发表机构
Texas A&M University（德克萨斯农工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文用共形球样条求解零亏格曲面上Laplace--Beltrami算子的Poisson方程和特征值问题，仅需共形因子描述几何，实验验证了最优收敛阶并优于参数曲面有限元。
AI 中文摘要
每个光滑的零亏格闭黎曼曲面都等距于带有共形度量的单位球面，即球面度量乘以一个正因子。本文使用光滑球样条求解此类度量的Poisson方程和Laplace--Beltrami算子的特征值问题，仅以共形因子作为几何的唯一描述。由于Dirichlet能量在二维中是共形不变的，刚度矩阵对于每个共形度量都是球面矩阵，因子仅进入载荷向量、均值约束和加权质量矩阵。嵌入空间且具有共形参数化的曲面、旋转曲面以及仅以三角网格给出的曲面是同一问题的不同实例，区别仅在于因子的获取方式；本文处理前两种，并在离散化任何方程之前，通过面积恒等式检查因子。未知量是固定球面三角剖分上的样条，采用分片Bernstein--Bézier形式，通过边泛函代数地施加$C^r$光滑性，共形空间由零空间矩阵实现，该矩阵计算一次并对每个度量重复使用。计算的每一步都给出所需公式。在球面度量、长球面、旋转哑铃、Evans--Fung红细胞轮廓以及无嵌入的指定因子上进行的实验表明：样条次数$d$在$L^2$中收敛阶为$d+1$，在能量中为$d$；特征值收敛阶为$2d$，且球面度量的偶阶特征值在$d$次时可重现至舍入误差；与相同三角剖分上的参数曲面有限元相比，在次数为四和六时，以相当的未知量数目获得更小的误差。
英文摘要
Every smooth closed Riemannian surface of genus zero is isometric to the unit sphere carrying a conformal metric, the round metric multiplied by a positive factor. This paper solves the Poisson equation and the eigenvalue problem of the Laplace--Beltrami operator of such a metric with smooth spherical splines, taking the conformal factor as the only description of the geometry. Because the Dirichlet energy is conformally invariant in two dimensions, the stiffness matrix is the round-sphere matrix for every conformal metric, and the factor enters only the load vector, the mean-value constraint and the weighted mass matrix. A surface embedded in space with a conformal parametrization, a surface of revolution, and a surface given only as a triangle mesh are instances of the same problem that differ in how the factor is obtained; the first two are treated here, and the factor is checked against an area identity before any equation is discretized. The unknown is a spline on a fixed spherical triangulation in broken Bernstein--Bézier form, with $C^r$ smoothness imposed algebraically through edge functionals and the conforming space realized by a null-space matrix computed once and reused for every metric. Each step of the computation is stated with the formulas it needs. Experiments on the round metric, a prolate spheroid, a dumbbell of revolution, an Evans--Fung red-blood-cell profile and a prescribed factor with no embedding show the rates $d+1$ in $L^2$ and $d$ in energy for spline degree $d$, the rate $2d$ for eigenvalues with reproduction of the even-order eigenvalues of the round metric up to degree $d$ to roundoff, and, against parametric surface finite elements on the same triangulations, smaller errors at comparable numbers of unknowns for degrees four and six.
