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This work addresses the problem of solving the Cahn-Hilliard equationnumerically. For that we introduce an abstract formulation for Cahn-Hilliardtype equations with dynamic boundary conditions, we conduct the spatialsemidiscretization via finite elements and prove error bounds based on thetechnique of energy estimates. The variational formulation forCahn-Hilliard/Cahn-Hilliard coupling, will apply to a larger abstract class ofproblems and is similar to the usual weak formulation of parabolic problems. Incontrast to problems with non dynamic boundary conditions, the Hilbert spaces$L^2(\Omega)$ and $H^1(\Omega)$ are exchanged with the spaces$L^2(\Omega)\times L^2(\Gamma)$ and $\lbrace v\in H^1(\Omega): \gamma v \inH^1(\Gamma)\rbrace$, respectively. Because we are considering a fourth-orderdifferential equation, which will be described by a system of two second-orderdifferential equations, the variational formulation also consists of a systemof two equations.

Abstract error analysis for Cahn--Hilliard type equations with dynamic boundary conditions