On Numerical Algorithms for the Solution of a Beltrami Equation

The paper concerns numerical algorithms for solving the Beltrami equation fz̄(z)=μ(z) fz(z) for a compactly supported μ. First, we study an efficient algorithm that has been proposed in [P. Daripa, J. Comput. Phys., 106 (1993), pp. 355–365] and [P. Daripa and D. Mashat, Numer. Algorithms, 18 (1998), pp. 133–157] and present its rigorous justification. We then propose a different scheme for solving the Beltrami equation which has a comparable speed and accuracy, but has the virtue of easier implementation by avoiding the use of the Hilbert transform. The present paper can also be viewed as a prologue to one important application of the Beltrami equation: it provides a detailed description of the algorithm that has been used in [D. Gaidashev, Nonlinearity, 20 (1998), pp. 713–741] and [D. Gaidashev and M. Yampolsky, Experiment. Math., 16 (2007), pp. 215–226] to address an important issue in complex dynamics—conjectural universality for Siegel disks.