Novosibirsk, Russia, May, 30 – June, 4, 2011

# International Conference "Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications", devoted to the 90th anniversary of professor Nikolai N. Yanenko

## Canonical domains for almost orthogonal quasi-isometric grids

### Reporter: Чумаков Г.А.

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.

 Abstracts file: chumakov_Nik-Nik-90.tex

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