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A Stabilized Trace FEM for Surface Cahn--Hilliard Equations: Analysis and Simulations

Deepika Garg, Maxim Olshanskii

TL;DR

This work develops and analyzes a stabilized, unfitted TraceFEM for the surface Cahn–Hilliard equation, combining a geometry-accurate level-set representation with a stabilization mechanism to obtain robust, energy-stable time stepping. The authors establish a discrete energy-dissipation law and derive optimal-order a priori error estimates for simplicial meshes with FE order $m\ge1$, accounting for geometric approximation errors. Numerical experiments on multiple surfaces confirm the predicted convergence rates and demonstrate realistic phase-separation dynamics on both simple and complex geometries. The method offers a flexible and reliable tool for simulating surface phase separation in thin films and membranes without requiring surface-fitted meshes.

Abstract

This paper addresses the analysis and numerical assessment of a computational method for solving the Cahn--Hilliard equation defined on a surface. The proposed approach combines the stabilized trace finite element method for spatial discretization with an implicit--explicit scheme for temporal discretization. The method belongs to a class of unfitted finite element methods that use a fixed background mesh and a level-set function for implicit surface representation. We establish the numerical stability of the discrete problem by showing a suitable energy dissipation law for it. We further derive optimal-order error estimates assuming simplicial background meshes and finite element spaces of order $m \geq 1$. The effectiveness of the method is demonstrated through numerical experiments on several two-dimensional closed surfaces, confirming the theoretical results and illustrating the robustness and convergence properties of the scheme.

A Stabilized Trace FEM for Surface Cahn--Hilliard Equations: Analysis and Simulations

TL;DR

This work develops and analyzes a stabilized, unfitted TraceFEM for the surface Cahn–Hilliard equation, combining a geometry-accurate level-set representation with a stabilization mechanism to obtain robust, energy-stable time stepping. The authors establish a discrete energy-dissipation law and derive optimal-order a priori error estimates for simplicial meshes with FE order , accounting for geometric approximation errors. Numerical experiments on multiple surfaces confirm the predicted convergence rates and demonstrate realistic phase-separation dynamics on both simple and complex geometries. The method offers a flexible and reliable tool for simulating surface phase separation in thin films and membranes without requiring surface-fitted meshes.

Abstract

This paper addresses the analysis and numerical assessment of a computational method for solving the Cahn--Hilliard equation defined on a surface. The proposed approach combines the stabilized trace finite element method for spatial discretization with an implicit--explicit scheme for temporal discretization. The method belongs to a class of unfitted finite element methods that use a fixed background mesh and a level-set function for implicit surface representation. We establish the numerical stability of the discrete problem by showing a suitable energy dissipation law for it. We further derive optimal-order error estimates assuming simplicial background meshes and finite element spaces of order . The effectiveness of the method is demonstrated through numerical experiments on several two-dimensional closed surfaces, confirming the theoretical results and illustrating the robustness and convergence properties of the scheme.
Paper Structure (9 sections, 2 theorems, 76 equations, 3 figures, 1 table)

This paper contains 9 sections, 2 theorems, 76 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Preliminaries
  4. Finite element method and stability estimate
  5. Error Analysis
  6. Numerical results
  7. Convergence Study
  8. Phase separation on a sphere
  9. Phase separation in a complex surface

Key Result

Theorem 4.1

Let be the modified discrete energy. Under the condition the system (dis_ch) obeys the following energy dissipation law In particular, this implies that the scheme (dis_ch) is energy stable in the sense that $E^{n+1}_h \le E^{n}_h$ (the discrete analogue of (eq_7)) for all $n = 0, 1, 2,\dots$.

Figures (3)

  • Figure 6.1: Figures (a)--(b) present the convergence behavior the numerical solutions of CH equations with the the exact solution defined in \ref{['eq95677']}.
  • Figure 6.2: Figures (a)--(l) present the phase separation with $\mathcal{P}_1$ approximation and $\epsilon=0.05$ and $\tau$ is given as in Table \ref{['stoke_table1_31']}.
  • Figure 6.3: Figures (a)--(l) present the phase separation with $\mathcal{P}_1$ approximation at $\epsilon=0.05$ with $\tau$ is given as in Table \ref{['stoke_table1_31']}.

Theorems & Definitions (4)

  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof