Divergence-free unfitted finite element discretisations for the Darcy problem

TL;DR

This paper introduces a novel unfitted finite element method for the Darcy problem that ensures mass conservation, remains stable with complex meshes, and provides optimal error estimates verified by numerical experiments.

Contribution

It develops a robust, mass-conserving unfitted finite element discretisation for Darcy flow with proven stability and error bounds, including an augmented Lagrangian variant.

Findings

Method achieves optimal convergence rates.

Ensures mass conservation up to solver tolerance.

Maintains stability independent of cut cell configurations.

Abstract

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.