\documentclass[12pt,times]{article}

% to do:
%   all these should be turned into giant itemized lists
%   fix the parens in the typo correction to the top of p303
%   \mathcal{\hat{P}} in the second note to p303 may need fixing

% begin preamble
% \usepackage{doublespace,epsfig}
\usepackage{amsmath,amscd,amsfonts}  % could also use \usepackage{amssymb} ???
\author{James M.\ Rath}
\title{Notes for ``A Mixed Finite Element Method for 2-nd Order Elliptic
Problems'' by Raviart and Thomas, 1977}
\date{\today}

% we might need to define \undertilde{arg} to be \stackrel{arg}{~}
%   or \underset{arg}{~} (is ~ special?).  For ``div'' and ``grad'' we want
%   \DeclareMathOperator{\div}{div} and the same for grad (but we want to
%   indicate that it's a vector...)
\DeclareMathOperator{\diver}{div}
\DeclareMathOperator{\grad}{\mathbf{grad}}
%\newcommand{\undertilde}[1]{\utilde{#1}}
%\newcommand{\undertilde}[1]{\underset{\~ }{#1}}
\newcommand{\undertilde}[1]{\underset{\sim}{#1}}
%\newcommand{\undertilde}[1]{\underset{\textasciitilde}{#1}}

\setlength{\parindent}{0pt}
\setlength{\parskip}{1ex plus 0.5ex minus 0.2ex}
% end preamble

\begin{document}

\maketitle
\begin{abstract}
Published in ``Mathematical aspects of finite element methods : proceedings of
the conference held in Rome, 10-12 December 1975'' by Springer-Verlag in 1977.
Part of the ``Lecture Notes in Mathematics'' series; it's number 606.
\end{abstract}
\newpage
\tableofcontents
\newpage


\section{Typographical Errors}

on page 293:
\begin{itemize}
\item line 3: ``ptoblem'' should read ``problem''
\end{itemize}

on page 295:
\begin{itemize}
\item line 2 of remark 1: ``seddle-point'' should read ``saddle-point''
\end{itemize}

on page 297:
\begin{itemize}
\item line 8: ``consrtuction'' should read ``constructing''
\item in equation 2.16: ``$K' = K_1 \subset K_2$'' should read ``$K' = K_1 \cap K_2$''
\end{itemize}

on page 298:
\begin{itemize}
\item in lemma 1: the function is uniquely determined by \emph{both} conditions
\textit{a} and \textit{b}
\end{itemize}

on page 299:
\begin{itemize}
\item at the bottom, in remark 3: ``\ldots one could have equivalently specified
the moments of order $\leq k$'' should read ``\ldots one could have equivalently
specified the moments of order $\leq k-1$ and \ldots''
\end{itemize}

on page 300:
\begin{itemize}
\item in the top half of equation 3.8: ``$\alpha_1 \xi^k$'' should read
``$\alpha_1 \xi^k \eta$''
\item in the bottom half of equation 3.8: the open parenthesis next to the
$\beta_1 \xi \eta^k$ term is spurious
\end{itemize}

on page 302:
\begin{itemize}
\item in equation 3.21: ``$\left\|B_K\right\|^2$'' should read
``$\left\|B_K\right\|^l$''
\end{itemize}

on page 303:
\begin{itemize}
\item in theorem 3: ``Then there exist an operator $\undertilde{\pi}_K \in
L((H^1(K))^2$'' should read ``Then there exists an operator $\undertilde{\pi}_K
\in L( (H^1(K))^2 )$ such that $\undertilde{\pi}_K: (H^1(K))^2 \rightarrow
\undertilde{Q}_K$''
\item equation 3.23 in theorem 3 should add the relation:
$\forall \undertilde{r} \in \left(P_{k-1}\right)^2, \int_K
\left(\undertilde{\pi}_K \undertilde{q} - \undertilde{q}\right) \cdot
\undertilde{r}\,dx=0$
\item in equation 3.26: the integral should be equal to 0
\item in the proof of theorem 3: ``\ldots there exists by Lemma 1 and Remark 1
\ldots'' should read ``\ldots there exists by Lemma 1 and Remark 3 \ldots''
\end{itemize}

on page 305:
\begin{itemize}
\item in the second line after equation 4.2: ``Note that the function \ldots''
should read ``Note that the functions \ldots''
\end{itemize}

on page 306:
\begin{itemize}
\item in equation 4.7: on the RHS of the inequality ``$u$'' should be ``$\mu$''
\end{itemize}

on page 308:
\begin{itemize}
\item equation 4.19 is missing a closing brace at the end
\end{itemize}

on page 311:
\begin{itemize}
\item in equation 4.38:
``$\left\| \undertilde{p}-p_h \right\|_{\undertilde{H}(\diver; \Omega)}$''
should read
``$\left\| \undertilde{p}-\undertilde{p}_h \right\|_{\undertilde{H}(\diver; \Omega)}$''
\item third line from the bottom: ``$\undertilde{\pi}_h \in \undertilde{Q}_h$''
should read ``$\undertilde{\pi}_h p \in \undertilde{Q}_h$''
\end{itemize}

on page 312:
\begin{itemize}
\item in the statement following equation 4.30:
``\ldots inequalities (2.11) \ldots''
should read ``\ldots inequalities (2.13) \ldots''
\end{itemize}

