Energy law preserving finite element scheme for the cahn-hilliard equation with dynamic boundary conditions

In this paper, we develop the energy law preserving method for a phase-field model of Cahn-Hilliard type describing binary mixtures. A new class of dynamic boundary conditions in a rather general setting proposed in [1] is adopted here. The model equations are discretized by a continuous finite element method in space and a midpoint scheme in time. The discrete energy law of the numerical method for the model with the dynamic boundary conditions is derived. By a few two-phase examples, we demonstrate the performance of the energy law preserving method for the computation of the phase-field model with the new class of dynamic boundary conditions, even in the case of relatively coarse mesh.