Domain decomposition methods for the Stokes-Biot model of fluid-poroelastic structure interaction

Abstract

We develop a non-overlapping domain decomposition method for the numerical solution of the Stokes-Biot model of fluid-poroelastic structure interaction in a mixed form. The model is based on a velocity-pressure formulation for the free fluid, a three-field stress-displacement-rotation formulation with weakly symmetric stress for the solid deformation, and a Darcy velocity-pressure formulation for the fluid in the poroelastic media. Mass conservation, balance of stress, and the Beavers-Joseph-Saffman slip with friction condition are imposed on the interface. The interface conditions are incorporated through Lagrange multipliers modeling the traces of the displacement and the Darcy pressure. The system is discretized using stable mixed finite element spaces for Stokes flow, elasticity, and Darcy flow. The domain is decomposed into a union of subdomains of either Stokes or Biot type with three types of interfaces: Stokes-Stokes, Biot-Biot, and Stokes-Biot. On the Stokes-Stokes interfaces, a normal stress Lagrange multiplier is introduced to impose weakly velocity continuity, while the Biot-Biot and Stokes-Biot interfaces are equipped with displacement and pressure Lagrange multipliers to impose weakly continuity of normal stress and normal velocity, respectively. The global problem is reduced via Schur complement to an interface problem for the Lagrange multipliers, which is solved by GMRES. Each iteration requires the solution of local Stokes or Biot problems, which can be performed in parallel. We show that the resulting interface operator is positive definite and analyze the convergence of the GMRES iteration through fields-of-value analysis. Numerical experiments are presented to illustrate the performance of the method.