The fixed-stress splitting scheme for Biot's equations as a modified Richardson iteration: Implications for optimal convergence

TL;DR

This paper analyzes the fixed-stress splitting scheme for Biot's equations, revealing how to optimally select the stabilization parameter to ensure convergence and improve computational efficiency.

Contribution

It provides a novel approach to determine the optimal stabilization parameter using matrix structure analysis in discretized Biot equations.

Findings

Optimal parameter choice improves convergence

Matrix structure analysis guides parameter selection

Enhanced stability in solving Biot's equations

Abstract

The fixed-stress splitting scheme is a popular method for iteratively solving the Biot equations. The method successively solves the flow and mechanic subproblems while adding a stabilizing term to the flow equation, which includes a parameter that can be chosen freely. However, the convergence properties of the scheme depend significantly on this parameter and choosing it carelessly might lead to a very slow, or even diverging, method. In this paper, we present a way to exploit the matrix structure arizing from discretizing the equations in the regime of impermeable porous media in order to obtain a priori knowledge of the optimal choice of this tuning/stabilization parameter.