关于多面体上调和坐标插值误差估计
Interpolation Error Estimates for Harmonic Coordinates On Polytopes
AI总结：
研究多面体上调和坐标的插值误差估计，分析几何质量对插值误差的影响，发现非凸多边形和多面体的插值误差限制与几何质量相关。
摘要：
在二维和三维空间中，建立了关于几何质量度量的调和坐标插值误差估计。首先推导了凸多边形上插值误差估计，该估计依赖于约束Delaunay三角剖分中三角形的几何质量。证明了当多边形的约束Delaunay三角剖分中存在质量差的三角形时，插值基本二次函数会产生较大误差。非凸多边形也表现出类似限制：由于顶点接近非相邻边导致的较大约束Delaunay三角形也会导致插值误差增大。尽管这种关系被推广到三维凸多面体，但约束Delaunay三角剖分中可能存在的细长四面体阻止了类似的估计精确反映实际插值误差。通过一个包含顶点接近非相邻面的多面体族的例子，证明了非凸多面体本质上是不同的，且插值误差保持有界。
英文摘要：
Interpolation error estimates in terms of geometric quality measures are established for harmonic coordinates on polytopes in two and three dimensions. First we derive interpolation error estimates over convex polygons that depend on the geometric quality of the triangles in the constrained Delaunay triangulation of the polygon. This characterization is sharp in the sense that families of polygons with poor quality triangles in their constrained Delaunay triangulations are shown to produce large error when interpolating a basic quadratic function. Non-convex polygons exhibit a similar limitation: large constrained Delaunay triangles caused by vertices approaching a non-adjacent edge also lead to large interpolation error. While this relationship is generalized to convex polyhedra in three dimensions, the possibility of sliver tetrahedra in the constrained Delaunay triangulation prevent the analogous estimate from sharply reflecting the actual interpolation error. Non-convex polyhedra are shown to be fundamentally different through an example of a family of polyhedra containing vertices which are arbitrarily close to non-adjacent faces yet the interpolation error remains bounded.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
