Structure-preserving approximation of the Cahn-Hilliard-Biot system

TL;DR

This paper introduces a structure-preserving numerical scheme for the Cahn-Hilliard-Biot system, ensuring thermodynamic consistency, mass conservation, and energy dissipation, with proven existence of solutions and effective decoupling of subsystems.

Contribution

It develops a novel conforming finite element discretisation that preserves key physical structures and allows for a decoupled, efficient solution approach.

Findings

The scheme maintains thermodynamic properties in simulations.

Existence of discrete solutions is rigorously proven.

Numerical tests confirm convergence and effectiveness.

Abstract

In this work, we propose a structure-preserving discretisation for the recently studied Cahn-Hilliard-Biot system using conforming finite elements in space and problem-adapted explicit-implicit Euler time integration. We prove that the scheme preserves the thermodynamic structure, that is, the balance of mass and volumetric fluid content and the energy dissipation balance. The existence of discrete solutions is established under suitable growth conditions. Furthermore, it is shown that the algorithm can be realised as a splitting method, that is, decoupling the Cahn-Hilliard subsystem from the poro-elasticity subsystem, while the first one is nonlinear and the second subsystem is linear. The schemes are illustrated by numerical examples and a convergence test.