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Divergence-free unfitted finite element discretisations for the Darcy problem

Santiago Badia, Anne Boschman, Alberto F. Martín, Erik Nilsson, Ricardo Ruiz-Baier, Sara Zahedi

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.

Divergence-free unfitted finite element discretisations for the Darcy problem

Abstract

We develop an unfitted compatible finite element discretisation for the Darcy problem based on -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 -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.

Paper Structure

This paper contains 25 sections, 7 theorems, 103 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Geometrical discretisation
  3. Problem formulation and preliminaries
  4. Discrete spaces
  5. Discrete formulation
  6. Stability analysis
  7. A priori error analysis
  8. Augmented Lagrangian formulation
  9. Stabilisation
  10. Cell-wise stabilisation
  11. Extension operator
  12. Aggregate-wise projection
  13. Stability and approximability properties
  14. Face-based stabilisation
  15. Numerical experiments
  16. ...and 10 more sections

Key Result

Lemma 3.3

If $\Gamma_p \neq \emptyset$, the discrete solution $u_h$ satisfies $\mathrm{div} \, u_h = -\pi_{h}^{0}(g)$ on $\Omega_h$. If $\Gamma_p = \emptyset$, there exists a constant $c_h \in \mathbb{R}$ such that $\mathrm{div} \, u_h = -\pi_{h}^{0}(g) + c_h$ on $\Omega_h$. For $g=0$, the discrete flux diver

Figures (6)

  • Figure 8.1: Cut square problem: $h$-convergence test using $\Gamma = \Gamma_p$. For the AL-BGP method, $\tau_{\mathrm{d}}= \tau_{\mathrm{0}}=10^{0}$ is used.
  • Figure 8.2: Cut square problem: $h$-convergence test using $\Gamma = \Gamma_{u}$. The stabilisation parameters are set to $\tau_{\bullet}=10^{0}$.
  • Figure 8.3: Cut square problem: pressure robustness $h$-convergence test using $\Gamma = \Gamma_p$. For the AL-BGP method, $\tau_{\mathrm{d}}= \tau_{\mathrm{0}}=10^{0}$ is used.
  • Figure 8.4: Cut square problem for varying cut length ratios $h_{\mathrm{cut}}/h$ using $\Gamma = \Gamma_p \cup \Gamma_{u}$$(n=32)$. The stabilisation parameters are set to $\tau_{\bullet}=10^{0}$ and the penalty parameter $\gamma=10^{0}$.
  • Figure 8.5: Rectangle problem. Unstabilised method (std) compared with the FGP stabilisation method. For the FGP method, $\tau_{\bullet}= 1$ and $\delta = 0.25$ are used.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Lemma 3.3: Discrete mass conservation
  • proof
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Remark 3.7
  • Theorem 4.1: Stability
  • proof
  • Theorem 5.2: Error estimate
  • proof
  • ...and 8 more