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Laplace--Beltrami Equations and Numerical Conformal Mappings on Surfaces

Harri Hakula, Antti Rasila

Abstract

The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and multiply connected domains. In this paper, the conjugate function method is extended to cover conformal mappings between Riemannian surfaces. The main challenge addressed here is the connection between Laplace--Beltrami equations on surfaces and the computation of the conformal modulus of a quadrilateral. We consider mappings of simply, doubly, and multiply connected domains. The numerical computation is based on an $hp$-adaptive finite element method. The key advantage of our approach is that it allows highly accurate computations of mappings on surfaces, including domains of complex boundary geometry involving strong singularities and cusps. The efficacy of the proposed method is illustrated via an extensive set of numerical experiments including error estimates.

Laplace--Beltrami Equations and Numerical Conformal Mappings on Surfaces

Abstract

The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and multiply connected domains. In this paper, the conjugate function method is extended to cover conformal mappings between Riemannian surfaces. The main challenge addressed here is the connection between Laplace--Beltrami equations on surfaces and the computation of the conformal modulus of a quadrilateral. We consider mappings of simply, doubly, and multiply connected domains. The numerical computation is based on an -adaptive finite element method. The key advantage of our approach is that it allows highly accurate computations of mappings on surfaces, including domains of complex boundary geometry involving strong singularities and cusps. The efficacy of the proposed method is illustrated via an extensive set of numerical experiments including error estimates.
Paper Structure (26 sections, 4 theorems, 34 equations, 14 figures, 1 table)

This paper contains 26 sections, 4 theorems, 34 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. PDEs on Surfaces
  3. Conformal Mappings on Surfaces
  4. Illustrative Example: Cartography
  5. Organization
  6. Preliminaries
  7. The Conjugate Function Method on Planar Domains
  8. High-Order Finite Element Method
  9. Error Estimation
  10. A Priori Refinement Strategies
  11. Conjugate Function Method on Surfaces
  12. Conformal Modulus and Generalized Quadrilaterals of Surfaces
  13. Laplace-Beltrami
  14. Helicoid and Catenoid: Isothermal Coordinates
  15. Catenoid
  16. ...and 11 more sections

Key Result

Theorem 1.1

Every simply connected Riemann surface is conformally equivalent to a disk, to the complex plane, or to the Riemann sphere.

Figures (14)

  • Figure 1: Haumea. (a) Artist's impression, (b) latitudes and longitudes, and (c) world map.
  • Figure 1: Rule based mesh refinement. Refinement to an edge coupled with corner refinement. (a) and (b): The first two steps starting from a regular grid. Notice that the process generates both triangles and quadrilaterals.
  • Figure 1: Catenoid and Helicoid: Conformal maps on the surfaces.
  • Figure 1: Mercator projection. (a) Convergence measured in $L^2$-norm (solid line) and $H^1$-seminorm (dashed line) vs polynomial order (loglog-plot). (b) Effect of flattening. Difference between the normalised Mercator projection and the observed one for the Earth ellipsoid along $\lambda = 0$.
  • Figure 2: Helicoid: (a) Maps on the parameter space. Isothermal coordinates: Solid lines, General: Dashed lines. (b) Convergence in the non isothermal parameterisation, error estimates vs. $p$. Reciprocal error estimate: Solid line, Auxiliary subspace error estimate: Dashed line.
  • ...and 9 more figures

Theorems & Definitions (5)

  • Theorem 1.1
  • Lemma 2.1: hqr
  • Theorem 2.2
  • Theorem 2.3: hno
  • Conjecture 5.1