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

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.

💡 Research Summary

The paper addresses the numerical solution of the coupled Cahn‑Hilliard‑Biot system, which models phase separation, fluid flow, and elastic deformation in porous media. Recognizing the challenges posed by the system’s nonlinearity, non‑convexity, and strong coupling, the authors develop a semi‑implicit time discretization that transforms each time step into a globally convex minimization problem. This is achieved by splitting the double‑well potential into convex and concave parts and treating the concave contribution explicitly, while the convex part and all other terms are handled implicitly.

With this reformulation, the authors apply an abstract convex‑optimization framework to devise an alternating minimization (AM) algorithm. The algorithm decouples the original system into a Cahn‑Hilliard‑like subproblem (for the phase field φ and chemical potential μ) and a Biot‑like subproblem (for displacement u, fluid content θ, and pressure p). Each subproblem remains convex, allowing the use of standard Newton solvers. The paper provides rigorous convergence proofs for the AM scheme under a set of realistic assumptions on material parameters (boundedness, Lipschitz continuity, and positivity) and requires only standard inverse inequalities for the chosen finite‑element spaces.

Two cases are examined: (i) state‑independent material coefficients, where the analysis is cleaner, and (ii) state‑dependent coefficients, where additional linearizations are introduced but the convex‑minimization structure is preserved. Spatial discretization employs conforming finite‑element spaces for φ, μ, u, θ, and p, with mean‑value constraints for the phase field and fluid content to ensure well‑posedness of the dual norms.

For the Biot subproblem, the authors compare two solution strategies: a monolithic approach that solves the heterogeneous Biot equations as a single block, and a splitting approach based on classical fixed‑stress or undrained schemes. Both are embedded within the AM loop; the splitting methods further decouple pressure from displacement, reducing assembly costs.

Numerical experiments in two and three dimensions validate the theory. Convergence studies confirm first‑order accuracy in time and optimal spatial rates. The AM algorithm dramatically reduces the number of nonlinear iterations compared with a fully coupled Newton method. Moreover, the fixed‑stress and undrained splits achieve comparable accuracy while cutting computational time by roughly 30‑45 % relative to the monolithic solve. The experiments also demonstrate robustness with respect to parameter variations and mesh refinement.

In conclusion, the paper presents a mathematically sound and practically efficient framework for solving the Cahn‑Hilliard‑Biot model. By leveraging convex splitting and alternating minimization, it offers a flexible platform that accommodates various spatial discretizations and can be extended to more complex boundary conditions, anisotropic materials, or multiscale couplings in future work.