Erik Nilsson

Santiago Badia, Anne Boschman, Alberto F. Martín et al.

We develop an unfitted compatible finite element discretisation for the Darcy problem based on $H(\mathrm{div})$-conforming flux spaces and discontinuous pressure spaces. The method is designed to preserve pointwise discrete mass conservation while remaining robust in the presence of arbitrarily small cut cells arising from unfitted meshes. Robustness is achieved by combining an $L^2$-stabilisation of the flux with an additional mixed-term stabilisation that enhances pressure control without destroying the local conservation structure. We consider both cell-wise (bulk) and face-based ghost-penalty realisations of the stabilisation. Mixed boundary conditions are handled by weak imposition of both flux and pressure traces on unfitted boundaries. We prove stability and optimal-order a priori error estimates with constants independent of the cut configuration, and establish pressure-robust flux error bounds in the case of pure pressure boundary conditions. We also introduce an augmented Lagrangian variant that improves control of the conservation constraint and is amenable to efficient preconditioning strategies. Numerical experiments for a range of cut configurations, boundary-condition regimes and parameter choices confirm the theoretical results, demonstrating optimal convergence, cut-independent conditioning and mass conservation up to solver tolerance.

Erik Nilsson, Silvano Pitassi

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.