On the convergence and efficiency of splitting schemes for the Cahn-Hilliard-Biot model

TL;DR

This paper introduces a new, efficient solution strategy for the complex, coupled Cahn-Hilliard-Biot model, combining convex minimization and alternating minimization to improve convergence and robustness in simulations.

Contribution

The work develops a semi-implicit discretization and proves convergence of an alternating minimization method for the coupled model, enhancing solution efficiency and robustness.

Findings

Convex minimization reformulation ensures stable discretization.

Convergence of the alternating minimization method is theoretically proven.

Numerical experiments demonstrate improved efficiency and robustness.

Abstract

In this paper, we present a novel solution strategy for the Cahn-Hilliard-Biot model, a three-way coupled system that features the interplay of solid phase separation, fluid dynamics, and elastic deformations in porous media. It is a phase-field model that combines the Cahn-Hilliard regularized interface equation and Biot's equations of poroelasticity. Solving the system poses significant challenges due to its coupled, nonlinear, and non-convex nature. The main goal of this work is to provide a consistent and efficient solution strategy. With this in mind, we introduce a semi-implicit time discretization such that the resulting discrete system is equivalent to a convex minimization problem. Then, using abstract theory for convex problems, we prove the convergence of an alternating minimization method to the time-discrete system. The solution strategy is relatively flexible in terms of spatial discretization, although we require standard inverse inequalities for the guaranteed convergence of the alternating minimization method. Finally, we perform some numerical experiments that show the promise of the proposed solution strategy, both in terms of efficiency and robustness.