The geometry of optimally transported meshes on the sphere
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Mesh redistribution methods provide meshes with increased resolution in particular regions of interest, while requiring that the connectivity of the mesh remains fixed. Some state of the art methods attempt to explicitly prescribe both the local cell size and their alignment to features of the solution. However, the resulting problem is overdetermined, necessitating a compromise between these conflicting requirements. An alternative approach, extolled by the current authors, is to prescribe only the local cell size and augment this with an optimal transport condition. It is well-known that the resulting problem is well-posed and has a unique solution, not just in Euclidean space but also on the sphere. Although the optimal transport approach abandons explicit control over mesh anisotropy, previous work has shown that, on the plane, the meshes produced are naturally aligned to various simple features. In this paper, we show that the same is true on the sphere. Furthermore, formal results on the regularity of solutions to optimal transport problems imply the sphere is actually more benign than the plane for mesh generation, due to its intrinsic curvature. We also see informal evidence of this. We provide a wide range of examples to showcase the behaviour of optimally transported meshes on the sphere, from axisymmetric cases that can be solved analytically to more general examples that are tackled numerically. Evaluation of the singular values and singular vectors of the mesh transformation provides a quantitative measure of the mesh anisotropy, and this is shown to match analytic predictions.
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