Uniformly Bounded Cochain Extensions and Uniform Poincar\'e Inequalities

TL;DR

This paper introduces a new bounded cochain extension operator for differential forms on Lipschitz domains, enabling uniform Poincaré inequalities and spectral bounds crucial for CutFEM analysis.

Contribution

It develops a globally bounded cochain extension that preserves exterior derivative commutation on arbitrary topology domains, enhancing analytical tools for CutFEM.

Findings

Constructed a global bounded cochain extension operator for differential forms.

Restored commutativity with the exterior derivative in the $H ext{Lambda}^k( ext{Omega})$ setting.

Established uniform Poincaré inequalities and eigenvalue bounds on non-convex domains.

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 $H\Lambda^k(\Omega)$ 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 Poincar\'e inequalities and lower bounds for the first Neumann eigenvalue on non-convex domains.