Quasi-optimal interpolation of gradients and vector-fields on protected Delaunay meshes in $\mathbb{R}^d$

TL;DR

This work addresses the gap in high-order interpolation of gradients and vector fields on Delaunay meshes in $\mathbb{R}^d$ by introducing quasi-optimal interpolation on protected Delaunay meshes. It develops a higher-dimensional roughness functional $\Psi_{\mathcal{T}}(\cdot)$ and proves its equivalence to the $L^2$ gradient norm, then derives mesh-dependent bounds that tie gradient interpolation error to the best $L^{\infty}$-approximation and Rajan’s radius-functional framework. By incorporating a sizing function $\mathcal{D}(x)$ and Lebesgue-optimized interpolation points, the authors obtain error controls that degrade gracefully with mesh slivers and are mitigated by protection level, particularly through a lower bound on element thickness $\Xi(K)$ and the min-containment radius. The results extend naturally to $L_2$ vector-field interpolation, providing a practical and theoretically grounded path to reliable high-order interpolants on higher-dimensional Delaunay meshes, with implications for finite-element analysis in complex geometries. The approach harmonizes geometric meshing theory (thickness, protection, and Rajan’s functionals) with classical interpolation theory to yield quasi-optimality guarantees in a setting where exact optimality is challenging.

Abstract

There are very few mathematical results governing the interpolation of functions or their gradients on Delaunay meshes in more than two dimensions. Unfortunately, the standard techniques for proving optimal interpolation properties are often limited to triangular meshes. Furthermore, the results which do exist, are tailored towards interpolation with piecewise linear polynomials. In fact, we are unaware of any results which govern the high-order, piecewise polynomial interpolation of functions or their gradients on Delaunay meshes. In order to address this issue, we prove that quasi-optimal, high-order, piecewise polynomial gradient interpolation can be successfully achieved on protected Delaunay meshes. In addition, we generalize our analysis beyond gradient interpolation, and prove quasi-optimal interpolation properties for sufficiently-smooth vector fields. Throughout the paper, we use the words 'quasi-optimal', because the quality of interpolation depends (in part) on the minimum thickness of simplicies in the mesh. Fortunately, the minimum thickness can be precisely controlled on protected Delaunay meshes in $\mathbb{R}^d$. Furthermore, the current best mathematical estimates for minimum thickness have been obtained on such meshes. In this sense, the proposed interpolation is optimal, although, we acknowledge that future work may reveal an alternative Delaunay meshing strategy with better control over the minimum thickness. With this caveat in mind, we refer to our interpolation on protected Delaunay meshes as quasi-optimal.

Figures (2)

• Figure 1: Left, the sizing function for a tetrahedron element $K$ evaluated at its vertices, $\bm{p}_{K,i}$. Here, the longest edge length between vertices is denoted by $\Delta(K)$. Right, the squared-reciprocal of the sizing function evaluated at the vertices, and its linear interpolant $\mathcal{L}_{[1/\mathcal{D}^2]}(\bm{x})$.
• Figure 2: Mapping between the reference element and the physical element $K$ in three dimensions.

Theorems & Definitions (18)

• Definition 1: Roughness Functional
• Definition 2: Gradient Norm
• Theorem 3: Equivalence of the Roughness Functional and the Gradient Norm
• Remark 4: Sliver-Detecting Constant
• Remark 5: Protected Delaunay Meshes and Thickness
• Remark 6: Maximum Protection
• Remark 7: Max Edge-Length Constant
• Lemma 8: Upper Bound for the Roughness Functional
• Remark 9: Min-Containment Radius Constant
• Remark 10: Rajan's Functional