A structure-preserving discretization for the coupled Stokes–Biot equations

Abstract

We develop and analyze a structure-preserving discretization for the coupled Stokes–Biot equations. The method is based on a fully-mixed formulation and computes directly the fluid and poroelastic stresses and the Darcy velocity. The scheme employs equal-order polynomial approximations for all unknowns, yet exhibits local momentum conservation, strongly symmetric stress fields, and exactly divergence-free velocity in the fluid domain. Transmission conditions enforcing mass conservation, stress balance, and the Beavers–Joseph–Saffman slip with friction law are incorporated directly into the formulation as natural interface conditions through careful balance of finite element spaces across subdomains. Well-posedness of the fully discrete scheme based on the backward Euler time-stepping method is established, followed by optimal convergence error estimates for all variables. The analysis shows that the method is locking-free in the regimes of small storativity and almost incompressible material. Several numerical experiments are carried out to validate the accuracy and robustness of the proposed scheme.