Numerical solution of the Beltrami equation via a purely linear system
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arXiv
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| Format: | Preprint |
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2014
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| _version_ | 1866909346899689472 |
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| author | Porter, R. Michael Shimauchi, Hirokazu |
| author_facet | Porter, R. Michael Shimauchi, Hirokazu |
| contents | An effective algorithm is presented for solving the Beltrami equation df/dz = mu (df/dzbar) 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1405_7359 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Numerical solution of the Beltrami equation via a purely linear system Porter, R. Michael Shimauchi, Hirokazu Complex Variables 30C62 An effective algorithm is presented for solving the Beltrami equation df/dz = mu (df/dzbar) 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. |
| title | Numerical solution of the Beltrami equation via a purely linear system |
| topic | Complex Variables 30C62 |
| url | https://arxiv.org/abs/1405.7359 |