theequation}{2.2}
\begin{equation}
c = c_{s} (u(x)) = 2 - s + s u(x), ~~ s = 0, 1.
\end{equation}
The corresponding solutions of ($P$) and ($P_h$) are
denoted by $u_{s}$ and $z_{s}$ respectively. It will be shown
that either $u_{0} (x_{1}) - z_{0} (x_{1})$ or
$u_{1} (x_{1}) - z_{1} (x_{1})$
does not converge to zero as $N\rightarrow \infty$ for the sequence of
problems with $\varepsilon N = 1$.
It is clear that the coefficient in (2.2) fulfills condition (2.1) for
the linear problem corresponding to $s=0$. That the same is true when
$0 < s \leq 1$ may be verified by a standard argument using the
maximum principle.
It is more convenient to work with the following auxiliary problems in
the semi-infinite domain $[0, \infty)$ :
\[
\varepsilon^{2} v_{s}^{''} (x) - c_{s} (v_{s} (x))v_{s} (x) = 0, ~~~~~
x \in [0,\infty) ,
\]
\[
v_{s} (0) = 1, ~~~~~ v_{s} (\infty) = 0.
\]
The exact solution of the linear problem
corresponding to $s=0$ is $v_{0} (x) = e^{-\sqrt{2}\varepsilon^{-1} x}$.
Again, using a standard maximum principle argument, it is not hard to
show that for $0 < s \leq 1$ the solutions $v_{s} (x)$ satisfy
\renewcommand{\theequation}{2.3}
\begin{equation}
e^{-\sqrt{2} \varepsilon^{-1}x} \leq
v_{s} (x) \leq e^{-\sqrt{2-s} \;\varepsilon^{-1} x}, x \in
[0,\infty).
\end{equation}
Moreover, on the interval $\bar{\Omega} = [0,1]$, the
difference between the solution $u_{s}$ and the solution $v_{s}$
of the corresponding auxiliary problem decreases as $\varepsilon \rightarrow 0$
in the sense that
\renewcommand{\theequation}{2.4}
\begin{equation}
|u_{s} (x) - v_{s} (x) | \leq \nu(\varepsilon) , ~~
x \in \bar{\Omega}, ~ s = 0, 1,
\end{equation}
where $\nu (\varepsilon) \rightarrow 0$ as $\varepsilon \rightarrow 0$.
Changing from $x$ to the new variable $\eta(x) = x/\varepsilon$ the
auxiliary problems become
\[
w_{s}'' (\eta) - c_{s} (w_{s} (\eta)) w_{s} (\eta) = 0, ~~
\eta \in [0, \infty),
\]
\[
w_{s} (0) = 1, ~~~ w_{s} (\infty) = 0,
\]
where $w_{s} (\eta) = v_{s} (x)$.
Letting $w_{s} (\eta)$ denote the solution of this problem,
the function $\beta (s)$ is chosen to satisfy the
difference equation
\[
\beta (s) \delta_{\eta}^{2} w_{s} (\eta) - c_{s}
(w_{s} (0)) w_{s} (\eta) = 0,
\]
at the point $\eta = \varepsilon^{-1}h$. Since
$w_{s} (0) = 1$ it follows that $c_{s} (w_{s}(0)) = c_{s} (1) = 2$
for all $ 0 \leq s \leq 1$ and $\beta (s)$ satisfies
\renewcommand{\theequation}{2.5}
\begin{equation}
\beta (s) [\delta_{\eta}^{2} w_{s} (\eta)]_{\eta = \ell} -
2 w_{s} (\ell) = 0,
\end{equation}
where $\ell = \varepsilon^{-1} h$.
To obtain an approximate expression for $\beta (s)$ as a power series in
$\ell$, the function $w_{s} (\eta)$ is expanded as a power series in $\eta$
\renewcommand{\theequation}{2.6}
\begin{equation}
w_{s} (\eta) = 1 + \dis{\sum_{r=1}^{\infty}} ~
k_{r} (s)\eta^{r}.
\end{equation}
Then, by (2.5) and (2.6),
\renewcommand{\theequation}{2.7}
\begin{eqnarray}
\beta (s) & =
& \frac{2(1+\sum_{r=1}^{\infty} k_{r} (s) \ell^{r})}{\sum_{r=2}^{\infty}
k_{r} (s) [\delta_{\eta}^{2} \eta^{r}]_{\eta = \ell}} \\
&\ &\ \nonumber \\
& =
& \frac{1+k_{1} (s) \ell + k_{2} (s) \ell^{2} + 0 (\ell^{3})}{k_{2} (s) +
3k_{3} (s) \ell + 7k_{4} (s) \ell^{2} + 0(\ell^{3})}.
\end{eqnarray}
Expressions for the functions $k_{r} (s)$ are now obtained by
substituting the expansion of $w_{s} (\eta)$ into the differential
equation satisfied by $w_{s} (\eta)$, namely
\[
w_{s}'' (\eta) =
(2-s+s w_{s} (\eta)) w_{s} (\eta),
\]
and so
\[
\left(
\sum_{r=0}^{\infty} k_{r} (s) \eta^{r}\right)'' = \left(
2-s+s\sum_{r=0}^{\infty} k_{r} (s) \eta^{r}\right) \left(
\sum_{r=0}^{\infty} k_{r} (s) \eta^{r} \right).
