A cut finite element method for the Biot system of poroelasticity

TL;DR

The Biot system models coupled poroelastic deformation and fluid flow on complex domains, where body-fitted meshes are challenging. This paper develops a CutFEM for Biot using a three-field displacement–pressure–total pressure formulation with $p_T = p_F + p_S$ and $p_S = -\lambda \nabla\cdot u$, achieving parameter-robust and geometrically robust stability through ghost-penalty stabilization and a modified inf-sup condition. Key contributions include a carefully constructed interpolant with boundary orthogonality, Biot-specific ghost penalties scaled to the governing bilinear forms, and a rigorous a priori error analysis yielding optimal convergence independent of cut size and material parameters. Numerical experiments in 2D and 3D, including a brain-mechanics example on realistic geometries, validate stability, accuracy, and geometric robustness, demonstrating the method’s practicality for complex poroelastic simulations. The approach enables robust unfitted simulations of poroelastic phenomena in irregular domains without body-fitted meshes, with potential for adaptive refinement and preconditioning in large-scale brain or geophysics applications.

Abstract

We propose a novel cut finite element method for the numerical solution of the Biot system of poroelasticity. The Biot system couples elastic deformation of a porous solid with viscous fluid flow and commonly arises on domains with complex geometries that make high-quality volumetric meshing challenging. To address this issue, we employ the cut finite element framework, where the domain boundary is represented independently of the background mesh, which significantly simplifies the meshing process. Our approach builds upon a parameter robust total pressure formulation of the Biot system, which we combine with the cut finite element method to develop a geometrically robust solution scheme, while preserving the parameter robustness. A key ingredient in the theoretical analysis is a modified inf-sup condition which also holds for mixed boundary conditions, leading to stability and optimal error estimates for the proposed formulation. Finally, we provide numerical evidence demonstrating the theoretical properties of the method and showcasing its capabilities by solving the Biot system on a realistic brain geometry.

Figures (7)

• Figure 3.1: Example of an unfitted mesh, with the domain $\Omega$ inside the boundary $\Gamma$.
• Figure 4.1: Example of a patch, with non-constant normal vector.
• Figure 6.1: Convergence analysis 2D. The displacement solution $\bu_h$ (magnitude) on the full computational domain and the fluid pressure solution $p_{F,h}$ on the physical domain for $N=32$ and $\lambda =1$ and $K=1$.
• Figure 6.2: Convergence analysis 2D. Left: Errors measured in both the discrete norms \ref{['eq:normstar_1']}-\ref{['eq:normstar_3']} and $L^2$-norm for $\lambda =1$ and $K=1$. Right: Errors measured in the discrete norms for different values of $\lambda$ and $K$.
• Figure 6.3: Geometrical robustness. Errors in both the discrete norms \ref{['eq:normstar_1']}-\ref{['eq:normstar_3']} and the $L^2$-norm for different cut configurations with and without stabilization.

Theorems & Definitions (22)

• Remark 3.1
• Lemma 4.1: Interpolation estimates
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 5.1
• proof
• Lemma 5.2
• proof