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A fully discrete evolving surface finite element method for the Cahn-Hilliard equation with a regular potential

Charles M. Elliott, Thomas Sales

TL;DR

This work develops and analyzes fully discrete evolving surface finite element methods for the Cahn–Hilliard equation on evolving surfaces with a regular potential. It introduces two time discretisations—the fully implicit backward Euler and a convex-splitting implicit–explicit scheme—along with isoparametric spatial discretisation, and proves optimal-order error bounds under suitable regularity and small-timestep conditions. The analysis covers both quadratic and general polynomial-growth potentials, employing Ritz projections, geometric perturbation estimates, and stability arguments; for non-quadratic cases a truncation strategy yields uniform bounds and preserves convergence rates. Numerical experiments on evolving spheres and tori corroborate the theoretical rates and reveal rich dynamics induced by surface evolution, including non-monotone energy behavior and a Mullins–Sekerka-like sharp-interface limit as $\varepsilon\to 0$. Overall, the paper provides the first rigorous fully discrete ESFEM error analysis for nonlinear fourth-order surface PDEs and offers practical guidance for accurate simulations of phase-field phenomena on evolving geometries.

Abstract

We study two fully discrete evolving surface finite element schemes for the Cahn-Hilliard equation on an evolving surface, given a smooth potential with polynomial growth. In particular we establish optimal order error bounds for a (fully implicit) backward Euler time-discretisation, and an implicit-explicit time-discretisation, with isoparametric surface finite elements discretising space.

A fully discrete evolving surface finite element method for the Cahn-Hilliard equation with a regular potential

TL;DR

This work develops and analyzes fully discrete evolving surface finite element methods for the Cahn–Hilliard equation on evolving surfaces with a regular potential. It introduces two time discretisations—the fully implicit backward Euler and a convex-splitting implicit–explicit scheme—along with isoparametric spatial discretisation, and proves optimal-order error bounds under suitable regularity and small-timestep conditions. The analysis covers both quadratic and general polynomial-growth potentials, employing Ritz projections, geometric perturbation estimates, and stability arguments; for non-quadratic cases a truncation strategy yields uniform bounds and preserves convergence rates. Numerical experiments on evolving spheres and tori corroborate the theoretical rates and reveal rich dynamics induced by surface evolution, including non-monotone energy behavior and a Mullins–Sekerka-like sharp-interface limit as . Overall, the paper provides the first rigorous fully discrete ESFEM error analysis for nonlinear fourth-order surface PDEs and offers practical guidance for accurate simulations of phase-field phenomena on evolving geometries.

Abstract

We study two fully discrete evolving surface finite element schemes for the Cahn-Hilliard equation on an evolving surface, given a smooth potential with polynomial growth. In particular we establish optimal order error bounds for a (fully implicit) backward Euler time-discretisation, and an implicit-explicit time-discretisation, with isoparametric surface finite elements discretising space.
Paper Structure (27 sections, 24 theorems, 240 equations, 5 figures, 4 tables)

This paper contains 27 sections, 24 theorems, 240 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some geometric analysis
  4. Sobolev spaces
  5. Evolving surfaces
  6. Time-dependent Lebesgue/Bochner spaces
  7. Triangulated surfaces
  8. Construction
  9. Lifts and interpolation
  10. Evolving triangulated surfaces
  11. Geometric perturbation estimates and the Ritz projection
  12. A fully implicit scheme
  13. Scheme and main results
  14. Well-posedness
  15. Stability
  16. ...and 12 more sections

Key Result

Proposition 2.7

Let $\eta, \zeta \in H^1_{H^{-1}} \cap L^2_{L^2}.$ Then $t \mapsto m(\eta(t), \zeta(t))$ is absolutely continuous and such that Moreover, if $\nabla_{\Gamma} \partial^{\bullet} \eta, \nabla_{\Gamma} \partial^{\bullet} \zeta \in L^2_{L^2(\Gamma)}$ then $t \mapsto a_S(\eta(t), \zeta(t))$ is absolutely continuous and such that Here where

Figures (5)

  • Figure 1: Plot of the Ginzburg-Landau functional (for the fully implicit scheme) on an evolving torus with constant surface area over $t\in [0,1]$.
  • Figure 2: Plot of the Ginzburg-Landau functional (for the fully implicit scheme) on a torus with periodic evolution over $t\in [0,1]$.
  • Figure 3: Evolution of $u$ (computed by the fully implicit scheme). Regions of blue correspond to a negative quantity, and red a positive quantity. Solution converges to an oscillatory function, as indicated by the last 3 images.
  • Figure 4: Convergence to a sharp interface at $t = 0.3$.
  • Figure 5: Propagation of a sharp interface on an evolving sphere for $\varepsilon = 0.05$, computed by the fully implicit scheme. The centre of the interface is given in green.

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: $C^{k+1}$ evolving surface
  • Definition 2.4
  • Definition 2.5: Strong material derivative
  • Definition 2.6: Weak material derivative
  • Proposition 2.7: dziuk2013finite, Lemma 5.2
  • Definition 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 45 more