\]
Equating
coefficients of $1, \eta$ and $\eta^{2}$ on both sides and simplifying,
the relations
\renewcommand{\theequation}{2.8}
\begin{equation}
k_{2} (s) = 1, ~~~
6k_{3} (s) = (2+s) k_{1} (s), ~~ 12k_{4} (s) = 2+s+s k_{1}^{2} (s)
\end{equation}
are obtained.
Writing $w_{s} (2\ell) = e^{-{\kappa} (s)\ell}$, for some ${\kappa} (s)$, and
expanding as a power series in $\ell$, we obtain
\[
w_{s} (2\ell) = 1 +
\sum_{r=1}^{\infty} (-1)^{r} \frac{{\kappa} ^{r} (s)\ell^{r}}{r!}.
\]
Comparing this with the expansion (2.6) when $\eta = 2\ell$, and using
$k_{2} (s) = 1$ from (2.8), it follows that
\renewcommand{\theequation}{2.9}
\begin{equation}
k_{1}(s) = -\frac{{\kappa} (s)}{2} +
\left( \frac{{\kappa} ^{2}(s)}{4} -2\right)\ell + O (\ell^{2}).
\end{equation}
Using
(2.8) and (2.9), the expression (2.7) for $\beta (s)$ becomes
\begin{eqnarray*}
\beta (s) & =
& \lfrac{1+[-\frac{{\kappa} (s)}{2} + (\frac{{\kappa} ^{2}(s)}{4} - 2) \ell]
\ell +
\ell^{2} + O (\ell^{3})}{1+\frac{1}{2} (2+s) [ -\frac{{\kappa} (s)}{2} +
(\frac{{\kappa} ^{2}(s)}{4} - 2)\ell] \ell + \frac{7}{12}
[2 + s + s \frac{{\kappa} ^{2}(s)}{4} ] \ell^{2} + O (\ell^{3})} \\
&\ &\ \\
& = & [1-\frac{{\kappa} (s)\ell}{2} + (-1+\frac{{\kappa} ^{2}(s)}{4})\ell^{2}
+ O (\ell^{3})] [ 1 - \frac{(2+s){\kappa} (s)\ell}{4} + \\
& & \hspace{.5in}+ (-\frac{5(2+s)}{12} + \frac{(12+13s) {\kappa} ^{2}(s)}{48})
\ell^{2} + O (\ell^{3})]^{-1} \\
&\ &\ \\
& = & [1-\frac{{\kappa} (s)\ell}{2} + (-1+\frac{{\kappa} ^{2}(s)}{4})\ell^{2}
+ O (\ell^{3})] [1 + \frac{(2+s){\kappa} (s)\ell}{4} - \\
& & \hspace{.5in}- (-\frac{5(2+s)}{12} + \frac{(12+13s){\kappa} ^{2}
(s)}{48})\ell^{2} +
\frac{(2+s)^{2}{\kappa} ^{2} (s) \ell^{2}}{16} + O (\ell^{3})] \\
&\ &\ \\
& = & 1 - \frac{\ell^{2}}{6} + \frac{\ell}{4} \left\{ s{\kappa} (s) + \ell
[\frac{5s}{3} + {\kappa} ^{2}(s) (-1 -\frac{13s}{12} + 1 + s +
\frac{s^{2}}{4} -1 -\frac{s}{2} + 1 )]\right\} \\
& & \hspace{.5in}+ O (\ell^{3}) \\
&\ &\ \\
& =
& 1 - \frac{\ell^{2}}{6} + \frac{\ell s}{4} \left\{ {\kappa} (s) + \ell
[ \frac{5}{3} - \frac{7 {\kappa} ^{2}(s)}{12} +
\frac{s{\kappa} ^{2}(s)}{4}]\right\} + O (\ell^{3}) \\
&\ &\ \\
& = & \bar{\beta} (s) + O (\ell^{3}),
\end{eqnarray*}
where
\renewcommand{\theequation}{2.10}
\begin{equation}
\bar{\beta} (s) = 1 -
\frac{\ell^{2}}{6} + \frac{\ell s}{4}\left\{ {\kappa} (s) + \ell
[ \frac{5}{3} - \frac{7}{12} {\kappa} ^{2} (s) +
\frac{s{\kappa} ^{2}(s)}{4}]\right\}.
\end{equation}
The following bounds for ${\kappa} (s)$ are obtained from (2.3)
\renewcommand{\theequation}{2.11}
\begin{equation}
2\sqrt{2-s} \leq {\kappa} (s) \leq 2\sqrt{2}.
\end{equation}
>From (2.10)
\begin{eqnarray*}
\bar{\beta} (0) & = & 1-\frac{\ell^{2}}{6}, \\
&\ &\ \\
\bar{\beta} (1) & = & 1-\frac{\ell^{2}}{6} +
\frac{\ell}{4} \left\{ {\kappa} (1) + \ell \left[ \frac{5}{3} - \frac{7}{12}
{\kappa} ^{2} (1) + \frac{{\kappa} ^{2}(1)}{4}\right] \right\}
\end{eqnarray*}
and so
\[
\bar{\beta} (1) - \bar{\beta} (0) = \frac{{\kappa} (1) \ell}{4} + O (\ell^{2}).
