A Lagrange Multiplier-based method for Stokes-linearized poro-hyperelastic interface problems

TL;DR

This paper introduces a novel Lagrange multiplier-based mixed finite element method for solving coupled fluid-poro-hyperelastic interface problems, with theoretical analysis and numerical validation in geophysical and biomechanical contexts.

Contribution

It develops a new numerical scheme for coupled fluid-poro-hyperelastic problems, including proof of well-posedness and error estimates, validated through numerical experiments.

Findings

The method achieves optimal convergence rates.

Numerical experiments confirm theoretical error estimates.

Simulations demonstrate applicability to geophysical and biomechanical problems.

Abstract

We propose a model for the coupling between free fluid and a linearized poro-hyperelastic body. In this model, the Brinkman equation is employed for fluid flow in the porous medium, incorporating inertial effects into the fluid dynamics. A generalized poromechanical framework is used, incorporating fluid inertial effects in accordance with thermodynamic principles. We carry out the analysis of the unique solvability of the governing equations, and the existence proof relies on an auxiliary multi-valued parabolic problem. We propose a Lagrange multiplier-based mixed finite element method for its numerical approximation and show the well-posedness of both semi-and fully-discrete problems. Then, a priori error estimates for both the semi- and fully-discrete schemes are derived. A series of numerical experiments is presented to confirm the theoretical convergence rates, and we also employ the proposed monolithic scheme to simulate 2D physical phenomena in geophysical fluids and biomechanics of the brain function.