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A least squares approach to Whitney forms

Ludovico Bruni Bruno, Giacomo Elefante

TL;DR

This work develops a least-squares framework to construct high-order Whitney differential forms by projecting onto $\mathcal{P}_r^- \Lambda^k (T)$ using weights defined via integration over small simplices. By replacing exact unisolvence with a normal-equation approach $W^T W \mathbf{a}=W^T \mathbf{f}$ (and its weighted variant), the method enables stable, high-order discretizations of differential forms, with conditioning analyzed through $\kappa_2(W)$ and mitigated by enrichment of the weight set. Numerical experiments on segments, triangles, and tetrahedra show that, for easy-to-capture forms, least-squares approximants closely track interpolants, while Runge-type forms require coordinated degree elevation and weight-set enrichment to achieve convergence. The framework offers a practical route to robust Whitney-form discretizations and highlights open questions regarding efficient computation, stabilization, and potential blends of interpolation and projection techniques.

Abstract

In this work we describe and test the construction of least squares Whitney forms based on weights. If, on the one hand, the relevance of such a family of differential forms is nowadays clear in numerical analysis, on the other hand the selection of performing sets of supports (hence of weights) for projecting onto high order Whitney forms turns often to be a rough task. As an account of this, it is worth mentioning that Runge-like phenomena have been observed but still not resolved completely. We hence move away from sharp results on unisolvence and consider a least squares approach, obtaining results that are consistent with the nodal literature and making some steps towards the resolution of the aforementioned Runge phenomenon for high order Whitney forms.

A least squares approach to Whitney forms

TL;DR

This work develops a least-squares framework to construct high-order Whitney differential forms by projecting onto using weights defined via integration over small simplices. By replacing exact unisolvence with a normal-equation approach (and its weighted variant), the method enables stable, high-order discretizations of differential forms, with conditioning analyzed through and mitigated by enrichment of the weight set. Numerical experiments on segments, triangles, and tetrahedra show that, for easy-to-capture forms, least-squares approximants closely track interpolants, while Runge-type forms require coordinated degree elevation and weight-set enrichment to achieve convergence. The framework offers a practical route to robust Whitney-form discretizations and highlights open questions regarding efficient computation, stabilization, and potential blends of interpolation and projection techniques.

Abstract

In this work we describe and test the construction of least squares Whitney forms based on weights. If, on the one hand, the relevance of such a family of differential forms is nowadays clear in numerical analysis, on the other hand the selection of performing sets of supports (hence of weights) for projecting onto high order Whitney forms turns often to be a rough task. As an account of this, it is worth mentioning that Runge-like phenomena have been observed but still not resolved completely. We hence move away from sharp results on unisolvence and consider a least squares approach, obtaining results that are consistent with the nodal literature and making some steps towards the resolution of the aforementioned Runge phenomenon for high order Whitney forms.
Paper Structure (16 sections, 3 theorems, 42 equations, 6 figures, 1 table)

This paper contains 16 sections, 3 theorems, 42 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Approximation by Whitney forms
  3. From low-order to high-order Whitney forms
  4. Weights
  5. Least squares approximation
  6. Weighted least squares
  7. Solving the system
  8. Computation of high order weights
  9. Edge least squares
  10. Face least squares
  11. Numerical experiments
  12. Choice of supports of approximation
  13. On the real line
  14. Segments in the triangle
  15. Faces in the tetrahedron
  16. ...and 1 more sections

Key Result

Theorem 1

For each $\sigma \in \mathfrak{S}_k^n$, define $\mathcal{I}_\sigma \doteq \{ \boldsymbol{\alpha} \in \mathcal{I}(n+1,r-1) \mid \alpha_i = 0 \quad \forall \,i < \sigma(0) \}$. The set is a basis for $\mathcal{P}_r^- \Lambda^k (T)$.

Figures (6)

  • Figure 1: Small simplices on a triangle for $r = 2$.
  • Figure 2: Conditioning of the problem for segments in $\mathbb{R}^2$ (left, $r = 4$) and faces in $\mathbb{R}^3$ (right, $r = 4)$.
  • Figure 3: Left: error of the least squares approximation (blue) on $X_r^1 (T)$ of the $1$-form $\omega = e^{x+y} \mathrm{d} x + \sin{(xy)} \mathrm{d} y$ vs error of the interpolated (red) on minimal small simplices $X_{r,min}^1 (T)$. Right: error of the least squares approximation (blue) of the Runge $1$-form for the set $X^1_{r+10} (T)$ vs error of the interpolated (red) on minimal small simplices $X_{r,min}^1 (T)$.
  • Figure 4: Left: the degree is fixed ($r = 5$) and the number of segments is increased. Right: the number of segments is fixed (and large), the degree $r$ is increased. Uniform segments are considered.
  • Figure 5: Zero-norm of the error with respect to the total polynomial degree. Left: results for the $1$-form $\omega_1$. Right: results for the Runge $1$-form.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Proposition 2
  • Theorem 3
  • Remark 4
  • Remark 5