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Linear variational principle for Riemann mappings and discrete conformality. (English) Zbl 1447.30001

In this paper, the authors provide a method for computing conformal mappings from bounded Lipschitz domains in the plane to a triangle. They show that the Riemann mapping, in this case, satisfies a variational principle, namely, it is the minimizer of the Dirichlet energy over an appropriate affine space. By discretizing the variational principle they obtain discrete conformal mappings which converge to the Riemann mapping in the \(H^1\) norm. In the case of the Delaunay triangulation, the discrete conformal mappings converge uniformly and are known to be bijective. As a consequence they show that the Riemann mapping between two bounded Lipschitz domains can be uniformly approximated by composing the discrete Riemann mappings between each Lipschitz domain and the triangle.

MSC:

30C70 Extremal problems for conformal and quasiconformal mappings, variational methods
30C20 Conformal mappings of special domains
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