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Uniformly Bounded Cochain Extensions and Uniform Poincaré Inequalities

Erik Nilsson, Silvano Pitassi

Abstract

In this paper, we construct a novel global bounded cochain extension operator for differential forms on Lipschitz domains. Building upon the classical universal extension of Hiptmair, Li, and Zou, our construction restores global commutativity with the exterior derivative in the natural $HΛ^k(Ω)$ setting. The construction applies to domains and ambient extension sets of arbitrary topology, with strict commutation holding on the orthogonal complement of harmonic forms, as dictated by the underlying topological obstruction. This provides a missing analytical tool for the rigorous foundation of Cut Finite Element Methods (CutFEM). We also obtain continuous uniform Poincaré inequalities and lower bounds for the first Neumann eigenvalue on non-convex domains.

Uniformly Bounded Cochain Extensions and Uniform Poincaré Inequalities

Abstract

In this paper, we construct a novel global bounded cochain extension operator for differential forms on Lipschitz domains. Building upon the classical universal extension of Hiptmair, Li, and Zou, our construction restores global commutativity with the exterior derivative in the natural setting. The construction applies to domains and ambient extension sets of arbitrary topology, with strict commutation holding on the orthogonal complement of harmonic forms, as dictated by the underlying topological obstruction. This provides a missing analytical tool for the rigorous foundation of Cut Finite Element Methods (CutFEM). We also obtain continuous uniform Poincaré inequalities and lower bounds for the first Neumann eigenvalue on non-convex domains.

Paper Structure

This paper contains 16 sections, 15 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Construction of the cochain extension operators
  4. Recursive definition and well-posedness
  5. Stability bounds and proof of the main theorem
  6. Topological considerations and further remarks
  7. Equivalent gauge formulation and connection with previous literature
  8. Stability of unfitted methods
  9. Uniform discrete Poincaré inequality
  10. Uniform ghost norm Poincaré inequality
  11. Additional applications
  12. Uniform Poincaré inequalities
  13. Uniform lower bound for first Neumann eigenvalue of the Hodge Laplacian on Lipschitz domains
  14. Further discussion and open questions
  15. Poincaré constants and boundary conditions
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain. For any form degree $k$ and any regularity parameter $s \ge 0$, there exists an extension operator $E_\mathrm{HLZ}^k : H^{(s,s)}\Lambda^k(\Omega) \to H^{(s,s)}\Lambda^k(\mathbb{R}^n)$ and a constant $C_\mathrm{HLZ}$, which depends on t $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Comparison between the present global extension setting and a collar-based construction. (a) A bounded Lipschitz domain $\Omega$ is contained in an ambient Lipschitz domain $K$, and the extension is constructed on the exterior region $A=K\setminus\overline{\Omega}$, allowing for arbitrary ambient topology. (b) In a collar-based setting, the ambient domain $\Omega_+\supset\Omega$ remains localised near $\Omega$ and has the same homotopy type.
  • Figure 2: Illustration of the convex hull of a collar neighbourhood $\Omega_+$.

Theorems & Definitions (38)

  • Theorem : Hiptmair, Li, Zou hiptmair2012universalextension
  • Theorem 1: Cochain extension operator
  • Lemma 2: Well-posedness of the recursive construction
  • proof
  • Definition 3: Extension operator
  • Lemma 4: Stability of the recursive Dirichlet step
  • proof
  • Remark 5: Dependence of the constant
  • Theorem 5: Cochain extension operator
  • proof
  • ...and 28 more