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A New Mixed Finite Element Method For The Cahn-Hilliard Equation

Zhen Liu, Rui Ma, Min Zhang

TL;DR

This work develops a new mixed finite element formulation for the Cahn–Hilliard equation by introducing an auxiliary tensor $σ$ in $H(divDiv, Ω; S)$ and showing its equivalence to the primal problem. A backward-Euler, semi-implicit time discretization is paired with a conforming spatial discretization on general meshes, yielding a linearized fully discrete scheme with a robust discrete inf-sup framework and projection-error control. Under standard regularity, the authors prove unique solvability for small mesh size and time step and derive an a priori error bound of order $O(τ^2 + h^{2k-2})$ for $k≥3$, along with $L^{∞}$-boundedness of the discrete solution. Numerical experiments in 2D and 3D, plus a postprocessing step, confirm the theoretical rates and demonstrate effectiveness on challenging domains such as the L-shaped region and drop coalescence, highlighting a unified, high-degree-capable framework.

Abstract

This paper presents a new mixed finite element method for the Cahn-Hilliard equation. The well-posedness of the mixed formulation is established and the error estimates for its linearized fully discrete scheme are provided. The new mixed finite element method provides a unified construction in two and three dimensions allowing for arbitrary polynomial degrees. Numerical experiments are given to validate the efficiency and accuracy of the theoretical results.

A New Mixed Finite Element Method For The Cahn-Hilliard Equation

TL;DR

This work develops a new mixed finite element formulation for the Cahn–Hilliard equation by introducing an auxiliary tensor in and showing its equivalence to the primal problem. A backward-Euler, semi-implicit time discretization is paired with a conforming spatial discretization on general meshes, yielding a linearized fully discrete scheme with a robust discrete inf-sup framework and projection-error control. Under standard regularity, the authors prove unique solvability for small mesh size and time step and derive an a priori error bound of order for , along with -boundedness of the discrete solution. Numerical experiments in 2D and 3D, plus a postprocessing step, confirm the theoretical rates and demonstrate effectiveness on challenging domains such as the L-shaped region and drop coalescence, highlighting a unified, high-degree-capable framework.

Abstract

This paper presents a new mixed finite element method for the Cahn-Hilliard equation. The well-posedness of the mixed formulation is established and the error estimates for its linearized fully discrete scheme are provided. The new mixed finite element method provides a unified construction in two and three dimensions allowing for arbitrary polynomial degrees. Numerical experiments are given to validate the efficiency and accuracy of the theoretical results.
Paper Structure (16 sections, 15 theorems, 97 equations, 4 figures, 4 tables)

This paper contains 16 sections, 15 theorems, 97 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mixed variational formulation
  4. Mixed finite element method
  5. Linearized fully discrete scheme
  6. Estimates of Lg and Lg
  7. Error Estimates
  8. Numerical Results
  9. Accuracy test in 2D
  10. Postprocessing
  11. Accuracy test on the L-shaped domain
  12. Accuracy test in 3D
  13. Coalescence of two drops
  14. Concluding remarks
  15. Declaration of competing interest
  16. ...and 1 more sections

Key Result

Lemma 2.1

Suppose $\Omega \subset \mathbb{R}^3$. Let $\boldsymbol{\tau} \in H(\operatorname{divDiv}, \Omega ; \mathbb{S}) \cap C^1(\bar{\Omega}; \mathbb{S})$ and $v \in H^2(\Omega)$. Then the following equality holds and furthermore

Figures (4)

  • Figure 1: Initial mesh for the square domain.
  • Figure 2: Initial mesh for the L-shaped domain.
  • Figure 3: Initial mesh for the cube domain.
  • Figure 5: Temporal sequence of snapshots for the coalescence of two drops. Results are obtained with $\tau = 1E-2$ and $h=\sqrt{2}/20$.

Theorems & Definitions (28)

  • Lemma 2.1: Green's identity in 3D chen2022finiteNiemi2019
  • Lemma 2.2: Discrete Gronwall's inequality Heywood1990
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Lemma 3.5
  • ...and 18 more