A cut finite element method for the Biot system of poroelasticity

TL;DR

This paper introduces a novel cut finite element method for solving the Biot system of poroelasticity, effectively handling complex geometries and ensuring stability and accuracy through a parameter robust formulation.

Contribution

The paper presents a new cut finite element approach for the Biot system that simplifies meshing for complex geometries while maintaining parameter robustness and stability.

Findings

Method achieves stability and optimal error estimates.

Numerical experiments confirm theoretical properties.

Successfully applied to realistic brain geometry.

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.