Structure-preserving approximation of the non-isothermal Cahn-Hilliard system based on the entropy equation

TL;DR

This paper introduces a finite element-based approximation method for the non-isothermal Cahn-Hilliard system that preserves key physical properties like mass, energy, and entropy, validated through analysis and numerical tests.

Contribution

It presents a novel structure-preserving numerical scheme based on a variational formulation and mixed discretization for the non-isothermal Cahn-Hilliard equation.

Findings

The method conserves mass and energy exactly.

Numerical tests confirm convergence and stability.

The approach effectively captures entropy production.

Abstract

We propose and analyze a structure-preserving approximation of the non-isothermal Cahn-Hilliard equation using conforming finite elements for the spatial discretization and a problem-specific mixed explicit-implicit approach for the temporal discretization. To ensure the preservation of structural properties, i.e. conservation of mass and internal energy as well as entropy production, we introduce a suitable variational formulation for the continuous problem, based on the entropy equation. Analytical findings are supported by numerical tests, including convergence analysis.