Part 1: The first part of the proof deals with
\(0<\rho\leq\frac{1}{2}\). So
$$ \frac{\rho}{1-\rho}\leq1. $$
(4.7)
For
\(n=1\), in domain 1
, the discrete analog
\(w_{h_{1}}\) of
\(u_{1} \) defined in (
4.1) considered as the upper bound of the set of discrete subsolutions [
16], satisfies
$$\begin{aligned}& b_{1} \bigl( w_{h_{1}},\varphi_{s}^{1} \bigr) \leq \bigl( f ( u_{1} ) ,\varphi_{s}^{1} \bigr) , \quad \forall s\in \bigl\{ 1,\ldots,m ( h_{1} ) \bigr\} , \\& w_{h_{1}} =\pi_{h_{1}}u_{2}\quad \text{on } \Gamma_{1}. \end{aligned}$$
Since the nonlinear functional is Lipschitz and according to (
4.3), we get
$$ f ( u_{1} ) -f ( w_{h_{1}} ) \leq kCh^{2}\vert \log h\vert ^{2}. $$
Then
$$\begin{aligned}& b_{1} \bigl( w_{h_{1}},\varphi_{s}^{1} \bigr) \leq \bigl( f ( u_{1} ) ,\varphi_{s}^{1} \bigr) \leq \bigl( f ( w_{h_{1}} ) +kCh^{2}\vert \log h\vert ^{2},\varphi_{s}^{1} \bigr), \\& w_{h_{1}} =\pi_{h_{1}}u_{2}\quad \text{on } \Gamma_{1}. \end{aligned}$$
Let
$$ W_{h_{1}}=\sigma_{h_{1}} \bigl( f ( w_{h_{1}} ) +kCh^{2} \vert \log h\vert ^{2},\pi_{h_{1}}u_{2} \bigr) ; $$
(4.8)
therefore,
\(w_{h_{1}}\) is a subsolution of
\(W_{h_{1}}\),
$$ w_{h_{1}}\leq W_{h_{1}}\quad \text{in }\Omega_{1}. $$
(4.9)
By applying (
2.19), we get
$$\begin{aligned} \begin{aligned} \bigl\Vert W_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}& \leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \bigl\Vert f ( w_{h_{1}} ) +kCh^{2}\vert \log h\vert ^{2}-f \bigl( u_{h_{1}}^{1} \bigr) \bigr\Vert _{1} ; \bigl\vert u_{2}-u_{h_{2}}^{0}\bigr\vert _{1} \biggr\} \\ & \leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \bigl\Vert f ( w_{h_{1}} ) -f \bigl( u_{h_{1}}^{1} \bigr) \bigr\Vert _{1}+ \biggl( \frac{k}{\beta} \biggr) Ch^{2}\vert \log h \vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \biggr\} . \end{aligned} \end{aligned}$$
So
$$ \bigl\Vert W_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\max \bigl\{ \rho \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \bigr\} . $$
(4.10)
On the other hand, (
4.9) generates two possibilities, that is,
$$ ( \mathrm{A}_{1} ) \mbox{:}\quad \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}\leq\bigl\Vert W_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1} $$
or
$$ ( \mathrm{A}_{2} )\mbox{:}\quad \bigl\Vert W_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}. $$
Case (A
1) in conjunction with (
4.10) implies that
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\max \bigl\{ \rho \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \bigr\} , $$
which lets us distinguish the following two cases:
$$\begin{aligned}& 1\mbox{:} \quad \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \bigr\} \\& \hphantom{1\mbox{:} \quad}\quad =\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} \end{aligned}$$
(4.11)
and
$$ 2\mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \bigr\} =\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}. $$
(4.12)
Equation (
4.11) implies that
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} $$
and
$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}. $$
Then
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{\rho }{1-\rho}Ch^{2}\vert \log h \vert ^{2} $$
and
$$\begin{aligned} \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}& \leq\frac{\rho ^{2}}{1-\rho }Ch^{2}\vert \log h\vert ^{2}+\rho Ch^{2}\vert \log h\vert ^{2} \\ & \leq\frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}\leq Ch^{2}\vert \log h\vert ^{2}, \end{aligned}$$
which coincides with (
4.5) and contradicts (
4.6). So, (
4.11) is only possible in situation (A). Equation (
4.12) implies that
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} $$
(4.13)
and
$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}. $$
So, by multiplying (
4.13) by
ρ and adding
\(\rho Ch^{2} \vert \log h\vert ^{2}\), we get
$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\rho \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}+\rho Ch^{2}\vert \log h \vert ^{2}. $$
Then
\(\rho \Vert w_{h_{1}}-u_{h_{1}}^{1}\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\) is bounded by both
\(\Vert u_{2}-u_{h_{2}}^{0}\Vert _{2}\) and
\(\rho \Vert u_{2}-u_{h_{2}}^{0}\Vert _{2}+\rho Ch^{2}\vert \log h \vert \), so
$$ ( \mathrm{a} ) \mbox{:}\quad \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}\leq \rho \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert $$
or