\]
It follows from (2.11) that ${\kappa} (1) \geq 2$ and thus, for all
sufficiently small $\ell$,
\renewcommand{\theequation}{2.12}
\begin{equation}
\bar{\beta} (1) - \bar{\beta} (0) \geq \frac{\ell}{4}.
\end{equation}
Putting
\renewcommand{\theequation}{2.13}
\begin{equation}
\beta^{\ast} = \bar{\beta} (0) + \frac{\ell}{8},
\end{equation}
the difference equation (2.5) can be rewritten as
\[
\frac{\beta^{\ast}}{2(\beta^{\ast} + \ell^{2})} ( w_{s} (0) + w_{s} (2\ell))
= w_{s} (\ell) + \frac{(\beta^{\ast} - \beta (s))\ell^{2}}{\beta (s) (\beta^{\ast}
+ \ell^{2})} w_{s} (\ell).
\]
Then, for all sufficiently small $\ell$ and some constant
$m_{0} > 0$, from (2.10), (2.13) and (2.3), it follows that
\renewcommand{\theequation}{2.14}
\begin{equation}
\frac{\beta^{\ast}}{2(\beta^{\ast} + \ell^{2})}
(w_{0} (0) + w_{0} (2\ell)) \geq w_{0} (\ell) + m_{0} \ell^{3}
\end{equation}
and, from (2.10), (2.12), (2.13) and (2.3),
\renewcommand{\theequation}{2.15}
\begin{equation}
\frac{\beta^{\ast}}{2(\beta^{\ast} + \ell^{2})} (w_{1} (0) + w_{1} (2\ell))
\leq w_{1} (\ell) - m_{0} \ell^{3} .
\end{equation}
Consider the difference scheme ($P_h$), with frozen fitting factor
$\tilde{\gamma}$, applied to problems ($P$) with the coefficient $c_{s}$
that is
\[
(\tilde{P}_h) \ \ \left\{ \begin{array}{l}
\varepsilon^{2} \tilde{\gamma} \delta_x^{2} z_s(x_{i}) - c_s(z_s (x_{i}))
z_s(x_{i}) = 0, ~~~ x_{i} \in \Omega_{N}, \\
~~~~~~ \\
z_s(0) = 1, ~~~z_s(1) = 0,
\end{array}\right.
\]
At the point $x_{1}$, this can be written in the form
\[
z_{s} (x_{1}) = \frac{\tilde{\gamma}}{2\tilde{\gamma} + \ell^{2}
(2-s+sz_{s} (x_{1}))} (z_{s} (x_{2}) + z_{s} (0)),
\]
where $\tilde{\gamma} = \tilde{\gamma} ( \varepsilon, h,
c_{s} (z_{s} (0)) = \tilde{\gamma} (\varepsilon, h, c_{s} (1)) =
\tilde{\gamma} (\varepsilon, h, 2)$.
Note that
$\tilde{\gamma}$
is independent of $s$. We first show that $\eps$-uniform convergence implies
$\tilde{\gamma} \ge 0$ for all sufficiently small $h$ and $\eps$, satisfying
$l=h/\eps = \mbox{const}$.
>From $\tilde{P}_h$, it is clear that
$\tilde{\gamma} \ge 0$, if
$z_0(x_1) \ge 0$ and $\delta_x^2 z_0(x_1) > 0$.
By the maximum principle, $u_0(x) \ge 0 , x \in (0,1/4]$.
Now, by uniform convergence,
\[
| u_s(x) - z_s(x) | \le \mu(h),
\]
where $\mu (h) \rightarrow 0$ as $h \rightarrow 0$.
Hence
\[
z_s(x_i) \ge 0 , \ \ x_i \in (0,1/4],\ \ \ h \ \mbox{\rm sufficiently small}.
\]
It remains to show $\delta_x^2 z_0(x_1) \ge 0$.
To do this rewrite the equations for $u_s$ and $z_s$ in scaled coordinates
$\eta(x) = x/\varepsilon$ thus:
\[
\frac{d^2}{d\eta^2} \tilde{u}_s(\eta) - c_s(\tilde{u}_s(\eta_i ) )
\tilde{u}_s(\eta) = 0,
\]
\[
\tilde{u}_s(0) = 1\ \ , \ \ \tilde{u}_s(1/\eps) = 0\ ,
\]
and
\[
\tilde{\gamma} \delta_\eta^2 \tilde{z}_s(\eta_i) - c_s(\tilde{z}_s(\eta_i))
\tilde{z}_s(\eta_i) = 0,
\]
\[
\tilde{z}_s(0) = 1\ \ , \ \ \tilde{z}_s(1/\eps) = 0\ .
