Numerical Solution of the Beltrami Equation Via a Purely Linear System

Abstract

An effective algorithm is presented for solving the Beltrami equation ∂f /∂z = µ ∂f /∂z 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.

A construction of trivial Beltrami coefficients

“…We observe that the boundary values are close to those of the identity map. This picture was created by Hirokazu Shimauchi based on the method given in his joint paper [9] with M. Porter. We note that their normalisation conditions are f (0) = 1 and f (1) = 1 for a quasiconformal automorphism of D. By the property w µ (0) = 0 for µ ∈ M a (D), their method works without renormalisation.…”