## Convergence in discrete Cauchy problems and applications to circle patterns

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**9**(2005), 1-23 Request permission

## 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.## References

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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
- MR Author ID: 722279
- Email: matthes@mathematik.uni-mainz.de
- 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
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Conform. Geom. Dyn.
**9**(2005), 1-23 - MSC (2000): Primary 30G25; Secondary 35A10, 52C15
- DOI: https://doi.org/10.1090/S1088-4173-05-00118-9
- MathSciNet review: 2133803