$$ ( \mathrm{b} ) \mbox{:}\quad \rho\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert \leq\bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}. $$
That is,
$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq\frac{\rho}{1-\rho} Ch^{2}\vert \log h\vert ^{2} $$
or
$$ \frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}\leq \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}. $$
Thus
$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq\frac{\rho}{1-\rho} Ch^{2}\vert \log h\vert ^{2}\leq Ch^{2}\vert \log h\vert ^{2} $$
or
$$ \frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}\leq \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2} \leq Ch^{2}\vert \log h \vert ^{2}. $$
So, the two cases (a) and (b) are true because they both coincide with (
4.5). Therefore, there is either a contradiction and thus (
4.12) is impossible or (
4.12) is possible only if
$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}=\frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}. $$
Then (
4.12) in situation (A) implies
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}=\frac{\rho}{1-\rho}Ch^{2} \vert \log h \vert ^{2}, $$
while in situation (B) only (b) is true and leads to
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}\quad \text{and}\quad \frac{\rho}{1-\rho}Ch^{2} \vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}. $$
Then
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{\rho }{1-\rho}Ch^{2}\vert \log h \vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} $$
or
$$ \frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}\leq \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1} \leq\bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}. $$
We remark that both possibilities are true. There is either a contradiction and (
4.12) is impossible or (
4.12) is possible only if
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}=\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
So, in the two situations (A) and (B) and in the two cases (
4.11) and (
4.12) of situation (A
1), we get
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{\rho }{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} , $$
(4.14)
which implies
$$ w_{h_{1}}-\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{1}}^{1}\leq w_{h_{1}}+ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
Let us denote
$$ \alpha_{h_{1}}=w_{h_{1}}-\frac{\rho}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2} $$
(4.15)
and
$$ \tilde{\alpha}_{h_{1}}=w_{h_{1}}+\frac{\rho}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2}. $$
(4.16)
Then
$$ \alpha_{h_{1}}\leq u_{h_{1}}^{1}\leq\tilde{ \alpha}_{h_{1}} $$
(4.17)
with
$$\begin{aligned} \Vert \alpha_{h_{1}}-u_{1}\Vert _{1}& =\biggl\Vert w_{h_{1}}- \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}-u_{1}\biggr\Vert _{1} \\ & \leq \Vert w_{h_{1}}-u_{1}\Vert _{1}+ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} \\ & \leq Ch^{2}\vert \log h\vert ^{2}+ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} \end{aligned}$$
by virtue of (
4.3). So
$$ \Vert \alpha_{h_{1}}-u_{1}\Vert _{1}\leq \frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
(4.18)
By using the same reasoning we see that (
4.16) implies
$$ \Vert \tilde{\alpha}_{h_{1}}-u_{1}\Vert _{1}\leq \frac {1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
(4.19)
On the other hand, (
4.17) implies
$$ \alpha_{h_{1}}-u_{1}\leq u_{h_{1}}^{1}-u_{1} \leq\tilde{\alpha}_{h_{1}}-u_{1} $$
(4.20)
so according to (
4.18) and (
4.19) we get
$$ -\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{1}}^{1}-u_{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2} $$
(4.21)
thus
$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
(4.22)
Case (A
2) in conjunction with (
4.10) implies that
\(\Vert W_{h_{1}}-u_{h_{1}}^{1}\Vert _{1}\) is bounded by the values
\(\Vert w_{h_{1}}-u_{h_{1}}^{1}\Vert _{1}\) and
\(\max \{ \rho \Vert w_{h_{1}}-u_{h_{1}}^{1}\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \Vert u_{2}-u_{h_{2}}^{0}\Vert _{2} \} \) which generates the two situations
$$ ( \mathrm{c} ) \mbox{:}\quad \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}\leq \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \bigr\} $$
or
$$ ( \mathrm{d} ) \mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \bigr\} \leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}. $$
(4.23)
It is clear that case (c) coincides with situation (A
1). Let us study case (d); as in case (A
1),
