Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media

TL;DR

This paper introduces robust fixed-stress splitting schemes combined with Anderson acceleration for efficient, stable, and decoupled simulation of nonlinear unsaturated poromechanics, validated by theoretical analysis and numerical experiments.

Contribution

It proposes novel fixed-stress splitting schemes with Anderson acceleration for nonlinear poromechanics, providing theoretical convergence guarantees and demonstrating improved robustness and efficiency.

Findings

The Fixed-Stress-L-scheme is simple, low-cost, and robust.

Anderson acceleration effectively speeds up convergence.

The combined Fixed-Stress-Newton scheme with Anderson acceleration is highly robust.

Abstract

In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, employing the equivalent pore pressure. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow the use of separate simulators, which is not met by the classical Newton method. We propose three different linearization schemes incorporating the fixed-stress splitting scheme, coupled with an L-scheme, Modified Picard and Newton linearization of the flow. All schemes allow the efficient and robust decoupling of mechanics and flow equations. In particular, the simplest scheme, the Fixed-Stress-L-scheme, employs solely constant diagonal stabilization, has low cost per iteration, and is very robust. Under mild, physical assumptions, it is theoretically shown to be a contraction. Due to possible break-down or slow convergence of all considered splitting schemes, Anderson acceleration is applied as post-processing. Based on a special case, we justify theoretically the general ability of the Anderson acceleration to effectively accelerate convergence and stabilize the underlying scheme, allowing even non-contractive fixed-point iterations to converge. To our knowledge, this is the first theoretical indication of this kind. Theoretical findings are confirmed by numerical results. In particular, Anderson acceleration has been demonstrated to be very effective for the considered Picard-type methods. Finally, the Fixed-Stress-Newton scheme combined with Anderson acceleration provides a robust linearization scheme, meeting the above criteria.