Numerical Methods for Fourth Order Nonlinear Degenerate Diffusion Problems

Numerical schemes are presented for a class of fourth order diffusion problems. These problems arise in lubrication theory for thin films of viscous fluids on surfaces. The equations being in general fourth order degenerate parabolic, additional singular terms of second order may occur to model effects of gravity, molecular interactions or thermocapillarity. Furthermore, we incorporate nonlinear surface tension terms. Finally, in the case of a thin film flow driven by a surface active agent (surfactant), the coupling of the thin film equation with an evolution equation for the surfactant density has to be considered. Discretizing the arising nonlinearities in a subtle way enables us to establish discrete counterparts of the essential integral estimates found in the continuous setting. As a consequence, the resulting algorithms are efficient, and results on convergence and nonnegativity or even strict positivity of discrete solutions follow in a natural way. The paper presents a finite element and a finite volume scheme and compares both approaches. Furthermore, an overview over qualitative properties of solutions is given, and various applications show the potential of the proposed approach.