\(\max \{ \rho \Vert w_{h_{1}}-u_{h_{1}}^{1}\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \Vert u_{2}-u_{h_{2}}^{0}\Vert _{2} \} \) lets us distinguish the two cases (
4.11) and (
4.12). Equation (
4.11) in conjunction with (d) implies
$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1} $$
and (
4.12) in conjunction with (d) implies
$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2} \leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}. $$
Then it is clear that in the two cases (
4.11) and (
4.12), we obtain
$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1} $$
(4.24)
with
$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}. $$
(4.25)
Thus,
\(\Vert w_{h_{1}}-u_{h_{1}}^{1}\Vert _{1}\) is bounded below by both
\(\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\) and
\(\Vert u_{2}-u_{h_{2}}^{0}\Vert _{2}\) so we distinguish the two following possibilities:
$$ ( \mathrm{e} ) \mbox{:}\quad \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}\leq \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}\leq Ch^{2}\vert \log h\vert ^{2} $$
or
$$ ( \mathrm{f} ) \mbox{:}\quad \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}\leq Ch^{2}\vert \log h\vert ^{2}. $$
So, the two cases (e) and (f) are true because they both coincide with (
4.5). Therefore, there is either a contradiction and thus cases (
4.11) and (
4.12) are impossible or the two cases (
4.11) and (
4.12) are possible in situation (A) only if
$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}=\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}, $$
while in situation (B) only the case (f) is true and leads to
$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2} \leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}. $$
In summary, in situation (A
2) and in the two cases (
4.11) and (
4.12) of situations (A) and (B), we get
$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}. $$
(4.26)
Let us decompose the subdomain
\(\Omega_{1}=\Omega_{1,1}\cup\Omega _{1,1}^{c}\) such that
$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\vert \quad \text{on }\Omega _{1,1} $$
(4.27)
and
$$ \bigl\vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\vert < \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\quad \text{on }\Omega_{1,1}^{c}. $$
(4.28)
We begin with
\(\Omega_{1,1}\). If
\(w_{h_{1}}-u_{h_{1}}^{1}\geq0\) on
\(\Omega_{1,1}\) then (
4.27) implies
\(u_{h_{1}}^{1}\leq \alpha_{h_{1}}\); thus,
$$ u_{h_{1}}^{1}-u_{1}\leq\alpha_{h_{1}}-u_{1} \leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$
(4.29)
by virtue of (
4.18). On the other hand, (
4.18) leads also to
$$ -\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq \alpha_{h_{1}}-u_{1}. $$
So,
\(\alpha_{h_{1}}-u_{1}\) is bounded below by both
\(u_{h_{1}}^{1}-u_{1}\) and
\(-\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}\), which lets us distinguish the two following possibilities:
$$ u_{h_{1}}^{1}-u_{1}\leq-\frac{1}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2} $$
or
$$ -\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{1}}^{1}-u_{1}. $$
Then
$$ u_{h_{1}}^{1}-u_{1}\leq-\frac{1}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2}\leq w_{h_{1}}-u_{1} $$
or
$$ -\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{1}}^{1}-u_{1}\leq w_{h_{1}}-u_{1}. $$
So, both possibilities are true because they coincide with (
4.3). So, there is either a contradiction and (
4.27) is impossible or (
4.27) is possible and we must have
$$ \bigl\Vert u_{h_{1}}^{1}-u_{1}\bigr\Vert _{L^{\infty} ( \Omega _{1,1} ) }=\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
(4.30)
The case
\(w_{h_{1}}-u_{h_{1}}^{1}<0\) on
\(\Omega_{1,1}\) is studied in a similar manner and leads to the same result (
4.30). Equation (
4.28) is studied in the same way as that for case (A
1) and leads to
$$ \bigl\Vert u_{h_{1}}^{1}-u_{1}\bigr\Vert _{L^{\infty} ( \Omega _{1,1}^{c} ) }\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
(4.31)
Equations (
4.30) and (
4.31) imply
$$ \bigl\Vert u_{h_{1}}^{1}-u_{1}\bigr\Vert _{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
(4.32)
Finally, in the two cases (A
1) and (A
2) and in the two situations (A) and (B), we get
$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
(4.33)
For
\(n=1\) in domain 2, the discrete analog
\(w_{h_{2}}\) of
\(u_{2}\), defined in (
4.2) and considered as the upper bound of the set of discrete subsolutions [
16], satisfies
