Linear variational principle for Riemann mappings and discrete conformality.


Journal Article

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 [Formula: see text], 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 discrete Riemann mappings between each Lipschitz domain and the triangle.

Full Text

Duke Authors

Cited Authors

  • Dym, N; Slutsky, R; Lipman, Y

Published Date

  • January 2019

Published In

Volume / Issue

  • 116 / 3

Start / End Page

  • 732 - 737

PubMed ID

  • 30593564

Pubmed Central ID

  • 30593564

Electronic International Standard Serial Number (EISSN)

  • 1091-6490

International Standard Serial Number (ISSN)

  • 0027-8424

Digital Object Identifier (DOI)

  • 10.1073/pnas.1809731116


  • eng