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 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.