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.

翻译：本文针对近期研究的Cahn-Hilliard-Biot系统，提出了一种结构保持的离散化方案，采用空间上的协调有限元与问题适配的显-隐式欧拉时间积分。我们证明该格式能够保持热力学结构，即质量与体积流体含量的平衡以及能量耗散平衡。在适当的增长条件下，建立了离散解的存在性。此外，研究表明该算法可实现为分裂方法，即将Cahn-Hilliard子系统与多孔弹性子系统解耦，其中前者为非线性系统，后者为线性系统。通过数值算例和收敛性测试对所提格式进行了验证。