$$\begin{aligned}& b_{2} \bigl( w_{h_{2}},\varphi_{s}^{2} \bigr) \leq \bigl( f ( u_{2} ) ,\varphi_{s}^{2} \bigr), \quad \forall s\in \bigl\{ 1,\ldots,m ( h_{2} ) \bigr\} , \\& w_{h_{2}} =\pi_{h_{2}}u_{1}\quad \text{on } \Gamma_{2}. \end{aligned}$$
The nonlinear functional is Lipschitz and according to (
4.3)
$$ f ( u_{2} ) -f ( w_{h_{2}} ) \leq kCh^{2}\vert \log h\vert ^{2}. $$
Then
$$\begin{aligned}& b_{2} \bigl( w_{h_{2}},\varphi_{s}^{2} \bigr) \leq \bigl( f ( u_{2} ) ,\varphi_{s}^{2} \bigr) \leq \bigl( f ( w_{h_{2}} ) +kCh^{2}\vert \log h\vert ^{2},\varphi_{s}^{2} \bigr) , \\& w_{h_{2}} =\pi_{h_{2}}u_{1}\quad \text{on } \Gamma_{2}. \end{aligned}$$
Let
$$ W_{h_{2}}=\sigma_{h_{2}} \bigl( f ( w_{h_{2}} ) +kCh^{2} \vert \log h\vert ^{2},\pi_{h_{2}}u_{1} \bigr) ; $$
(4.34)
therefore,
\(w_{h_{2}}\) is a subsolution of
\(W_{h_{2}}\), so
$$ w_{h_{2}}\leq W_{h_{2}}\quad \text{in }\Omega_{2}. $$
(4.35)
By applying (
2.19), we get
$$\begin{aligned} \bigl\Vert W_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}& \leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \bigl\Vert f ( w_{h_{2}} ) +kCh^{2}\vert \log h\vert -f \bigl( u_{h_{2}}^{1} \bigr) \bigr\Vert _{2} ; \bigl\vert u_{1}-u_{h_{1}}^{1}\bigr\vert _{2} \biggr\} \\ & \leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \bigl\Vert f ( w_{h_{2}} ) -f \bigl( u_{h_{2}}^{1} \bigr) \bigr\Vert _{2}+ \biggl( \frac{k}{\beta} \biggr) Ch^{2}\vert \log h \vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1} \biggr\} . \end{aligned}$$
So
$$ \bigl\Vert W_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\max \bigl\{ \rho \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \bigr\} . $$
(4.36)
On the other hand, (
4.35) generates two possibilities, that is,
$$ ( \mathrm{B}_{1} ) \mbox{:}\quad \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}\leq\bigl\Vert W_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2} $$
or
$$ ( \mathrm{B}_{2} ) \mbox{:}\quad \bigl\Vert W_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}\leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}. $$
Case (B
1) in conjunction with (
4.36) implies that
$$ \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\max \bigl\{ \rho \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \bigr\} , $$
which lets us distinguish the following two cases:
$$\begin{aligned}& 1\mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \bigr\} \\& \hphantom{1\mbox{:}\quad}\quad =\rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} \end{aligned}$$
(4.37)
and
$$ 2\mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \bigr\} =\bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}. $$
(4.38)
Equation (
4.37) implies that
$$ \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} $$
and
$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2}. $$
Then
$$ \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\frac{\rho }{1-\rho}Ch^{2}\vert \log h \vert ^{2} $$
and
$$\begin{aligned} \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}& \leq\frac{\rho ^{2}}{1-\rho }Ch^{2}\vert \log h\vert ^{2}+\rho Ch^{2}\vert \log h\vert ^{2} \\ & \leq\frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}< \frac{1}{1-\rho}Ch^{2}\vert \log h\vert ^{2}, \end{aligned}$$
which coincides with (
4.33). Equation (
4.38) implies that
$$ \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1} $$
(4.39)
and
$$ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}. $$
So, by multiplying (
4.39) by
ρ and adding
\(\rho Ch^{2} \vert \log h\vert ^{2}\) we get
$$ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\rho \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h \vert ^{2}; $$
then
\(\rho \Vert w_{h_{2}}-u_{h_{2}}^{1}\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2}\) is bounded above by
\(\Vert u_{1}-u_{h_{1}}^{1}\Vert _{1}\) and
\(\rho \Vert u_{1}-u_{h_{1}}^{1}\Vert _{1}+\rho Ch^{2}\vert \log h \vert \). So, either
$$ ( \mathrm{a} ) \mathrm{:}\quad \bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}\leq \rho \bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert $$
or
$$ ( \mathrm{b} ) \mbox{:}\quad \rho\bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert \leq\bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}, $$
that is,
$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{\rho}{1-\rho} Ch^{2}\vert \log h\vert ^{2}< \frac{1}{1-\rho}Ch^{2} \vert \log h\vert ^{2} $$
or
$$ \frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}\leq \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \leq\frac{1}{1-\rho}Ch^{2} \vert \log h\vert ^{2}. $$
