A Linear Variational Principle for Riemann Mapping and Discrete Conformality
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We consider Riemann mappings from bounded Lipschitz domains in the plane to a triangle. We show that in this case the Riemann mapping has a linear variational principle: it is the minimizer of the Dirichlet energy over an appropriate affine space. By discretizing the variational principle in a natural way we obtain discrete conformal maps which can be computed by solving a sparse linear system. We show that these discrete conformal maps converge to the Riemann mapping uniformly and in H^1. Additionally, the discrete conformal maps are known to be bijective, and we show that the Riemann mapping between two bounded Lipschitz domains can be approximated arbitrarily well by composing the Riemann mapping between each Lipschitz domain and the triangle.
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