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Conformal Geometry and Dynamics

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Convergence in discrete Cauchy problems and applications to circle patterns

Author: D. Matthes
Journal: Conform. Geom. Dyn. 9 (2005), 1-23
MSC (2000): Primary 30G25; Secondary 35A10, 52C15
Published electronically: February 9, 2005
MathSciNet review: 2133803
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Abstract: A lattice-discretization of analytic Cauchy problems in two dimensions is presented. It is proven that the discrete solutions converge to a smooth solution of the original problem as the mesh size $\varepsilon$ tends to zero. The convergence is in $C^\infty$ and the approximation error for arbitrary derivatives is quadratic in $\varepsilon$. In application, $C^\infty$-approximation of conformal maps by Schramm's orthogonal circle patterns and lattices of cross-ratio minus one is shown.

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Additional Information

D. Matthes
Affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17.Juni 136, 10623 Berlin, Germany
Address at time of publication: Institut für Mathematik, Johannes Gutenberg Universität Mainz, Staudingerweg 9, 55128 Mainz, Germany

Received by editor(s): March 19, 2004
Received by editor(s) in revised form: November 16, 2004
Published electronically: February 9, 2005
Additional Notes: Supported by the SFB 288 “Differential Geometry and Quantum Physics” of the Deutsche Forschungsgemeinschaft
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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