Skip to content
DraconiticResearch
Search Sources API
201,818 studies31 sourcesUpdated 32 min ago
PreprintarXiv licencearXiv

Whitney--Lagrange Decomposition for the First Family of Finite Element Differential Forms

  • Long Chen
  • Xuehai Huang
Open at sourcePDFDOICite

Abstract

We derive polynomial bubble duality in barycentric coordinates and construct explicit nodal bases for the first family of finite element differential forms. Logarithmic coordinates give an elementary proof of the Whitney--bubble decomposition. The Hessian of negative entropy defines a metric whose normalized Hodge star maps bubble generators to complementary-degree polynomial forms. This gives a direct geometric description of the known dualities and proves nondegeneracy of their integral pairings. Combining Whitney forms with shifted Lagrange polynomials then gives bases dual to tangential point evaluations in every dimension and order. The construction extends to scalar nodes that are unisolvent on every face. The interpolant is independent of the roots and commutes with face restriction.

Related

  1. A robust finite element method for incompressible magnetohydrodynamics using Whitney forms
    Tianwei Yu
    arXivPreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
  2. Sparsely connected neural network representation of Lagrange finite element function
    Jiaxiong Hao, Yunqing Huang, Nianyu Yi
    arXivPreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
  3. Quantum Domain Decomposition for Preconditioning the Finite Element Method
    Elise Fressart, Michel Nowak, Nicole Spillane
    HALPreprintCC BY 4.0Full textNumerical analysis & scientific computingQuantum physics & quantum informationHAL
  4. Finite element exterior calculus for spectra and pseudospectra of advection-diffusion of differential forms
    Daniele Boffi, Kaibo Hu, Yizhou Liang +1
    arXivPreprintCC BY 4.0Full textNumerical analysis & scientific computingarXiv
  5. A family of symmetric counterexamples to the bent sum decomposition conjecture
    Máté Róbert Kepes, Anita Kercsó-Molnár, Viczián András
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{chen2026whitneylagrange,
  title = {{Whitney--Lagrange Decomposition for the First Family of Finite Element Differential Forms}},
  author = {Long Chen and Xuehai Huang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15477},
  eprint = {2610.15477},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15477},
}
RIS
TY  - UNPB
TI  - Whitney--Lagrange Decomposition for the First Family of Finite Element Differential Forms
AU  - Long Chen
AU  - Xuehai Huang
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15477
UR  - https://arxiv.org/abs/2610.15477
AB  - We derive polynomial bubble duality in barycentric coordinates and construct explicit nodal bases for the first family of finite element differential forms. Logarithmic coordinates give an elementary proof of the Whitney--bubble decomposition. The Hessian of negative entropy defines a metric whose normalized Hodge star maps bubble generators to complementary-degree polynomial forms. This gives a direct geometric description of the known dualities and proves nondegeneracy of their integral pairings. Combining Whitney forms with shifted Lagrange polynomials then gives bases dual to tangential point evaluations in every dimension and order. The construction extends to scalar nodes that are unisolvent on every face. The interpolant is independent of the roots and commutes with face restriction.
ER  -