\]
Recalling $l = h/\eps$, we have
\renewcommand{\theequation}{2.16}
\begin{eqnarray}
\delta_\eta^2 \tilde{z}_s(\eta_i) & = & \{ \tilde{z}_s(\eta_i+l) - 2
\tilde{z}_s(\eta_i) + \tilde{z}_s(\eta_i-l) \} / l^2 \nonumber
\\
&= & \{ \tilde{z}_s(\eta_i+l) - \tilde{u}_s(\eta_i+l) \}/l^2
- 2 \{ \tilde{z}_s(\eta_i) - \tilde{u}_s(\eta_i) \}/l^2 \nonumber
\\
&&+ \{ \tilde{z}_s(\eta_i-l) - \tilde{u}_s(\eta_i-l) \} / l^2
+ \delta_\eta^2 \tilde{u}_s(\eta_i) \nonumber
\\
& \ge & - 4 \mu(h)/l^2 + \delta_\eta^2 \tilde{u}_s(\eta_i). \label{2.16}
\end{eqnarray}
Now
\renewcommand{\theequation}{2.17}
\begin{equation}
\delta_\eta^2 \tilde{u}_s(\eta_i) =
\frac{d^2}{d\eta^2} \tilde{u}_s(\eta_i + \theta_i)
= c_s(\tilde{u}_s ) \tilde{u}_s(\eta_i + \theta_i)
\ge \alpha \tilde{u}_s(\eta_i + \theta_i),\ \ \ 0\le\theta_i\le l .
\label{2.17}
\end{equation}
Using (2.4), we have
\renewcommand{\theequation}{2.18}
\begin{equation}
| \tilde{u}_s(\eta) - w_s(\eta) | =
| u_s(x) - v_s(x) | \le \nu(\eps), \label{2.18}
\end{equation}
where $\nu (\varepsilon) \rightarrow 0$ as $\varepsilon \rightarrow 0$.
We now restrict our attention to the case required, that is $u_0(x)$.
We have
\[
w_0(\eta_1+\theta _1) = v_0(x_1/\eps~+~\theta _1) \le v_0(2h/\eps) = v_0(2l) = e^{-2\sqrt{2} l}.
\]
Using (\ref{2.18}), we now have
\[
\tilde{u}_0(\eta_1+\theta _1) \ge w_0(\eta_1+\theta _1) - \nu(\eps) = e^{-2\sqrt{2} l} - \nu(\eps).
\]
Thus, for $\eps$ sufficiently small,
\renewcommand{\theequation}{2.19}
\begin{equation}
\tilde{u}_0(\eta_1+\theta_1) \ge e^{-4l} > 0. \label{2.19}
\end{equation}
Hence, from (\ref{2.16}), (\ref{2.17}) and (\ref{2.19}), since $l$ is a
constant, we have
\[
\delta_\eta^2 \tilde{z}_0(\eta_1)
\ge - 4 \mu(h)/l^2 + e^{-4l} > 0.
\]
for $h$ sufficiently small, and the required result,
$\tilde{\gamma} \ge 0$ follows.
The argument is now divided into the two possible cases
$\tilde{\gamma} \geq \beta^{\ast}$ and $\tilde{\gamma} \leq \beta^{\ast}$.
Suppose first that $\tilde{\gamma} \geq \beta^{\ast}$ and consider problems
corresponding to $s = 0$. Then, for all sufficiently small $\ell$, using the
assumption of $\varepsilon$-uniform convergence, (2.4) and (2.14) we obtain
\begin{eqnarray*}
z_{0} (x_{1}) & = & \frac{\tilde{\gamma}}{2(\tilde{\gamma} + \ell^{2})} (z_{0}(x_{2}) +
z_{0}(0)) \\
&\ &\ \\
& \geq & \frac{\tilde{\gamma}}{2(\tilde{\gamma} + \ell^{2})} (u_{0} (x_{2}) +
u_{0} (0) - \mu (h)) \\
&\ &\ \\
& \geq & \frac{\tilde{\gamma}}{2(\tilde{\gamma} + \ell^{2})} (v_{0}
(x_{2}) + v_{0} (0) - \nu(\varepsilon) - \mu (h)) \\
&\ &\ \\
& = & \frac{\tilde{\gamma}}{2(\tilde{\gamma} + \ell^{2})} (w_{0} (x_{2}) +
w_{0} (0) - \nu(\varepsilon) - \mu (h)) \\
&\ &\ \\
& \geq & \frac{\tilde{\gamma}}{2(\tilde{\gamma} + \ell^{2})}
\left[ \frac{2(\beta^{\ast} + \ell^{2})}{\beta^{\ast}} (w_{0} (\ell) +
m_{0} \ell^{3}) - \nu(\varepsilon) - \mu (h)\right] \\
&\ &\ \\
& \geq & w_{0} (\ell) + m_{0} \ell^{3} - \nu(\varepsilon) - \mu (h) \\
&\ &\ \\
& \geq & u_{0} (x_{1}) + m_{0}\ell^{3} - \nu(\varepsilon) - \mu (h),
\end{eqnarray*}
since $\tilde{\gamma} \geq \beta^{\ast}$ implies that
$\frac{\tilde{\gamma} (\beta^{\ast} + \ell^{2})}{(\tilde{\gamma} +
\ell^{2})\beta^{\ast}} \geq 1$.
