On Space-Time-Adaptive Convergent Finite-Element Schemes for a General Class of Lubrication-Type Equations

In this contribution, we summarize the key ideas of recently developed finite element schemes for fourthorder degenerate parabolic equations like the thin film equation. Our approach is inspired both by the physics to be modeled and by analytical methods used in the theory of these partial differential equations. As a consequence, the scheme preserves non-negativity or positivity of solutions in a natural way, it is convergent in all the physically relevant space dimensions, and moving contact lines are tracked efficiently. In a second part, we present some characteristic numerical simulations for thin film flow. We consider wetting solely induced by capillary forces and dewetting caused by instabilities due to long-range intermolecular forces. By dimensionalized numerical experiments, we investigate the stability of receding contact lines during dewetting processes. G. Grün, J. Becker, M. Rumpf