Abstract
An effective algorithm is presented for solving the Beltrami equation $$\partial f / \partial \overline{z } =\mu \,\partial f/\partial z$$ 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 Teichmuller space of a Fuchsian group.
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