Fixing $\ell$ sufficiently small, and considering the sequence of
problems corresponding to $\varepsilon = \frac{h}{\ell} = \frac{1}{N\ell}$, it
follows that
\[
z_{0} (x_{1}) - u_{0} (x_{1}) \geq \frac{1}{2} m_{0} \ell^{3},
\]
for all sufficiently small $h$, which
contradicts the assumption of $\varepsilon$-uniform convergence of the method
for these problems.
On the other hand if $\tilde{\gamma} \leq \beta^{\ast}$,
using the assumption of $\varepsilon$-uniform convergence, (2.4) and (2.15), a
similar argument for problems corresponding to $s=1$ gives for all
sufficiently small $\ell$ that
\begin{eqnarray*}
z_{1} (x_{1}) & = & \frac{\tilde{\gamma}}{2\tilde{\gamma} + \ell^{2}
(1+z_{1} (x_{1}))} (z_{1} (x_{2}) + z_{1} (0)) \\
&\ &\ \\
& \leq & \frac{\tilde{\gamma}}{2\tilde{\gamma} + \ell^{2}
(1+z_{1} (x_{1}))}
\left[ \frac{2(\beta^{\ast} + \ell^{2})}{\beta^{\ast}}
(w_{1} (\ell) - m_{0} \ell^{3} ) +
\nu(\varepsilon) + \mu (h)\right] \\
&\ &\ \\
& \leq & u_{1} (\ell) - m_{0} \ell^{3} + \nu(\varepsilon) + \mu (h),
\end{eqnarray*}
since $\tilde{\gamma} \leq \beta^{\ast}$ implies that
$\frac{\tilde{\gamma} (\beta^{\ast} + \ell^{2})}{(\tilde{\gamma} +
\ell^{2})\beta^{\ast}} \leq 1$.
Again, fixing $\ell$ sufficiently small, and considering the
sequence of problems corresponding to
$\varepsilon = \frac{h}{\ell} = \frac{1}{N\ell}$,
it follows that
\[
u_{1} (x_{1}) - z_{1} (x_{1}) \geq
m_{0} \ell^{3} - \nu(\varepsilon) - \mu (h)
\geq \frac{1}{2} m_{0} \ell^{3},
\]
for all sufficiently small $h$, which contradicts the assumption of
$\varepsilon$-uniform convergence.
Thus it has been shown that the assumption of $\varepsilon$-uniform
convergence leads to a contradiction in all cases, which completes
the proof of the theorem.
\end{proof}
\section{Numerical Results}
We shall now examine numerically a number of fitted schemes on uniform
meshes for the continuous problem ($P$) and related problems with
two boundary layers. We shall first consider schemes of the form
\[
(P_h)\ \ \ \left\{ \begin{array}{l}
\varepsilon^{2} \gamma_{i} \delta_x^{2} z_N(x_{i}) - c(z_N (x_{i})) z_N(x_{i})
= 0, ~~~ x_{i} \in \Omega_{N}, \\
~~~~~~ \\
z_N(0) = 1, ~~~z_N(1) = 0 ,
\end{array}\right.
\]
where $\gamma_{i} \equiv \gamma(\varepsilon, h, c(z_N(0)))$
is the frozen fitting factor.
\par
The nonlinear finite difference method
$(P_h)$
is linearized using a continuation method of the form :
$$
L^h_t u_N \equiv \ve ^2 \gamma_{i} \delta^2_x u_N(x,t_j) -
c(u_N (x,t_{j-1})) u_N(x,t_j) -
D^-_t u_N(x, t_j) = 0, \ j = 1, \ldots K,$$
$$u_N(0, t_j) = u(0) , \
u_N(1, t_j) = u(1) \ {\rm for \ all\ } j ,
$$
$$
u_N(x,0) = u_{init}(x),$$
where $\gamma_{i} \equiv \gamma(\varepsilon, h, c(u_N(0,t_{j-1}))) \equiv
\gamma(\varepsilon, h, c(u(0)))$ is the frozen fitting factor.
Various starting values $u_{init}(x)$ are chosen. The number of
iterations $K$ and the choice of uniform time step
$h_t = t_j - t_{j-1} $ are discussed below. With the definition
$$
e (j) \equiv \max _{1 \leq i \leq N} \vert u_N(x_i,t_j)- u_N(x_i,t_{j-1})
\vert / h_t, \ {\rm for\ } j =1,2, \ldots , K ,
$$
the time step $h_t$ is chosen sufficiently small so that
\renewcommand{\theequation}{3.1}
\begin{equation}
e (j) \leq e (j-1), \ {\rm for\ } 1 < j \leq K, \label{n1}
\end{equation}
and the number of iterations $K$ is chosen such that
\renewcommand{\theequation}{3.2}
\begin{equation}
e (K) \leq \ {\rm TOL \ }, \label{n2}
\end{equation}
where TOL is some prescribed small tolerance.
\par\noindent
The numerical solution is obtained as follows:
\par
Start with $h_t =0.0625$. If, at some value of $j$,
(\ref{n1}) is not satisfied
then halve the time step
until (\ref{n1}) is satisfied.
