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

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.