精确球面三角形上的各向异性拉格朗日插值

中文摘要 AI 辅助

我们研究了由弦三角形径向投影得到的精确测地线球面三角形上的线性拉格朗日插值。将单元映射分解为仿射映射和径向修正，从而将弦向各向异性与径向畸变分离开来。我们推导了曲面测度、Dirichlet能量和方向导数的精确恒等式，并利用它们将平面各向异性插值估计转移到球面上。所得局部误差估计保留了弦向的两个方向长度尺度，并保持几何因子显式，常数与单元尺寸、形状和球面半径无关。在梯度估计中，最大角因子仅乘以较长的方向尺度。四个显式三角形族量化了用直径替换这些尺度所造成的损失，并确立了尖锐的最大角依赖性。它们还表明，在与单元无关的常数的同一两项方向界内，最大角因子不能移到较短的尺度上，也不能将其指数降低到1以下。第四个族证明了平方根径向传递因子的最优性，同时保留了相同的传递方向尺度。局部估计可推广到相容的球面三角剖分。作为应用，它们为精确积分下均值零Laplace--Beltrami问题的协调逼近提供了各向异性能量误差估计。

英文摘要

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.

发表机构

• Team FEM（FEM团队）

机构由 AI 辅助整理，请以论文原文为准。