Continue the
iterations until either (\ref{n2}) is satisfied or until $K=90$.
If (\ref{n2}) is not
satisfied, then repeat the entire process starting with
$h_t = 0.03125$ .
The resulting values of $u_N(x,K)$ are taken as approximations to the solution of the
continuous problem.
The problem is solved on a sequence of meshes, with $N = 8$, $16$, $32$, $64$,
$128$, $256$, $512$, $1024$ and for $\ve = 2^{-n}, n=1,2,\ldots j_{red}$,
where $j_{red}$ is chosen so that $\varepsilon$ is a value at which the rate
of convergence stabilizes, which normally occurs when, to machine
accuracy, we are solving the reduced problem.
The errors $|u_N(x_i,K) - u(x_i)|$ are approximated on each mesh
for successive values of $\ve$ by
$e_{\ve ,N}(i)=|u_N(x_i,K)-u^I(x_i,K)|$, where $u^I(x,K)$ is
defined by linear interpolation on each subinterval $[y_{j-1},y_j]$ by
$$
u^I(x,K) = u^*(y_{j-1},K)+ (u^*(y_j,K) - u^*(y_{j-1},K)) \frac{x - y_{j-1}}
{y_j - y_{j-1}},\ 1 \leq j \leq 1024,
$$
where the nodal values $ \{ u^*(y_j,K) \} _{j=0}^{1024} $ are
obtained from the solution of the finite difference method
$ L^h_t $ with $N=1024$.
For each $\ve$ and each $N$ the maximum nodal error is approximated by
$$
E_{\ve,N}=\max_{i}e_{\ve ,N}(i).
$$
For each $N$, the $\ve$-uniform maximum nodal error is approximated by
$$
E_{N}=\max_{\ve}E_{\ve,N}.
$$
A numerical method for solving $(P)$ is $\ve$-uniform of order $p$
on the mesh $\Omega _N = \{ x_i, i = 0,1,\ldots ,N \}$ if
$$
\sup_{0<\ve\le 1} \max_{\Omega _N} \vert u(x) - u_N(x,K) \vert \leq CN^{-p} ,
$$
where $u$ is the solution of $(P)$, $u_N$ is the numerical approximation to $u$,
$C$ and $p > 0$ are independent of $\ve$ and $N$.
An approximation to $p$, the
$\ve $-uniform rate of convergence, was determined using a variation of the
double mesh method described in \cite{farheg}.
This involves calculating the double mesh error
$$
D_{\ve ,N}= \max _{\Omega _N}|u_N(x_i,K)-u^I_{2N}(x_i,K)| ,
$$
which is the difference between the values of the solution on
a mesh of $N$ points and the interpolated value for the
solution, at the same point, on a mesh of $2N$ points.
For each value of $N$ the quantities
$$
D_{N}=\max_{\ve}D_{\ve,N} , \quad
p_{N}=\log_2\left( \frac{D_{N}}{D_{2N}} \right),
$$
are computed.
The values of $p_N$ are the approximations to $p$.
We now present numerical results, firstly for a
problem of the form $(P)$, and secondly for a generalization of that
problem. All calculations were carried out in
double-precision FORTRAN 77 on an Hewlett-Packard/Apollo 730.
The first scheme we consider is the unfitted
central difference scheme, where $\gamma_i \equiv 1$, and the second
is a modification to the single layer case of the
constant fitting factor version of the scheme of Miller
\cite{miller1} proposed in \cite[Ch. 10, p. 156]{DMS}. This gives
a frozen fitting factor method for the problem ($P$) with boundary layer
at $x=0$. The frozen fitting factor is given by:
\renewcommand{\theequation}{3.3}
\begin{equation}
\gamma_{i} = \frac{ c(u (0))h^2}{4\ve}
\mbox{\rm sinh}^{-2} \frac{\sqrt{c(u (0))}h}{2\sqrt{\ve}}. \label{frfit1}
\end{equation}
\par
Table \ref{t3.1} gives uniform errors and rates of uniform convergence for the
centered difference method for the problem
\renewcommand{\theequation}{3.4}
\begin{equation}
\ve ^ 2 { \frac{d ^ 2}{d x ^ 2} } u ( x ) - u - u^2 = 0 ,
\ x \in (0,1), \label{test1}
\end{equation}
$$u(0) = 1, \quad u(1) = 0.$$
\begin{table}[hbt]
\centering
\begin{tabular}{|l|r|r|r|r|r|r|} \hline
\multicolumn{7}{||c||}{Boundary Conditions: \ \ $u(0) = 1.0\ , \ u(1) = 0.0$}\\
\multicolumn{7}{||c||}{Initial Guess : \ \ \ \ \ \ \ $u_{init} = u(x) = u(0) + ( u(1) - u(0) )x$} \\ \hline
$N$ & 8 & 16 & 32 & 64 & 128 & 256 \\ \hline
$E_N$ & .048205 & .048192 & .048137 & .047921 & .047143 & .043894\\ \hline
$p_N$ & .00 & .00 & .00 & .00 & .00 & .00\\ \hline\end{tabular}
\caption{Maximum errors $E_N$ and rate of convergence $p_N$ for the Centered Difference scheme}
\label{t3.1}
\end{table}
Table \ref{t3.2} and \ref{t3.3} give errors and rates of uniform convergence
for the scheme with frozen fitting factor given by (\ref{frfit1}).
