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

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 an energy estimate and the novel approach of exploiting the almost mass conservation of the error equations to derive a Poincaré-type inequality. We demonstrate how this approach can be generalized to other almost mass conserving problems. To this end we prove optimal-order fully discrete error estimates for the Cahn–Hilliard equation on evolving surfaces. We illustrate and complement our findings by numerical experiments.