Polynomial Interpolation of a Vector Field on a Convex Polyhedral Domain

We develop a method for reconstructing polynomial vector fields from discrete velocity samples on a convex polyhedral domain under an exact no-penetration boundary condition. For a prescribed degree bound, the method computes a polynomial vector field that fits the sampled data in the least-squares sense while satisfying the tangency condition identically on every boundary facet. The central ingredient is an explicit algebraic characterization of the space of polynomial vector fields tangent to the boundary, obtained from the theory of logarithmic derivations of hyperplane arrangements. This characterization reduces the constrained reconstruction problem to finite-dimensional linear algebra. We also discuss extensions incorporating additional linear differential constraints, such as incompressibility, and piecewise polynomial constructions on non-convex polyhedral domains with prescribed smoothness across cell interfaces.