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

TL;DR

This paper establishes optimal error estimates for a numerical scheme solving the bulk-surface Cahn--Hilliard system with dynamic boundary conditions, using a novel approach based on almost mass conservation.

Contribution

It introduces a new error analysis method leveraging almost mass conservation, applicable to mass conserving problems, for finite element discretizations of complex PDE systems.

Findings

Optimal error estimates are proven for the discretization scheme.

The approach can be generalized to other mass conserving problems.

Numerical experiments confirm the theoretical results.

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 Poincar\'e-type inequality. We also outline how this approach can be generalized to other mass conserving problems and illustrate our findings by numerical experiments.