Numerical Solution of the Beltrami Equation Via a Purely Linear System

An effective algorithm is presented for solving the Beltrami equation ∂f/∂z¯=μ∂f/∂z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial f / \partial \overline{z } =\mu \,\partial f/\partial z$$\end{document} 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.