Stability and Error Analysis of a Partitioned Robin-Robin Method for Fluid Poroelastic Structure Interaction

Abstract Efficient and accurate numerical solvers for fluid-poroelastic structure interaction problems are critical to a variety of applications in geomechanics, biomedical engineering, and subsurface flow modeling. This work introduces and analyzes a partitioned method based on Robin-Robin coupling conditions for fluid-poroelastic structure interaction. We describe the fluid subproblem using the time dependent Stokes equations, and the poroelasticity subproblem using the Biot model. We propose to solve the fluid subproblem separately from the Biot subproblem, and to further decompose the Biot subproblem into a mechanics subproblem and a Darcy subproblem. We first consider a numerical method based on the Backward Euler time discretization, and show that it is conditionally stable, and sub-optimally convergent, with the rate of $$\mathcal {O}(\tau ^\frac{1}{2})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>τ</mml:mi> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . To improve the rate of convergence, we introduce a second numerical method based on Cauchy’s ‘ $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> -like’ scheme combined with Robin-Robin coupling conditions. This method consists of an iterative Backward Euler step followed by a Forward Euler step which is equivalent to a linear extrapolation. The iterative procedure is proved to be convergent assuming certain conditions on the time step and material parameters, under which the converged solution has also been shown to be stable. We further investigate the convergence rates and the impact of the Robin-Robin coupling parameter, L , using numerical examples. These examples show that the sub-optimal convergence of the first method is only observed when L is small. They also show that the iterative procedure decreases errors and increases the rates of convergence.