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Error estimates for the interpolation and approximation of gradients and vector fields on protected Delaunay meshes in $\mathbb{R}^d$

David M. Williams, Mathijs Wintraecken

TL;DR

This work provides geometrically explicit error estimates for high-order gradient interpolation and gradient/ vector-field approximation on protected Delaunay meshes in $\mathbb{R}^d$. By extending the roughness framework to higher dimensions and coupling it with classical interpolation theory, the authors derive $L_2$ and $L_{\lambda}$ bounds that explicitly involve mesh quantities such as thickness $C_{\Xi}$, Rajan’s functional $\Theta$, and the max min-containment radius $R_{\max}$. They also extend the analysis to gradient approximation in elliptic problems and to vector-field interpolation, demonstrating that protecting meshes yields controllable, dimension-robust error constants. The results highlight the practical impact of mesh protection: enforcing a positive protection $\delta$ leads to lower bounds on slivers and tighter error bounds, enabling reliable high-order interpolation and approximation in higher dimensions. Overall, the paper provides a rigorous, dimension-agnostic framework linking interpolation accuracy to concrete geometric mesh properties, with protected Delaunay meshes offering a viable path to high-quality, high-order finite element interpolation.

Abstract

One frequently needs to interpolate or approximate gradients on simplicial meshes. Unfortunately, there are very few explicit mathematical results governing the interpolation or approximation of vector-valued functions on Delaunay meshes in more than two dimensions. Most of the existing results are tailored towards interpolation with piecewise linear polynomials. In contrast, interpolation with piecewise high-order polynomials is not well understood. In particular, the results in this area are sometimes difficult to immediately interpret, or to specialize to the Delaunay setting. In order to address this issue, we derive explicit error estimates for high-order, piecewise polynomial gradient interpolation and approximation on protected Delaunay meshes. In addition, we generalize our analysis beyond gradients, and obtain error estimates for sufficiently-smooth vector fields. Throughout the paper, we show that the quality of interpolation and approximation often depends (in part) on the minimum thickness of simplices in the mesh. Fortunately, the minimum thickness can be precisely controlled on protected Delaunay meshes in $\mathbb{R}^d$.

Error estimates for the interpolation and approximation of gradients and vector fields on protected Delaunay meshes in $\mathbb{R}^d$

TL;DR

This work provides geometrically explicit error estimates for high-order gradient interpolation and gradient/ vector-field approximation on protected Delaunay meshes in . By extending the roughness framework to higher dimensions and coupling it with classical interpolation theory, the authors derive and bounds that explicitly involve mesh quantities such as thickness , Rajan’s functional , and the max min-containment radius . They also extend the analysis to gradient approximation in elliptic problems and to vector-field interpolation, demonstrating that protecting meshes yields controllable, dimension-robust error constants. The results highlight the practical impact of mesh protection: enforcing a positive protection leads to lower bounds on slivers and tighter error bounds, enabling reliable high-order interpolation and approximation in higher dimensions. Overall, the paper provides a rigorous, dimension-agnostic framework linking interpolation accuracy to concrete geometric mesh properties, with protected Delaunay meshes offering a viable path to high-quality, high-order finite element interpolation.

Abstract

One frequently needs to interpolate or approximate gradients on simplicial meshes. Unfortunately, there are very few explicit mathematical results governing the interpolation or approximation of vector-valued functions on Delaunay meshes in more than two dimensions. Most of the existing results are tailored towards interpolation with piecewise linear polynomials. In contrast, interpolation with piecewise high-order polynomials is not well understood. In particular, the results in this area are sometimes difficult to immediately interpret, or to specialize to the Delaunay setting. In order to address this issue, we derive explicit error estimates for high-order, piecewise polynomial gradient interpolation and approximation on protected Delaunay meshes. In addition, we generalize our analysis beyond gradients, and obtain error estimates for sufficiently-smooth vector fields. Throughout the paper, we show that the quality of interpolation and approximation often depends (in part) on the minimum thickness of simplices in the mesh. Fortunately, the minimum thickness can be precisely controlled on protected Delaunay meshes in .

Paper Structure

This paper contains 24 sections, 9 theorems, 127 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background: Properties of Standard Delaunay Meshes
  3. The Two-Dimensional Case
  4. The $d$-Dimensional Case
  5. Summary of Existing Literature and New Contributions
  6. Paper Approach and Outline
  7. Mathematical Preliminaries
  8. Notation
  9. Elliptic Problem in 4D
  10. Roughness in Rd
  11. Alternative Strategy
  12. Fundamentals of Meshing
  13. General Properties
  14. Sizing Parameters
  15. Delaunay Properties
  16. ...and 9 more sections

Key Result

Lemma 1

The functional in Definition isotropic_functional and the gradient norm in Definition isotropic_gradient_norm are equivalent in the following sense where $v$ resides in the space of piecewise-$H^1$-scalar fields on $\Omega$, $C_{\Xi}$ and $C_{\Upsilon}$ are constants that depend on the mesh, and

Figures (2)

  • Figure 1: A diagram of the relationship between standard Delaunay meshes and protected Delaunay meshes. The protection parameter is denoted by $\delta$.
  • Figure 2: Mapping between the reference element and the physical element $K$ in three dimensions.

Theorems & Definitions (30)

  • Definition 1: Rajan's Functional
  • Definition 2: Max Edge-Length Functional
  • Remark 1: Protected Delaunay Meshes and Thickness
  • Remark 2: Protected Delaunay Meshes and Regularity
  • Remark 3: Maximum Protection
  • Definition 3: Roughness Functional
  • Definition 4: Gradient Norm
  • Lemma 1: Equivalence of the Roughness Functional and the Gradient Norm
  • proof
  • Lemma 2: Upper Bound for the Roughness Functional
  • ...and 20 more