arXiv subjects

Lishuo Dong

Publications and source records attributed to Lishuo Dong.

2 recordsLinked to original sources

arXiv2026

Divergence-free interpolation of tangential vector fields via matrix-valued kernels

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.

math.NA↗

arXiv2026

Error estimates for vector field interpolation based on generalized matrix-valued kernels

Matrix-valued kernels provide a flexible framework for approximating vector fields from scattered data, especially when structural constraints such as divergence-free or curl-free conditions must be preserved. Classical potential-based constructions enforce these constraints naturally, but they typically require the generating scalar function to possess relatively high smoothness. We develop an operator-based framework for constructing div-free and curl-free matrix-valued kernels using integral and differential operators, which substantially relaxes the regularity requirements of the potential approach. Using dimension-walking techniques, we show that the resulting native spaces are norm-equivalent to appropriate vector-valued Sobolev spaces. Another main contribution of the paper is a sharp error analysis for the corresponding kernel matrix-valued interpolation problem. We derive direct Sobolev error estimates that allow fractional regularity of the target field, and we establish Bernstein-type inequalities for the associated kernel trial spaces. These results lead to a complete inverse theorem. We also investigate stability by proving lower bounds for the smallest eigenvalues of the interpolation matrices. Numerical experiments are included to verify the theoretical results.

math.NA↗