So, the two cases (a) and (b) are true because they both coincide with (
4.33). Therefore, there is either a contradiction and thus (
4.38) is impossible or (
4.38) is possible only if
$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}=\frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2} $$
thus
$$ \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}=\frac{\rho}{1-\rho}Ch^{2} \vert \log h \vert ^{2}. $$
In summary, in the two cases (
4.37) and (
4.38) of situation (B
1), we get
$$ \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\frac{\rho }{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$
so
$$ w_{h_{2}}-\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{2}}^{1}\leq w_{h_{2}}+ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
Let us denote
$$ \alpha_{h_{2}}=w_{h_{2}}-\frac{\rho}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2} $$
(4.40)
and
$$ \tilde{\alpha}_{h_{2}}=w_{h_{2}}+\frac{\rho}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2} ; $$
(4.41)
then
$$ \alpha_{h_{2}}\leq u_{h_{2}}^{1}\leq\tilde{ \alpha}_{h_{2}}. $$
(4.42)
By using a same reasoning as adopted in subdomain
\(\Omega_{1}\) for (
4.15) and (
4.16), we get
$$ \Vert \alpha_{h_{2}}-u_{2}\Vert _{2}\leq \frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$
(4.43)
and
$$ \Vert \tilde{\alpha}_{h_{2}}-u_{2}\Vert _{2}\leq \frac {1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
(4.44)
Equation (
4.42) implies
$$ \alpha_{h_{2}}-u_{2}\leq u_{h_{2}}^{1}-u_{2} \leq\tilde{\alpha}_{h_{2}}-u_{2} $$
and according to (
4.43) and (
4.44), we obtain
$$ -\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{2}}^{1}-u_{2}\leq\frac{1}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2} , $$
(4.45)
that is,
$$ \bigl\Vert u_{2}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
Case (B
2) in conjunction with (
4.36) implies that
\(\Vert W_{h_{2}}-u_{h_{2}}^{1}\Vert _{2}\) is bounded by the values
\(\Vert w_{h_{2}}-u_{h_{2}}^{1}\Vert _{2}\) and
\(\max \{ \rho \Vert w_{h_{2}}-u_{h_{2}}^{1}\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \Vert u_{1}-u_{h_{1}}^{1}\Vert _{1} \} \), which generates two situations,
$$ ( \mathrm{c} ) \mbox{:}\quad \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}\leq \max \bigl\{ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \bigr\} $$
or
$$ ( \mathrm{d} ) \mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \bigr\} \leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}. $$
It is clear that case (c) coincides with case (B
1). Let us study case (d); as in case (B
1)
\(\max \{ \rho \Vert w_{h_{2}}-u_{h_{2}}^{1}\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \Vert u_{1}-u_{h_{1}}^{1}\Vert _{1} \} \) lets us distinguish the two cases (
4.37) and (
4.38). Equation (
4.37) in conjunction with (d) implies
$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2} $$
and (
4.38) in conjunction with (d) implies
$$ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}. $$
Then the two cases (
4.37) and (
4.38) imply
$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2} $$
and
$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}. $$
\(\Vert w_{h_{2}}-u_{h_{2}}^{1}\Vert _{2}\) is bounded below by
\(\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\) and
\(\Vert u_{1}-u_{h_{1}}^{1}\Vert _{1}\) so we distinguish the two following possibilities:
$$ ( \mathrm{e} ) \mbox{:}\quad \bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}\leq \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}< \frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$
or
$$ ( \mathrm{f} ) \mbox{:} \quad \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}\leq\bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}\leq \frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}. $$
So, the two cases (e) and (f) are true because they both coincide with (
4.33). Therefore, there is either a contradiction and thus cases (
4.37) and (
4.38) are impossible or the two cases (
4.37) and (
4.38) are possible only if
$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}=\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}. $$
So, in the two cases (
4.37) and (
4.38) of situation (B
2), we get
$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}. $$
The remainder of the proof related to situation (B
2) rests on the same arguments used in subdomain
\(\Omega_{1}\) for situation (A
2) that is, on a decomposition of
