Error estimates for the Cahn-Hilliard equation with dynamic boundary conditions

A proof of convergence is given for a bulk-surface finite element semidiscretisation of the Cahn-Hilliard equation with Cahn-Hilliard-type dynamic boundary conditions in a smooth domain. The semidiscretisation is studied in an abstract weak formulation as a second-order system. Optimal-order uniform-in-time error estimates are shown in the L-2- and H-1-norms. The error estimates are based on a consistency and stability analysis. The proof of stability is performed in an abstract framework, based on energy estimates exploiting the anti-symmetric structure of the second-order system. Numerical experiments illustrate the theoretical results.