\section{Annotations}

on page 294:
\begin{itemize}
\item equation 2.4: recall that $\mathbf{p} = \grad{u}$ or
$\mathbf{p} = \mathbf{\nabla}u$
\item equation 2.5: recall that $-\diver{\mathbf{p}} = f$ or
$-\mathbf{\nabla}\cdot\mathbf{p} = f$
\end{itemize}

on page 295:
\begin{itemize}
\item regarding the top two lines: that is, the divergence operator maps
$\undertilde{H}(\diver, \Omega)$ \textit{onto} $L^2(\Omega)$
\end{itemize}

on page 296:
\begin{itemize}
\item equation 2.11 $\Rightarrow$ H2
\item equation 2.12 $\Leftrightarrow$ H1
\end{itemize}

on page 298:
\begin{itemize}
\item equation 3.1 is an approximation condition
\item equation 3.2 is an unisolvance condition
\item equation 3.3 is equivalent to equation 2.11
\item equation 3.4 is a conforming condition
\item equation 3.5 is an unisolvance condition
\end{itemize}

on page 300:
\begin{itemize}
\item the top half of equation 3.8 is equal to
$p_k + \sum_{j=0}^{k/2} \alpha_j \xi^{k+1-j} \eta^j$
\item the bottom half of equation 3.8 is equal to
$p_k + \sum_{j=0}^{k/2} \beta_j \eta^{k+1-j} \xi^j$
\end{itemize}

on page 301:
\begin{itemize}
\item the top half of equation 3.11 is equal to
$p_k + \sum_{j=0}^{\frac{k+1}{2}} \alpha_j \xi^{k+1-j} \eta^j$
\item the bottom half of equation 3.11 is equal to
$p_k + \sum_{j=0}^{\frac{k+1}{2}} \beta_j \eta^{k+1-j} \xi^j$
\end{itemize}

on page 303:
\begin{itemize}
\item The relations defined in equation 3.23 (as amended above) are the defining
relations of the FEM used.

\item in equation 3.28:
$\left\| \diver\left(\undertilde{\hat{\pi}}\undertilde{\hat{q}}-
\undertilde{\hat{q}}\right) \right\|_{0,\hat{K}} =
\left\| \mathcal{\hat{P}}\left(\diver\undertilde{\hat{q}}\right)-
\left(\diver\undertilde{\hat{q}}\right) \right\|_{0,\hat{K}}$

\item regarding the spaces defined on this page, is the following commutative (!)
diagram:
$\begin{CD}
\left(H^1(\hat{K})\right)^2 @>{\diver}>> L^2 @. \\
@V{\undertilde{\hat{\pi}}}VV @V{\mathcal{\hat{P}}=}V{L^2\mathrm{-proj}}V @. \\
\undertilde{\hat{Q}} @>{\diver}>> \hat{P}_K @>>> 0 \\
\end{CD}$
\end{itemize}

on page 304:
\begin{itemize}
\item As a corrolary to theorem 3 on p303, we have the following commutative (!)
diagram:
$\begin{CD}
H^1(K) @>{\diver}>> L^2(K) @>>> 0 \\
@V{\undertilde{\pi}}VV @V{\mathcal{P}=}V{L^2\mathrm{-proj}}V @. \\
\undertilde{Q} @>{\diver}>> P_k(K) @>>{\mathrm{see 4.3}}> 0 \\
@VVV @VVV @. \\
0 @. 0 @. \\
\end{CD}$
and that $\undertilde{\nabla}\cdot\undertilde{\pi}\undertilde{q} = 
\mathcal{P}\left(\undertilde{\nabla}\cdot\undertilde{q}\right)$.
\end{itemize}

on page 309:
\begin{itemize}
\item regarding equations 4.23, 4.24, and the following choice of the
characteristic function: this is a hybrid method for the PDE
\begin{displaymath}
\left\{ \begin{array}{rcll}
-\Delta \phi & = & v & \mathrm{in} \ \Omega \\
\phi & = & 0 & \mathrm{on} \ \partial\Omega \\
\end{array} \right.
\end{displaymath}
\end{itemize}

on page 310:
\begin{itemize}
\item in the proof at the bottom: recall that
$\begin{CD}
\undertilde{Q}_h @>{\diver}>{\mathrm{(onto)}}> V_h @>>> 0
\end{CD}$
($\diver$ is bounded below for an appropriate inverse)
\end{itemize}

\end{document}
