Convergence in discrete cauchy problems and applications to circle patterns

A lattice-discretization of analytic Cauchy problems in two di-mensions is presented. It is proven that the discrete solutions converge to a smooth solution of the original problem as the mesh size ε tends to zero. The convergence is in C∞ and the approximation error for arbitrary derivatives is quadratic in ε. In application, C∞-approximation of conformal maps by Schramm’s orthogonal circle patterns and lattices of cross-ratio minus one is shown.