\(\Omega _{2}=\Omega _{2,1}\cup\Omega_{2,1}^{c}\) and showing that
$$ \bigl\Vert u_{2}-u_{h_{2}}^{1}\bigr\Vert _{L^{\infty} ( \Omega _{2,1} ) }\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$
and
$$ \bigl\Vert u_{2}-u_{h_{2}}^{1} \bigr\Vert _{L^{\infty} ( \Omega_{2,1}^{c} ) }\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}. $$
Finally, in the two situations (B
1) and (B
2) we get
$$ \bigl\Vert u_{2}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
(4.46)
Now, let us assume that
$$ \begin{aligned} &\bigl\Vert u_{1}-u_{h_{1}}^{n}\bigr\Vert _{1} \leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} , \\ &\bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} , \end{aligned} $$
(4.47)
and prove that
$$ \begin{aligned} &\bigl\Vert u_{1}-u_{h_{1}}^{n+1}\bigr\Vert _{1} \leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert , \\ &\bigl\Vert u_{2}-u_{h_{2}}^{n+1}\bigr\Vert _{2} \leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert . \end{aligned} $$
(4.48)
By using the definition of
\(W_{h_{1}}\) in (
4.8) and by applying (
2.19), we get
$$\begin{aligned} \bigl\Vert W_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}& \leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \bigl\Vert f ( w_{h_{1}} ) +kCh^{2}\vert \log h\vert -f \bigl( u_{h_{1}}^{n+1} \bigr) \bigr\Vert _{1} ; \bigl\vert u_{2}-u_{h_{2}}^{n}\bigr\vert _{1} \biggr\} \\ & \leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \bigl\Vert f ( w_{h_{1}} ) -f \bigl( u_{h_{1}}^{n+1} \bigr) \bigr\Vert _{1}+ \biggl( \frac{k}{\beta} \biggr) Ch^{2}\vert \log h \vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2} \biggr\} \end{aligned}$$
so
$$ \bigl\Vert W_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\max \bigl\{ \rho \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2} \vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \bigr\} . $$
(4.49)
On the other hand, (
4.9) generates two possibilities, that is
$$ ( \mathrm{C}_{1} ) \mbox{:} \quad \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}\leq\bigl\Vert W_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1} $$
or
$$ ( \mathrm{C}_{2} ) \mbox{:} \quad \bigl\Vert W_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}. $$
Case (C
1) in conjunction with (
4.49) implies that
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\max \bigl\{ \rho \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2} \vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \bigr\} , $$
which lets us distinguish the following two cases:
$$\begin{aligned} \begin{aligned}[b] &1\mbox{:} \quad \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \bigr\} \\ &\hphantom{1\mbox{:} \quad}\quad =\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} \end{aligned} \end{aligned}$$
(4.50)
and
$$ 2\mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \bigr\} =\bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}. $$
(4.51)
Equation (
4.50) implies that
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} $$
and
$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}. $$
Then
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\frac{\rho }{1-\rho }Ch^{2}\vert \log h\vert ^{2} $$
and
$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}\leq\frac{\rho}{1-\rho} Ch^{2}\vert \log h\vert ^{2}< \frac{1}{1-\rho}Ch^{2} \vert \log h\vert ^{2}, $$
which coincides with (
4.47). Equation (
4.51) implies that
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2} $$
(4.52)
and
$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}. $$
So, by multiplying (
4.52) by
ρ and adding
\(\rho Ch^{2}\vert \log h\vert ^{2}\) we get
$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\rho \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}+\rho Ch^{2}\vert \log h \vert ^{2}; $$
then
\(\rho \Vert w_{h_{1}}-u_{h_{1}}^{n+1}\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\) is bounded by both
\(\Vert u_{2}-u_{h_{2}}^{n}\Vert _{2}\) and
\(\rho \Vert u_{2}-u_{h_{2}}^{n}\Vert _{2}+\rho Ch^{2}\vert \log h \vert \). So
$$ ( \mathrm{a} ) \mbox{:}\quad \bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}\leq \rho \bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert $$
or
$$ ( \mathrm{b} ) \mbox{:}\quad \rho\bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert \leq\bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}. $$
Thus
$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}\leq\frac{\rho}{1-\rho} Ch^{2}\vert \log h\vert ^{2}< \frac{1}{1-\rho}Ch^{2} \vert \log h\vert ^{2} $$
