Numerical solution of the Beltrami equation via a purely linear system

TL;DR

This paper introduces a linear algebra-based algorithm for solving the Beltrami equation on planar disks, avoiding complex integrals and iterations, and demonstrates its effectiveness through numerical examples including Teichmüller space deformations.

Contribution

The paper presents a novel linear system approach for solving the Beltrami equation, simplifying computations by eliminating nonlinear boundary conditions and avoiding iterative methods.

Findings

The algorithm produces accurate solutions for the Beltrami equation.

It efficiently handles boundary conditions via symmetry construction.

Numerical examples include deformations in Teichmüller space.

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 Teichm\"uller space of a Fuchsian group.