A PDAE formulation of parabolic problems with dynamic boundary conditions (Q1726610)

The authors present a new abstract formulation of parabolic initial-boundary value problems with dynamic boundary conditions. This formulation includes locally reacting Wentzell boundary conditions and nonlocal kinetic boundary conditions modelling a diffusion on the surface of the computational domain. A formulation as partial differential-algebraic equation (PDAE) is derived where the boundary condition is treated in the form of a coupling condition. This constraint is included to the system as an additional equation and incorporated with the help of Lagrange multiplier. The resulting structure is similar to a parabolic problem with pure Dirichlet boundary conditions. The proposed formulation as PDAE provides more freedom within the discretization which may lead to a new class of robust numerical methods for such systems. The well-posedness of the formulation is shown.