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

TL;DR

This work addresses the numerical analysis of a Stokes–Brinkman–poro-hyperelastic interface problem, introducing a Lagrange multiplier–based mixed FEM to enforce interfacial mass-conservation. By reformulating the problem as a parabolic-type mixed system and establishing energy estimates, the authors prove existence, uniqueness, and stability for both the mixed and original formulations, and derive semi- and fully-discrete error bounds. The semi-discrete and fully discrete analyses reveal suboptimal convergence for certain interfacial velocity-displacement components, while achieving optimal rates for many fields; a weakly-continuous interface interpolant is constructed to handle nonmatching grids. Numerical experiments in geophysics and brain biomechanics validate the theoretical results and demonstrate the method’s applicability to complex interfacial flows with heterogeneous porosity and material properties, highlighting practical impact for subsurface flows and biomedical applications. Future work includes extending to nonlinear regimes and developing robust block preconditioners for 3D 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.

Figures (6)

• Figure 1: Material properties (porosity $\phi(\boldsymbol{x})$, permeability $\kappa(\boldsymbol{x})$, and Young modulus $E(\boldsymbol{x})$) from layer 80 of the $\mathrm{SPE10}$ benchmark dataset for reservoir simulations, herein projected onto a $\mathbb{P}_1$ field for the poro-hyperelastic sub-domain.
• Figure 2: Snapshots of the approximate solutions for fluid injection into a fracture porous medium using the SPE10-based benchmark test.
• Figure 3: Filtration into a deformable porous medium. All snapshots are taken at time $t = 2$ with $\tau = 0.1$, and the black outer line indicates the location of the undeformed domain.
• Figure 4: Zoom of the meshes on the interface at times $t = 0$, $t = 1$, and $t = 2$. Effect of using or not the harmonic extension to move the fluid domain (bottom and top, respectively).
• Figure 5: Snapshots of the approximate solutions for the interfacial flow in an idealized geometry at $T=1$ with $dt = 0.005$. The traction boundary conditions in the top right and bottom left corners (axial slices), respectively.

Theorems & Definitions (58)

• Remark 3.1
• Lemma 1
• Proof 1
• Remark 4.1
• Remark 4.2
• Lemma 1
• Proof 2
• Lemma 2
• Proof 3
• Lemma 3