Well-posedness of a novel Lagrange multiplier formulation for fluid-poroelastic interaction

TL;DR

This work introduces a novel monolithic fluid-poroelastic interaction formulation that uses three Lagrange multipliers to couple the time-dependent Stokes equations with the fully dynamic two-field Biot model. By carefully structuring the interface conditions into a saddle-point problem and selecting an appropriate grouping of variables, the authors prove well-posedness for both semi-discrete and fully discrete formulations and establish stability via energy estimates and Gronwall-type arguments. A stabilization term on the interface is incorporated to guarantee coercivity and enable a robust analysis, while the framework is designed to support a true partitioned, domain-decomposed solution approach in future work. The results lay the mathematical foundation for parallel, subdomain updates of Stokes and Biot regions, with planned error analysis and Schur-complement-based decoupling to realize a partitioned solver.

Abstract

We introduce a novel monolithic formulation that employs Lagrange multipliers (LMs) to couple a fluid flow governed by the time-dependent Stokes equations with a poroelastic structure described by the Biot equations. The formulation is developed in detail, and we establish the well-posedness of both the semi-discrete and fully discrete saddle point problems. We further prove the stability of the fully discrete system. This saddle point formulation, which utilizes three LMs, is designed to enable a partitioned approach that completely decouples the Stokes and Biot subdomains, and this approach will be explored in a subsequent work.

Theorems & Definitions (13)

• Remark
• Remark
• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 3.1
• proof