or
$$ \frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}\leq \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \leq\frac{1}{1-\rho}Ch^{2} \vert \log h\vert ^{2}. $$
So, the two cases (a) and (b) are true because they both coincide with (
4.47). Therefore, there is either a contradiction and thus (
4.51) is impossible or (
4.51) is possible only if
$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}=\frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}. $$
Then (
4.51) implies
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}=\frac{\rho}{1-\rho}Ch^{2} \vert \log h \vert ^{2}. $$
Thus, in situation (C
1) and in the two cases (
4.50) and (
4.51), we get
$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\frac{\rho }{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$
so
$$ \alpha_{h_{1}}\leq u_{h_{1}}^{n+1}\leq\tilde{ \alpha}_{h_{1}} $$
(4.53)
and
$$ \alpha_{h_{1}}-u_{1}\leq u_{h_{1}}^{n+1}-u_{1} \leq\tilde{\alpha}_{h_{1}}-u_{1}. $$
So, according to (
4.18) and (
4.19), we get
$$ -\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{1}}^{n+1}-u_{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2}. $$
Thus
$$ \bigl\Vert u_{1}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
Case (C
2) in conjunction with (
4.52) implies that
\(\Vert W_{h_{1}}-u_{h_{1}}^{n+1}\Vert _{1} \) is bounded by the values
\(\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\Vert _{1}\) and
\(\max \{ \rho \Vert w_{h_{1}}-u_{h_{1}}^{n+1}\Vert _{1}+\rho Ch^{2} \vert \log h\vert ^{2} ; \Vert u_{2}-u_{h_{2}}^{n}\Vert _{2} \} \), which generates two situations,
$$ ( \mathrm{c} ) \mbox{:}\quad \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}\leq \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \bigr\} $$
or
$$ ( \mathrm{d} ) \mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \bigr\} \leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}. $$
It is clear that case (c) coincides with case (C
1). Let us study case (d); as in case (C
1),
\(\max \{ \rho \Vert w_{h_{1}}-u_{h_{1}}^{n+1}\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \Vert u_{2}-u_{h_{2}}^{n}\Vert _{2} \} \) lets us distinguish the two cases (
4.50) and (
4.51). Equation (
4.50) in conjunction with (d) implies
$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1} $$
and (
4.51) in conjunction with (d) implies
$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}. $$
Then in the two cases (
4.50) and (
4.51), we get
$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1} $$
and
$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}. $$
Hence,
\(\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\Vert _{1}\) is bounded below by both
\(\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\) and
\(\Vert u_{2}-u_{h_{2}}^{n}\Vert _{2}\) so we distinguish the two following possibilities:
$$ ( \mathrm{e} ) \mbox{:}\quad \bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}\leq \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}< \frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$
or
$$ ( \mathrm{f} ) \mbox{:}\quad \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}\leq \frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}. $$
So, the two cases (e) and (f) are true because they both coincide with (
4.47). Therefore, there is either a contradiction and the two cases (
4.50) and (
4.51) are impossible or the two cases (
4.50) and (
4.51) are possible only if
$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}=\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}; $$
thus, in the two cases (
4.50) and (
4.51) of situation (C
2), we get
$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}. $$
The remainder of the proof related to situation (C
2) rests on the same arguments used in subdomain
\(\Omega_{1}\) for situation (A
2) at iteration
\(n=1\), that is, on a decomposition of
\(\Omega_{1}=\Omega_{1,1}\cup\Omega_{1,1}^{c}\) and on showing that
$$ \bigl\Vert u_{1}-u_{h_{1}}^{n+1}\bigr\Vert _{L^{\infty} ( \Omega _{1,1} ) }\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$
and
$$ \bigl\Vert u_{1}-u_{h_{1}}^{n+1} \bigr\Vert _{L^{\infty} ( \Omega_{1,1}^{c} ) }\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}. $$
Finally, in the two situations (C
1) and (C
2) we get the desired result,
$$ \bigl\Vert u_{1}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$
(4.54)
Estimate (
4.48) in domain 2 can be proved similarly using estimate (
4.54).