A Lagrange multiplier-based method for Stokes–generalized poroelasticity interface problems

Abstract We study a mathematical model describing the interaction between a fluid and a poroelastic structure, along with its numerical approximation. The fluid domain is governed by the unsteady incompressible Stokes equations, while the poroelastic region is modeled using the linearized poro-hyperelastic equations. Within this region, the Brinkman equation is employed to describe fluid flow through the porous medium, incorporating inertial effects into the fluid dynamics. A generalized poromechanical framework is adopted to incorporate these inertial effects in accordance with thermodynamic principles. An alternative formulation is used in which the primary variables are the elastic stress and structural velocity. This formulation serves as a mathematical tool to establish the unique solvability of the governing equations, with the existence proof relying on an auxiliary multi-valued parabolic problem. For the numerical approximation, we propose a Lagrange multiplier-based mixed finite element method and demonstrate the well-posedness of both semidiscrete and fully discrete problems. Furthermore, we derive a priori error estimates for both discretization schemes. Numerical experiments validate the theoretical convergence rates. Finally, we apply the proposed monolithic scheme to simulate two-dimensional phenomena arising in geophysical flows and brain biomechanics.