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

Aaron Brunk, Dennis Höhn, Mária Lukáčová-Medvidová

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.

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

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.

Paper Structure

This paper contains 13 sections, 6 theorems, 60 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Assumptions
  5. Structural identities
  6. Variational structure
  7. Fully discrete scheme for the non-isothermal Cahn-Hilliard equation
  8. Structure-preserving property
  9. Existence of positive solutions
  10. Numerical experiments
  11. Convergence test
  12. Illustrating example
  13. Conclusion & Outlook

Key Result

Theorem 1

Let N and M be two positive integers and $C_1, C_2$ and $\varepsilon$ three positive constants. We define Here, the notation $\mathbf{y}>c$ means that each component of $\mathbf{y}$ is greater than a constant $c\in\mathbb{R}$. Further $\|\cdot\|$ is a norm defined over $\mathbb{R}^N$. Let $\mathbf{b}\in\mathbb{R}^N\times\mathbb{R}^M$, $\mathbf{g}$ and $\mathbf{G}$ be two continuous functions, res

Figures (4)

  • Figure 1: The driving potential $f$ for different $\vartheta$
  • Figure 2: Snapshots of volume fraction $\phi$ for different cross-coupling matrices $\mathbf{C}$: (Top) $\mathbf{C}=0\cdot\mathbf{I}$; (Bottom) $\mathbf{C}=10^{-4}\cdot\mathbf{I}$.
  • Figure 3: Snapshots of temperature $\vartheta$ for different cross-coupling matrices $\mathbf{C}$: (Top) $\mathbf{C}=0\cdot\mathbf{I}$; (Bottom) $\mathbf{C}=10^{-4}\cdot\mathbf{I}$.
  • Figure 4: Structure preserving properties over time (Top left): Mass conservation error; (Top right): Energy dissipation; (Bottom) Entropy production

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 2
  • proof : Proof of \ref{['th:spp']}
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 6
  • proof
  • Remark 7
  • ...and 3 more