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Error estimates for the Cahn--Hilliard equation with dynamic boundary conditions
A proof of convergence is given for bulk--surface finite element
semi-discretisation of the Cahn--Hilliard equation with Cahn--Hilliard-type
dynamic boundary conditions in a smooth domain. The semi-discretisation is
studied in the weak formulation as a second order system. Optimal-order
uniform-in-time error estimates are shown in the and 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