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This paper studies bulk-surface splitting methods of first order for(semi-linear) parabolic partial differential equations with dynamic boundaryconditions. The proposed Lie splitting scheme is based on a reformulation ofthe problem as a coupled partial differential-algebraic equation system, i.e.,the boundary conditions are considered as a second dynamic equation which iscoupled to the bulk problem. The splitting approach is combined withbulk-surface finite elements and an implicit Euler discretization of the twosubsystems. We prove first-order convergence of the resulting fully discretescheme in the presence of a weak CFL condition of the form $\tau \leq c h$ forsome constant $c>0$. The convergence is also illustrated numerically usingdynamic boundary conditions of Allen-Cahn-type.

Bulk-surface Lie splitting for parabolic problems with dynamic boundary conditions