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Linear variational principle for Riemann mappings and discrete conformality.

Nadav Dym1, Raz Slutsky2, Yaron Lipman3

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This study introduces a discrete conformal mapping method for Riemann mappings between bounded Lipschitz domains and triangles. The method discretizes a variational principle, enabling computation via sparse linear systems and ensuring convergence for accurate approximations.

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Area of Science:

  • Complex Analysis
  • Geometric Function Theory
  • Numerical Analysis

Background:

  • Riemann mappings are fundamental in complex analysis for mapping simply connected regions.
  • Lipschitz domains and triangles are specific geometric domains with important applications.
  • Variational principles offer powerful tools for analyzing and constructing mappings.

Purpose of the Study:

  • To establish a linear variational principle for Riemann mappings from bounded Lipschitz domains to a triangle.
  • To develop a computational method for discrete conformal maps based on this principle.
  • To analyze the convergence properties of these discrete conformal maps.

Main Methods:

  • Formulating a linear variational principle for Riemann mappings to a triangle.
  • Discretizing the variational principle to obtain discrete conformal maps.
  • Solving sparse linear systems for computational efficiency.
  • Analyzing convergence in L2 and uniform convergence for Delaunay triangulations.

Main Results:

  • Demonstrated that Riemann mappings from bounded Lipschitz domains to a triangle satisfy a linear variational principle.
  • Developed discrete conformal maps computable via sparse linear systems.
  • Proved convergence of discrete maps to the Riemann mapping in L2, even for non-Delaunay triangulations.
  • Established uniform convergence and bijectivity for Delaunay triangulations.

Conclusions:

  • The discrete conformal mapping approach provides a robust computational method for Riemann mappings.
  • The method allows for uniform approximation of Riemann mappings between Lipschitz domains by composing discrete maps.
  • This work bridges theoretical concepts of conformal mappings with practical numerical computation.