Numerical Approximation of Degenerate Fourth Order

A. description In recent years several mathematical models in uid dynamics, materials science and plasticity have lead to fourth order nonlinear degenerate parabolic equations. We mention for example the lubrication approximation for thin viscous lms (cf. and models that describe dislocation densities in plasticity (cf. Gr un (1995)). Although there has been a considerable eeort to understand degenerate parabolic equations of higher order analytically, there was no work from the numerical analysis viewpoint until Barrett, Blowey and Garcke (1998a) and Barrett, Blowey and Garcke (1998b). There are a few papers which include numerical experiments on the thin viscous lm equation in one space dimension (cf. and even fewer on the Cahn-Hilliard equation with a degenerate mobility (see Puri et al. (1997) and the references cited therein). In all of the above papers no attempts were made to analyse their nite diierence approaches. In Barrett, Blowey and Garcke (1998a) we developed and analysed a fully practical numerical scheme that is well-posed in all space dimensions for the degenerate parabolic problem: (P) Find a function u : 0; T] ! R such that @u @t + r:(b(u)ru) = 0 in T := (0; T); (1) u(x; 0) = u 0 (x) 8 x 2 ; @u @ = b(u) @u @ = 0 on @ (0; T); (2) where is a bounded domain in R d ; d63, with a Lipschitz boundary @ and is normal to @. This problem appears in the lubrication approximation of a thin viscous lm, which is driven by surface tension. In this context d = 1 (assuming the height is constant in one direction) or d = 2 and u describes the height of a liquid lm which spreads on a solid surface. The lubrication approximation with a no-slip condition at the interface leads to the diiusional mobility b(u) := c 1 juj 3 ; c 1 > 0: But a no-slip condition implies an innnite force at the contact line where the solid, liquid and air meet (cf. Huh and Scriven (1971) and Dussan and Davis (1974)). Therefore several variants of this condition, allowing slip, have appeared. They lead to a diiusional mobility of the form b(u) := c 1 juj 3 + c 2 juj p with c 1 ; c 2 > 0 and p 2 (0; 3) (cf. Greenspan (1978) and Hocking (1981)). To simplify our presentation we restrict ourselves to the …