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Error estimates for full discretization by an almost mass conservation technique for Cahn--Hilliard systems with dynamic boundary conditions

Nils Bullerjahn

TL;DR

This work proves optimal-order error estimates for the fully discrete bulk–surface Cahn–Hilliard system with dynamic boundary conditions, using linear bulk–surface finite elements in space and a q-step linearly implicit BDF method in time. A central advancement is a discrete Poincaré–Wirtinger inequality derived from an almost mass-conservation property of the error, enabling robust energy estimates without relying on anti-symmetric structure. The analysis covers a broad range of coupling parameters $(K,L)$, establishes stability and consistency, and yields rigorous $O(h^2+\tau^q)$ convergence with $q\in\{1,\dots,5\}$ under mild time-step restrictions, along with a generalization path to evolving-surface CH equations. Numerical experiments corroborate the theory, demonstrating convergence rates and illustrating how transmission and reaction rates shape boundary dynamics, mass conservation, and droplet evolution. Overall, the results provide rigorous guarantees for accurate and physically faithful simulations of coupled bulk–surface phase-field models with mass-transfer across the boundary.

Abstract

A proof of optimal-order error estimates is given for the full discretization of the bulk--surface Cahn--Hilliard system with 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 one to five in time. The error estimates are obtained by a consistency and stability analysis, based on energy estimates and the novel approach of exploiting the almost mass conservation of the error equations to derive a Poincaré-type inequality. We also outline how this approach can be generalized to other mass conserving problems and illustrate our findings by numerical experiments.

Error estimates for full discretization by an almost mass conservation technique for Cahn--Hilliard systems with dynamic boundary conditions

TL;DR

This work proves optimal-order error estimates for the fully discrete bulk–surface Cahn–Hilliard system with dynamic boundary conditions, using linear bulk–surface finite elements in space and a q-step linearly implicit BDF method in time. A central advancement is a discrete Poincaré–Wirtinger inequality derived from an almost mass-conservation property of the error, enabling robust energy estimates without relying on anti-symmetric structure. The analysis covers a broad range of coupling parameters , establishes stability and consistency, and yields rigorous convergence with under mild time-step restrictions, along with a generalization path to evolving-surface CH equations. Numerical experiments corroborate the theory, demonstrating convergence rates and illustrating how transmission and reaction rates shape boundary dynamics, mass conservation, and droplet evolution. Overall, the results provide rigorous guarantees for accurate and physically faithful simulations of coupled bulk–surface phase-field models with mass-transfer across the boundary.

Abstract

A proof of optimal-order error estimates is given for the full discretization of the bulk--surface Cahn--Hilliard system with 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 one to five in time. The error estimates are obtained by a consistency and stability analysis, based on energy estimates and the novel approach of exploiting the almost mass conservation of the error equations to derive a Poincaré-type inequality. We also outline how this approach can be generalized to other mass conserving problems and illustrate our findings by numerical experiments.

Paper Structure

This paper contains 28 sections, 12 theorems, 107 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Bulk--surface Cahn--Hilliard system with dynamic boundary conditions
  3. Notation and Preliminaries
  4. Full discretization of the bulk--surface Cahn--Hilliard system with dynamic boundary conditions
  5. Bulk--surface finite elements
  6. Linearly implicit backward difference time discretization
  7. The fully discrete scheme in abstract form
  8. Main result: Optimal-order fully discrete error estimates
  9. Bulk--surface Poincaré--Wirtinger inequality
  10. A continuous perturbation result
  11. Stability of the numerical scheme
  12. Ritz map and error equation
  13. An $L^\infty$ bound for the Ritz map
  14. Results by Dahlquist and Nevanlinna $\&$ Odeh
  15. Stability result for $K \in [0,\infty)$ and $L\in (0,\infty)$
  16. ...and 13 more sections

Key Result

Theorem 5.1

Let $1\leq q \leq 5$ and $(u,\psi,\mu,\theta)$ be a sufficiently smooth solution (see eq:regularityAss) of the bulk--surface Cahn--Hilliard system with dynamic boundary conditions eq:CHweak, with eq:AssumptionPara and nonlinear potentials satisfying eq:RegAssPot. Then there exists $h_0>0$ such that provided the error in the starting values $I_h^{q-1}$, defined in eq:DefIhq, is $O((\tau^q+h^2)^2)$

Figures (10)

  • Figure 1: Sketch of the different types of dynamic boundary conditions corresponding to the parameter-choices $K,L \in [0,\infty]$. The main result of this paper, Theorem \ref{['thm:OptimalOrderErrorEst']} shows optimal-order error estimates for the range of parameters marked in gray (the domain $[0,\infty]^2$ except circles and dashed lines).
  • Figure 2: General scheme of the new stability proof.
  • Figure 3: Spatial convergence plot for the linear bulk--surface FEM/BDF2 approximation to the bulk--surface Cahn--Hilliard system with dynamic boundary conditions for $K=0$ and $L=100$
  • Figure 4: Spatial convergence plot for the linear bulk--surface FEM/BDF2 approximation to the bulk--surface Cahn--Hilliard system with dynamic boundary conditions for $K=10$ and $L=0.01$.
  • Figure 5: Temporal convergence plot for the linear bulk--surface FEM/BDF2 approximation to the bulk--surface Cahn--Hilliard system with dynamic boundary conditions for $K=0$ and $L=100$
  • ...and 5 more figures

Theorems & Definitions (22)

  • Theorem 5.1
  • Remark 5.2
  • Lemma 6.1: Poincaré--Wirtinger inequality for bulk and surface
  • proof
  • Lemma 8.1
  • Lemma 8.2: NevanlinnaOdeh1981
  • Proposition 8.3
  • proof
  • Proposition 8.4
  • proof
  • ...and 12 more