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

TL;DR

This paper proves optimal error estimates for a numerical scheme discretizing the Cahn--Hilliard equation with dynamic boundary conditions, combining finite element spatial discretization and implicit time-stepping.

Contribution

It provides the first rigorous proof of optimal-order error estimates for this complex full discretization scheme.

Findings

Optimal error estimates are established.

The method achieves high-order accuracy in time.

Energy-based analysis ensures stability and consistency.

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.