\begin{table}[hbt]
\centering
\begin{tabular}{||r||r|r|r|r|r|r|r||} \hline
\multicolumn{8}{||c||}{Boundary Conditions: \ \ $u(0) = 1.0\ , \ u(1) = 0.0$}\\
\multicolumn{8}{||c||}{Initial Guess : \ \ \ \ \ \ \ $u_{init} = u(x) = u(0) + ( u(1) - u(0) )x$} \\ \hline
& \multicolumn{7}{c||}{Number of Mesh Points $N$} \\ \cline{2-8}
$\epsilon$ & 8 & 16 & 32 & 64 & 128 & 256 & 512 \\ \hline
1/ 2 & .000464 & .000119 & .000030 & .000007 & .000002 & .000000 & .000000 \\
1/ 4 & .000818 & .000211 & .000054 & .000013 & .000003 & .000001 & .000000 \\
1/ 8 & .001462 & .000391 & .000101 & .000025 & .000006 & .000001 & .000000 \\
1/ 16 & .002616 & .000746 & .000196 & .000049 & .000012 & .000003 & .000001 \\
1/ 32 & .003747 & .001435 & .000384 & .000098 & .000024 & .000006 & .000001 \\
1/ 64 & .003195 & .002608 & .000744 & .000195 & .000049 & .000012 & .000002 \\
1/ 128 & .003579 & .003623 & .001434 & .000383 & .000097 & .000023 & .000005 \\
1/ 256 & .008040 & .003184 & .002508 & .000742 & .000193 & .000046 & .000009 \\
1/ 512 & .011498 & .003579 & .003613 & .001430 & .000379 & .000093 & .000019 \\
1/ 1024 & .007142 & .008042 & .003175 & .002495 & .000733 & .000184 & .000037 \\
1/ 2048 & .001996 & .011497 & .003577 & .003595 & .001338 & .000360 & .000074 \\
1/ 4096 & .000231 & .007139 & .008043 & .003151 & .002457 & .000696 & .000147 \\
1/ 8192 & .000009 & .001994 & .011491 & .003570 & .003531 & .001263 & .000287 \\
1/ 16384 & .000000 & .000231 & .007127 & .008044 & .003066 & .002312 & .000461 \\
1/ 32768 & .000000 & .000009 & .001985 & .011461 & .003539 & .003287 & .000976 \\
1/ 65536 & .000000 & .000000 & .000227 & .007066 & .007966 & .002862 & .001851 \\
1/ 131072 & .000000 & .000000 & .000009 & .001948 & .011293 & .003372 & .002515 \\
1/ 262144 & .000000 & .000000 & .000000 & .000217 & .006867 & .007867 & .001835 \\
1/ 524288 & .000000 & .000000 & .000000 & .000008 & .001811 & .010684 & .002762 \\
1/ 1048576 & .000000 & .000000 & .000000 & .000000 & .000179 & .006060 & .006746 \\
1/ 2097152 & .000000 & .000000 & .000000 & .000000 & .000005 & .001360 & .007922 \\
1/ 4194304 & .000000 & .000000 & .000000 & .000000 & .000000 & .000094 & .003552 \\
1/ 8388608 & .000000 & .000000 & .000000 & .000000 & .000000 & .000001 & .000540 \\
1/ 16777216 & .000000 & .000000 & .000000 & .000000 & .000000 & .000000 & .000023 \\
1/ 33554432 & .000000 & .000000 & .000000 & .000000 & .000000 & .000000 & .000000 \\
\hline $E_N$ & .011498 & .011497 & .011491 & .011461 & .011293 & .010684 & .007922\\ \hline
\end{tabular}
\caption{Errors $E_{\ve,N}$ and $E_N$ for Frozen Fitting Factor scheme
(\protect{\ref{frfit1}})}
\label{t3.2}
\end{table}
\begin{table}[hbt]
\centering
\begin{tabular}{|l|r|r|r|r|r|r|} \hline
\multicolumn{7}{||c||}{Boundary Conditions: \ \ $u(0) = 1.0\ , \ u(1) = 0.0$}\\
\multicolumn{7}{||c||}{Initial Guess : \ \ \ \ \ \ \ $u_{init} = u(x) = u(0) + ( u(1) - u(0) )x$} \\ \hline
$N$ & 8 & 16 & 32 & 64 & 128 & 256 \\ \hline
$E_N$ & .011498 & .011497 & .011491 & .011461 & .011293 & .010684\\ \hline
$p_N$ & .00 & .00 & .00 & .00 & .00 & .00\\ \hline\end{tabular}
\caption{Maximum errors $E_N$ and rate of convergence $p_N$ for Frozen Fitting Factor scheme (\protect{\ref{frfit1}})}
\label{t3.3}
\end{table}
We now show numerically that the result of Theorem 2.1 also holds
for fitted methods with frozen and non-frozen fitting factors for
the problem
\[
(P2) \ \ \ \left\{ \begin{array}{c}
\varepsilon^{2} u''(x) - c(u(x)) u (x) = 0, ~~ x \in \Omega, \\
u(0) = A, ~~~ u (1) = B, \label{P2}
\end{array}\right.
