Numerical approximations and error analysis of the Cahn-Hilliard equation with reaction rate dependent dynamic boundary conditions

TL;DR

This paper develops a first-order, energy-stable numerical scheme for the Cahn-Hilliard equation with dynamic boundary conditions, providing error estimates and validating accuracy through numerical experiments.

Contribution

It introduces a new stabilized linearly implicit scheme with error analysis for the Cahn-Hilliard equation with reaction rate dependent boundary conditions.

Findings

The scheme is energy stable and first-order in time.

Numerical experiments confirm the scheme's accuracy and error estimates.

Convergence results are demonstrated for different relaxation parameters.

Abstract

We consider numerical approximations and error analysis for the Cahn-Hilliard equation with reaction rate dependent dynamic boundary conditions (P. Knopf et. al., arXiv, 2020). Based on the stabilized linearly implicit approach, a first-order in time, linear and energy stable scheme for solving this model is proposed. The corresponding semi-discretized-in-time error estimates for the scheme are also derived. Numerical experiments, including the comparison with the former work, the convergence results for the relaxation parameter $K\rightarrow0$ and $K\rightarrow\infty$ and the accuracy tests with respect to the time step size, are performed to validate the accuracy of the proposed scheme and the error analysis.