Numerical investigation on the fixed-stress splitting scheme for Biot's equations: Optimality of the tuning parameter

TL;DR

This paper numerically investigates the fixed-stress splitting scheme for Biot's equations, focusing on optimizing the tuning parameter by considering boundary conditions and flow parameters, beyond traditional material-based values.

Contribution

It demonstrates that the optimal tuning parameter depends on boundary and flow parameters, highlighting the need for integrated analysis for better optimization.

Findings

Optimal parameter depends on boundary conditions.

Flow and boundary parameters influence tuning.

Traditional values may be suboptimal.

Abstract

We study the numerical solution of the quasi-static linear Biot's equations solved iteratively by the fixed-stress splitting scheme. In each iteration the mechanical and flow problems are decoupled, where the flow problem is solved by keeping an artificial mean stress fixed. This introduces a numerical tuning parameter which can be optimized. We investigate numerically the optimality of the parameter and compare our results with physically and mathematically motivated values from the literature, which commonly only depend on mechanical material parameters. We demonstrate, that the optimal value of the tuning parameter is also affected by the boundary conditions and material parameters associated to the fluid flow problem suggesting the need for the integration of those in further mathematical analyses optimizing the tuning parameter.