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arXiv · 2602.01803

Polynomial Interpolation of a Vector Field on a Convex Polyhedral Domain

Junyan Chu·Shizuo Kaji
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Abstract

We present a computational method for reconstructing a vector field on a convex polytope $\mathcal{P} \subset \mathbb{R}^d$ of arbitrary dimension from discrete samples. We specifically address the scenario where the vector field is subject to a no-penetration (slip) boundary condition, requiring it to be tangent to the boundary $\partial \mathcal{P}$. Given a degree bound $k$, our algorithm computes a polynomial vector field of degree at most $k$ that fits the observed data in the least-squares sense while exactly satisfying the tangency constraints. Central to our approach is an explicit characterization of the module of polynomial vector fields tangent to $\partial \mathcal{P}$, derived using algebraic concepts from the theory of hyperplane arrangements.

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Junyan Chu, Shizuo Kaji. 2026-02-02. Polynomial Interpolation of a Vector Field on a Convex Polyhedral Domain. https://arxiv.org/abs/2602.01803

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