Department of Computer Science and Applied Mathematics, Faculty of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 7610001, Israel;
Department of Mathematics, Faculty of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 7610001, Israel.
Proc Natl Acad Sci U S A. 2019 Jan 15;116(3):732-737. doi: 10.1073/pnas.1809731116. Epub 2018 Dec 28.
We consider Riemann mappings from bounded Lipschitz domains in the plane to a triangle. We show that in this case the Riemann mapping has a linear variational principle: It is the minimizer of the Dirichlet energy over an appropriate affine space. By discretizing the variational principle in a natural way we obtain discrete conformal maps which can be computed by solving a sparse linear system. We show that these discrete conformal maps converge to the Riemann mapping in [Formula: see text], even for non-Delaunay triangulations. Additionally, for Delaunay triangulations the discrete conformal maps converge uniformly and are known to be bijective. As a consequence we show that the Riemann mapping between two bounded Lipschitz domains can be uniformly approximated by composing the discrete Riemann mappings between each Lipschitz domain and the triangle.
我们考虑平面有界 Lipschitz 域到三角形的黎曼映射。我们证明，在这种情况下，黎曼映射具有线性变分原理：它是适当仿射空间上狄利克雷能量的极小值。通过以自然的方式离散变分原理，我们得到离散共形映射，通过求解稀疏线性系统可以计算出离散共形映射。我们证明，即使对于非 Delaunay 三角剖分，这些离散共形映射也在 [公式：见文本] 中收敛于黎曼映射。此外，对于 Delaunay 三角剖分，离散共形映射一致收敛，并且已知是双射的。因此，我们证明两个有界 Lipschitz 域之间的黎曼映射可以通过组合每个 Lipschitz 域和三角形之间的离散黎曼映射来均匀逼近。
