Numerical Solution of the Beltrami Equation

TL;DR

This paper introduces an efficient algorithm for solving the Beltrami equation in a planar disk, avoiding singular integral evaluations by using a ring-based approach and conformal mapping corrections.

Contribution

It presents a novel, practical algorithm that constructs piecewise linear mu-conformal mappings and corrects images without singular integral computations.

Findings

Algorithm effectively solves the Beltrami equation numerically.

No singular integrals are evaluated during the process.

Computational complexity is analyzed and demonstrated.

Abstract

An effective algorithm is presented for solving the Beltrami equation fzbar = mu fz in a planar disk. The algorithm involves no evaluation of singular integrals. The strategy, working in concentric rings, is to construct a piecewise linear mu-conformal mapping and then correct the image using a known algorithm for conformal mappings. Numerical examples are provided and the computational complexity is analyzed.