Structure-preserving approximation for the non-isothermal Cahn-Hilliard-Navier-Stokes system

TL;DR

The paper tackles the non-isothermal Cahn-Hilliard-Navier-Stokes (CHNST) system by developing a structure-preserving, fully discrete finite element method built on a variational reformulation and a convex-concave splitting of the Cahn-Hilliard component. The scheme employs conforming spatial discretization and implicit time stepping to preserve physical structure, yielding discrete mass and energy conservation and nonnegative entropy production through a numerical dissipation term $\mathcal{D}_{num}^n \ge 0$. The authors provide a comprehensive numerical test showing expected convergence orders and discuss the method's stability and thermodynamic consistency in a periodic setting. The work lays a foundation for rigorous error analysis and extensions to more complex, density-dependent phase-field models in future studies.

Abstract

In this work we propose and analyse a structure-preserving approximation of the non-isothermal Cahn-Hilliard-Navier-Stokes system using conforming finite elements in space and implicit time discretisation with convex-concave splitting. The system is first reformulated into a variational form which reveal the structure of the equations, which is then used in the subsequent approximation.

Theorems & Definitions (8)

• proposition 1
• definition 1
• lemma 1
• proof
• lemma 2
• proof
• theorem 1
• proof