\documentclass[11pt]{article}
\usepackage{fullpage}
\usepackage{graphicx}
\begin{document}
\thispagestyle{empty}
\begin{center}
{\Large \bf Canonical domains for almost orthogonal quasi-isometric grids}\\
\medskip
G.A. Chumakov$^1$ \& S.G. Chumakov$^2$\\
\medskip
{\small $^1$ Sobolev Institute of Mathematics, Novosibirsk, 630090 Russia,
{\rm e-mail:} chumakov@math.nsc.ru \\
$^2$ Center for Turbulence Research, Stanford University, CA, USA,
{\rm e-mail:} chumakov@stanford.edu\\}
\end{center}
\noindent A special class of canonical domains is discussed for the
generation of quasi-isometric grids. The base computational strategy
of our approach is that the physical domain is decomposed into five
non-overlapping blocks, which are automatically generated by solving
a variational problem. Four of these blocks---the ones that contain
the corners---are conformally equivalent to geodesic quadrangles on
surfaces of constant curvature, while the fifth block is a conformal
image of a non-convex polygon composed of five planar rectangles (or
a large rectangle with four small rectangles cut out of its
corners). To ensure that the angles of the physical and canonical
domains coincide and the conformal modules are the same, the four
corner blocks are taken to be geodesic quadrangles on surfaces of
constant curvature, namely, spherical, planar or Lobachevsky plane,
depending on the angles of the physical domain.
Within each of these
blocks a quasi-isometric grid is generated. Orthogonality of
coordinate lines holds in the fifth, central block.
We present an
algorithm for automated construction of one-parameter family of such
canonical domains. The parameter $\delta$ is defined in such a way
that, according to a theorem that we have proved, for any physical
domain there exists a unique value of $\delta$ for which the mapping
from the canonical domain onto physical region is conformal and its
derivative is bounded. Application of such a mapping results in a
grid inside the physical region that is orthogonal far from the
corners. This strategy ensures the existence of such canonical
domain (the possibility to generate the grid) and the uniqueness of
the mapping, i.e., our algorithm cannot converge to two different
solutions. Note that the grid lines are the images of the geodesics
in corresponding metrics.
\end{document}