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

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.