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Numerical Solution of the Beltrami Equation Via a Purely Linear System

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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 Teichmüller space of a Fuchsian group.

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Acknowledgments

The authors are grateful to T. Sugawa for many critical and useful comments in the preparation of this work, as well as to the referees who uncovered some significant errors in the first version and made numerous useful suggestions (in particular the use of [18, Section 7.1]). The first author is also grateful to D. Marshall for pointing out several fundamental errors in the approach proposed in [23]. This paper forms part of the second author’s Ph.D. thesis.

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Authors and Affiliations

  1. Department of Mathematics, Centro de Investigación y de Estudios Avanzados del I.P.N., Apdo. Postal 1-798, Arteaga 5, 76000, Querétaro, QRO, Mexico

    R. Michael Porter

  2. Division of Mathematics, Graduate School of Information Sciences, Tohoku University, 6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai, 980-8579, Japan

    Hirokazu Shimauchi

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  1. R. Michael Porter
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  2. Hirokazu Shimauchi
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Correspondence to R. Michael Porter.

Additional information

Communicated by Stephan Ruscheweyh.

The first author was partially supported by CONACyT Grant 166183. The second author was supported by International Advanced Research and Education Organization in Tohoku University.

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Porter, R.M., Shimauchi, H. Numerical Solution of the Beltrami Equation Via a Purely Linear System. Constr Approx 43, 371–407 (2016). https://doi.org/10.1007/s00365-016-9334-6

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  • DOI: https://doi.org/10.1007/s00365-016-9334-6

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