Scattered data interpolation on the torus by compactly supported multinode Shepard operators

Abstract

We introduce a compactly supported multinode Shepard operator for the interpolation of scattered data on the torus embedded in $\mathbb{R}^3$. The method combines local polynomial interpolation of total degree $d\in \mathbb{N}$ with compactly supported Shepard-type weights, so that the approximation at each evaluation point depends only on neighbouring stencils of nodes. The torus is treated as an algebraic surface defined by a quartic polynomial, and Gröbner bases are used to construct reduced polynomial spaces on the surface by removing the redundancy induced by the defining equation. This yields local Vandermonde systems adapted to the toroidal geometry. We discuss the metric structure of the torus, show the local equivalence between the periodic parameter distance and the Euclidean distance inherited from $\mathbb{R}^3$, and use this equivalence to motivate the compact support construction. We establish a uniform error estimate in terms of the maximal support radius and the local Lebesgue constants. Under uniform locality and stability assumptions, the method converges with order $d+1$ with respect to the fill distance. Numerical experiments on analytical test functions confirm polynomial reproduction and exhibit an error decay consistent with the theoretical analysis. The approach is also tested on Computational Fluid Dynamics data mapped onto the torus, including the interpolation of the velocity components, the reconstruction of the velocity magnitude from the interpolated components, and the reconstruction of a tangent velocity field through an orthonormal lifting of the interpolated components.