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Numerical solution of the Beltrami equation via a purely linear system

R. Michael Porter, Hirokazu Shimauchi

Abstract

An effective algorithm is presented for solving the Beltrami equation df/dz = mu (df/dzbar) in a planar disk. The disk is triangulated in a simple way and f is approximated by piecewise linear mappings; the images of the vertices of the triangles are defined by an overdetermined system of linear equations. (Certain apparently nonlinear conditions on the boundary are eliminated by means of a symmetry construction.) The linear system is sparse and its solution is obtained by standard least-squares, so the algorithm involves no evaluation of singular integrals nor any iterative procedure for obtaining a single approximation of f. Numerical examples are provided, including a deformation in a Teichmüller space of a Fuchsian group.

Numerical solution of the Beltrami equation via a purely linear system

Abstract

An effective algorithm is presented for solving the Beltrami equation df/dz = mu (df/dzbar) in a planar disk. The disk is triangulated in a simple way and f is approximated by piecewise linear mappings; the images of the vertices of the triangles are defined by an overdetermined system of linear equations. (Certain apparently nonlinear conditions on the boundary are eliminated by means of a symmetry construction.) The linear system is sparse and its solution is obtained by standard least-squares, so the algorithm involves no evaluation of singular integrals nor any iterative procedure for obtaining a single approximation of f. Numerical examples are provided, including a deformation in a Teichmüller space of a Fuchsian group.

Paper Structure

This paper contains 18 sections, 11 theorems, 81 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Affine linear quasiconformal mappings
  4. Closeness to similarity
  5. Quasiconformal mappings
  6. Context for discrete Beltrami equation
  7. Logarithmic coordinates
  8. Triangulation
  9. Boundary conditions
  10. Discrete Beltrami equation
  11. Triangle equations
  12. Boundary equations
  13. Statement and proof of main theorem
  14. Statement of theorem
  15. Lemmas
  16. ...and 3 more sections

Key Result

Proposition 2.1

Given $z_1$, $z_2$ distinct and $w_1$, $w_2$ distinct, together with $|\mu|<1$, there is a unique $\mu$-conformal affine linear mapping $B=B_{\mu;\,z_1,z_2;\,w_1,w_2}$ which sends $z_1$ to $w_1$ and $z_2$ to $w_2$. This mapping is given explicitly by

Figures (13)

  • Figure 1: Basic mesh in $W$-plane, with its reflection in the imaginary axis.
  • Figure 2: Different constants in upper and lower $w$-half-planes. Note the normalizations $f(0)=0$, $f(1)-1$.
  • Figure 3: Different constants in upper and lower $W$-half-planes. Note the lifted ellipses at left and right extremes of the boundary.
  • Figure 4: $W$-image containing points far from the origin. The lifted ellipse and its reflection are drawn as thicker curves for emphasis.
  • Figure 5: Image for constant Beltrami derivative $\mu=0.5$ and $(M,N)=(52,64)$. Observe the "crowding phenomenon" at the boundary.
  • ...and 8 more figures

Theorems & Definitions (11)

  • Proposition 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 5.1
  • Lemma 5.2
  • Lemma 5.3
  • Lemma 5.4
  • ...and 1 more