Anisotropic Lagrange interpolation on exact spherical triangles
Abstract
We study linear Lagrange interpolation on exact geodesic spherical triangles obtained by radial projection of chordal triangles. Factorising the element map into an affine map and a radial correction separates chordal anisotropy from radial distortion. We derive exact identities for the surface measure, Dirichlet energy, and directional derivatives, and use them to transfer planar anisotropic interpolation estimates to the sphere. The resulting local error estimates retain the two chordal directional length scales and keep the geometric factors explicit, with constants independent of the element size, shape, and sphere radius. In the gradient estimate, the maximum-angle factor multiplies only the longer directional scale. Four explicit triangle families quantify the loss caused by replacing these scales with the diameter and establish the sharp maximum-angle dependence. They also show that, within the same two-term directional bound with a constant independent of the element, the maximum-angle factor cannot be moved to the shorter scale or have its exponent reduced below one. The fourth family proves the optimality of the square-root radial-transfer factor while retaining the same transported directional scale. The local estimates extend to compatible spherical triangulations. As an application, they yield an anisotropic energy-error estimate for the conforming approximation of the mean-zero Laplace--Beltrami problem with exact integration.
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BibTeX
@misc{ishizaka2026anisotropic,
title = {{Anisotropic Lagrange interpolation on exact spherical triangles}},
author = {Hiroki Ishizaka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09686},
eprint = {2610.09686},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09686},
}
RIS
TY - UNPB
TI - Anisotropic Lagrange interpolation on exact spherical triangles
AU - Hiroki Ishizaka
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09686
UR - https://arxiv.org/abs/2610.09686
AB - We study linear Lagrange interpolation on exact geodesic spherical triangles obtained by radial projection of chordal triangles. Factorising the element map into an affine map and a radial correction separates chordal anisotropy from radial distortion. We derive exact identities for the surface measure, Dirichlet energy, and directional derivatives, and use them to transfer planar anisotropic interpolation estimates to the sphere. The resulting local error estimates retain the two chordal directional length scales and keep the geometric factors explicit, with constants independent of the element size, shape, and sphere radius. In the gradient estimate, the maximum-angle factor multiplies only the longer directional scale. Four explicit triangle families quantify the loss caused by replacing these scales with the diameter and establish the sharp maximum-angle dependence. They also show that, within the same two-term directional bound with a constant independent of the element, the maximum-angle factor cannot be moved to the shorter scale or have its exponent reduced below one. The fourth family proves the optimality of the square-root radial-transfer factor while retaining the same transported directional scale. The local estimates extend to compatible spherical triangulations. As an application, they yield an anisotropic energy-error estimate for the conforming approximation of the mean-zero Laplace--Beltrami problem with exact integration.
ER -