A Linear Variational Principle for Riemann Mappings and Discrete Conformality

TL;DR

This paper introduces a linear variational principle for Riemann mappings from Lipschitz domains to triangles, enabling efficient computation of discrete conformal maps that converge to the continuous Riemann mapping.

Contribution

It establishes a linear variational formulation for Riemann mappings and demonstrates convergence of discrete conformal maps to the continuous mapping, including uniform convergence for Delaunay triangulations.

Findings

Discrete conformal maps converge in H^1 norm for non-Delaunay triangulations.

For Delaunay triangulations, convergence is uniform and maps are bijective.

Riemann mappings can be approximated by compositions of mappings to a triangle.

Abstract

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 in $H^1$, even for non-Delaunay triangulations. Additionally, for Delaunay triangulations the discrete conformal maps converge uniformly and are known to be bijective. As a consequence we show that the Riemann mapping between two bounded Lipschitz domains can be uniformly approximated by composing the Riemann mappings between each Lipschitz domain and the triangle.