\]
where $c$ satisfies (\ref{2.1}).
This has, in general,
boundary layers at both $x=0$ and $x=1$. The nature of these
boundary layers and the behavior of the derivatives of $u$ in the
neighborhood of $x=0$ and $x=1$ is similar to that of the layer in ($P$).
We present numerical results for the analog of (\ref{test1}) above, that is
\renewcommand{\theequation}{3.5}
\begin{equation}
\ve ^ 2 { \frac{d ^ 2}{d x ^ 2} } u ( x ) - u - u^2 = 0 ,
\ x \in (0,1),
\end{equation}
$$u(0) = A, \quad u(1) = B.$$
We first consider the method proposed in \cite[Ch. 10, p. 159]{DMS}.
This gives
a piecewise constant
frozen fitting factor method for the problem (P2).
The fitting factor is given by:
\renewcommand{\theequation}{3.6}
\begin{eqnarray}
\gamma_{i}& =& \frac{ c(u (0))h^2}{4\ve}
\mbox{\rm sinh}^{-2} \frac{\sqrt{c(u (0))}h}{2\sqrt{\ve}}, \ \ \
0 \le x_i \le 1/2 , \nonumber \\
\gamma_{i}& =& \frac{ c(u (1))h^2}{4\ve}
\mbox{\rm sinh}^{-2} \frac{\sqrt{c(u (1))}h}{2\sqrt{\ve}}, \ \ \
1/2 < x_i \le 1.
\label{frfit2}
\end{eqnarray}
Table \ref{t3.4} gives uniform convergence rates for this scheme.
\begin{table}[hbt]
\centering
\begin{tabular}{|l|r|r|r|r|r|r|} \hline
\multicolumn{7}{||c||}{Boundary Conditions: \ \ $u(0) = 0.5\ , \ u(1) = 0.7$}\\
\multicolumn{7}{||c||}{Initial Guess : \ \ \ \ \ \ \ $u_{init} = u(x) = u(0) + ( u(1) - u(0) )x$} \\ \hline
$N$ & 8 & 16 & 32 & 64 & 128 & 256 \\ \hline
$E_N$ & .006172 & .006171 & .006168 & .006154 & .006094 & .005807\\ \hline
$p_N$ & .00 & .00 & .00 & .00 & .00 & .00\\ \hline\end{tabular}
\caption{Maximum errors $E_N$ and rate of convergence $p_N$ for Frozen Fitting Factor scheme (\protect{\ref{frfit2}})}
\label{t3.4}
\end{table}
Table \ref{t3.5} gives rates for the scheme of Miller
\cite{miller1} discussed in \cite[Ch. 6]{DMS} which has the following
(non-frozen) fitting factor:
\renewcommand{\theequation}{3.7}
\begin{equation}
\gamma_{i} = \frac{ c(u (x_{i}))h^2}{4\ve}
\mbox{\rm sinh}^{-2} \frac{\sqrt{c(u (x_{i}))}h}{2\sqrt{\ve}}.
\label{miller}
\end{equation}
\begin{table}[hbt]
\centering
\begin{tabular}{|l|r|r|r|r|r|r|} \hline
\multicolumn{7}{||c||}{Boundary Conditions: \ \ $u(0) = 0.5\ , \ u(1) = 0.7$}\\
\multicolumn{7}{||c||}{Initial Guess : \ \ \ \ \ \ \ $u_{init} = u(x) = u(0) + ( u(1) - u(0) )x$} \\ \hline
$N$ & 8 & 16 & 32 & 64 & 128 & 256 \\ \hline
$E_N$ & .007933 & .007932 & .007923 & .007887 & .007746 & .007201\\ \hline
$p_N$ & .00 & .00 & .00 & .00 & .00 & .00\\ \hline
\end{tabular}
\caption{Maximum errors $E_N$ and rate of convergence $p_N$ for the scheme of Miller (\protect{\ref{miller}})}
\label{t3.5}
\end{table}
As can be seen from these tables, none of the standard fitted schemes
from the literature, on uniform meshes, are uniformly $\ve$-convergent
for the test problems. As remarked in the introduction,
numerical methods, $\ve$-uniform
in the maximum norm,
were constructed in \cite{fmors1b} for a class of
semilinear problems, which includes the class of problems considered here.
These use classical finite difference operators on
special piecewise-uniform meshes condensed or refined in the boundary layers.
Thus $\ve$-uniform methods can be constructed on special piecewise uniform
meshes although it is not possible on uniform meshes.
\bibliographystyle{amsplain}
\begin{thebibliography}{99}
\bibitem{brandt-yavneh}
A. Brandt, I. Yavneh,
{\em Inadequacy of First-order Upwind Difference Schemes for some
Recirculating Flows}, J. Comput. Phys., 93, (1991) pp. 128-143.
\bibitem{DMS} E.P. Doolan, J.J.H. Miller, W.H.A. Schilders, {\em
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\end{document}