Convergence in discrete Cauchy problems and applications to circle patterns

Matthes, D.

doi:10.1090/S1088-4173-05-00118-9

Convergence in discrete Cauchy problems and applications to circle patterns
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by D. Matthes

Conform. Geom. Dyn. 9 (2005), 1-23

DOI: https://doi.org/10.1090/S1088-4173-05-00118-9

Published electronically: February 9, 2005

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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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