Uniformly Bounded Cochain Extensions and Uniform Poincaré Inequalities
This addresses a foundational gap in the rigorous mathematical underpinning of CutFEM, a domain-specific computational method for partial differential equations.
The paper constructs a globally bounded cochain extension operator for differential forms on Lipschitz domains that restores commutativity with the exterior derivative, providing a key analytical tool for Cut Finite Element Methods (CutFEM). It also derives continuous uniform Poincaré inequalities and lower bounds for the first Neumann eigenvalue on non-convex domains.
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 $HÎ^k(Ω)$ 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.