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Error estimates for full discretization of Cahn--Hilliard equation with dynamic boundary conditions

Nils Bullerjahn, Balázs Kovács

TL;DR

This work addresses the fully discrete approximation of the Cahn–Hilliard equation with Cahn–Hilliard-type dynamic boundary conditions by combining linear bulk–surface finite elements in space with linearly implicit $q$-step BDF time stepping ($q\in\{1,\dots,5\}$). By reformulating the problem as a second-order system and employing an energy-based stability analysis together with geometric, temporal, and discretization consistency estimates, the authors prove optimal-order error bounds $O(h^2+\tau^q)$ for the lifted bulk–surface variables and the associated chemical potential, under mild step-size restrictions. The analysis leverages $G$-stability and Nevanlinna–Odeh multipliers, along with a Ritz-map-based scheme initialization, to control nonlinearities under local Lipschitz conditions on the potentials. Numerical experiments on the unit disk and unit sphere corroborate the theoretical rates and demonstrate mass conservation and energy dissipation, confirming the practical reliability of the method for bulk–surface phase-field simulations.

Abstract

A proof of optimal-order error estimates is given for the full discretization of the Cahn--Hilliard equation with Cahn--Hilliard-type dynamic boundary conditions in a smooth domain. The numerical method combines a linear bulk--surface finite element discretization in space and linearly implicit backward difference formulae of order 1 to 5 in time. Optimal-order error estimates are proven. The error estimates are based on a consistency and stability analysis in an abstract framework, based on energy estimates exploiting the anti-symmetric structure of the second-order system.

Error estimates for full discretization of Cahn--Hilliard equation with dynamic boundary conditions

TL;DR

This work addresses the fully discrete approximation of the Cahn–Hilliard equation with Cahn–Hilliard-type dynamic boundary conditions by combining linear bulk–surface finite elements in space with linearly implicit -step BDF time stepping (). By reformulating the problem as a second-order system and employing an energy-based stability analysis together with geometric, temporal, and discretization consistency estimates, the authors prove optimal-order error bounds for the lifted bulk–surface variables and the associated chemical potential, under mild step-size restrictions. The analysis leverages -stability and Nevanlinna–Odeh multipliers, along with a Ritz-map-based scheme initialization, to control nonlinearities under local Lipschitz conditions on the potentials. Numerical experiments on the unit disk and unit sphere corroborate the theoretical rates and demonstrate mass conservation and energy dissipation, confirming the practical reliability of the method for bulk–surface phase-field simulations.

Abstract

A proof of optimal-order error estimates is given for the full discretization of the Cahn--Hilliard equation with Cahn--Hilliard-type dynamic boundary conditions in a smooth domain. The numerical method combines a linear bulk--surface finite element discretization in space and linearly implicit backward difference formulae of order 1 to 5 in time. Optimal-order error estimates are proven. The error estimates are based on a consistency and stability analysis in an abstract framework, based on energy estimates exploiting the anti-symmetric structure of the second-order system.
Paper Structure (28 sections, 12 theorems, 169 equations, 7 figures)

This paper contains 28 sections, 12 theorems, 169 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Cahn--Hilliard equation with dynamic boundary conditions
  3. Weak formulation as a second-order system
  4. Abstract formulation
  5. Semi-discretization of Cahn--Hilliard equation with dynamic boundary conditions
  6. Bulk--surface finite elements
  7. Discrete norms and bilinear forms
  8. A Ritz map
  9. The semi-discrete problem
  10. Matrix--vector formulation of the semi-discrete problem
  11. Linearly implicit backward difference time discretization
  12. A modified fully discrete problem
  13. Main results: optimal-order fully discrete error estimates
  14. Stability
  15. Error equations for the full discretization
  16. ...and 13 more sections

Key Result

Theorem 1

Let $u$ and $w$ be sufficiently smooth solutions (see eq:RegularityAss) of the Cahn--Hilliard equation with Cahn--Hilliard-type dynamic boundary conditions eq:strong2CH, with nonlinear potentials satisfying RegPotential. Then, there exists $h_0 > 0$ such that for all $h \leq h_0$ and $\tau > 0$, sat Further, the errors in the discrete time derivative eq:BDF diff and extrapolation, for finite eleme

Figures (7)

  • Figure 1: Sketch of the energy estimates for the fully discrete stability proof.
  • Figure 2: Spatial convergence plots for the linear bulk--surface FEM/BDF3 approximation to the Cahn--Hilliard equation with Cahn--Hilliard-type dynamic boundary conditions in the two-dimensional unit disk with free energy double well potentials.
  • Figure 3: Temporal convergence plots for the linear bulk--surface FEM/BDF3 approximation to the Cahn--Hilliard equation with Cahn--Hilliard-type dynamic boundary conditions in the two-dimensional unit disk with free energy double well potentials.
  • Figure 4: Spatial convergence plots for the linear bulk--surface FEM/BDF2 approximation to the Cahn--Hilliard equation with Cahn--Hilliard-type dynamic boundary conditions in the three-dimensional unit ball with free energy double well potentials.
  • Figure 5: Evolution plot on a cross-section of the three-dimensional unit ball, for the exact solution and numerical approximation by the bulk-surface finite element/BDF2 method of the convergence experiment, with $\tau=0.001$ and a mesh with dof$=1766$.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Lemma 2: Dahlquist1978
  • Lemma 3
  • Lemma 4: NevanlinnaOdeh1981
  • Lemma 5
  • proof
  • Proposition 6
  • ...and 13 more