AI 中文总结
该研究提出一种无需反复应用曲面微分算子的切向量场无散核插值方法，推导了显式核表示并验证了其收敛性与稳定性，可推广至非球面。
AI 中文摘要
我们开发并分析了一类用于单位球面上切向量场的无散核插值方法。从欧氏空间中的标量径向核出发，我们构造了切值、曲面无散的矩阵核，无需反复应用曲面微分算子。该构造将无散约束的几何实现与标量生成元的选择分离，与基于经典势的方法相比，允许具有降低正则性要求的低阶变体。利用向量球谐函数，我们推导了显式核表示并刻画了其傅里叶乘子。我们还引入了逆拉普拉斯-贝尔特拉米构造，其保留了底层标量带状核的乘子。对于散乱节点处的插值，我们建立了插值矩阵最小特征值的下界，并推导了逐点和索伯列夫误差估计，包括针对比原生空间更光滑目标的超收敛性。数值实验证实了理论收敛性和稳定性结果，而非球面上的额外示例则说明了核公式在球面之外的适用性。
英文摘要
We develop and analyze a family of divergence-free kernel interpolation methods for tangential vector fields on the unit sphere. Starting from scalar radial kernels in Euclidean space, we construct tangent-valued, surface divergence-free matrix kernels without repeatedly applying surface differential operators. The construction separates the geometric enforcement of the divergence-free constraint from the choice of scalar generator and admits lower-order variants with reduced regularity requirements compared with classical potential-based methods. Using vector spherical harmonics, we derive explicit kernel representations and characterize their Fourier multipliers. We also introduce an inverse Laplace-Beltrami construction that preserves the multipliers of the underlying scalar zonal kernel. For interpolation at scattered nodes, we establish a lower bound for the smallest eigenvalue of the interpolation matrix and derive pointwise and Sobolev error estimates, including superconvergence for targets smoother than the native space. Numerical experiments corroborate the theoretical convergence and stability results, while additional examples on nonspherical surfaces illustrate the applicability of the kernel formula beyond the sphere.
