The degenerate and non-degenerate deep quench obstacle problem: a numerical comparison

The deep quench obstacle problem

(DQ) {partial derivative u/partial derivative t = del . M(u)del w,

w + epsilon(2)Delta u + u epsilon partial derivative Gamma(u),

for (x, t) is an element of Omega x (0, T), models phase separation at low temperatures. In (DQ), epsilon > 0, partial derivative Gamma(.) is the sub-differential of the indicator function I-[- 1,I-1](.), and u(x, t) should satisfy nu . del u = 0 on the "free boundary" where u = +/- 1. We shall assume that u is sufficiently smooth to make these notions well-defined. The problem (DQ) corresponds to the zero temperature "deep quench" limit of the Cahn-Hilliard equation. We focus here on a degenerate variant of (DQ) in which M(u) = 1-u(2), as well as on a constant mobility non-degenerate variant in which M(u) = 1. Although historically more emphasis has been placed on models with non-degenerate mobilities, degenerate mobilities capture some of the underlying physics more accurately. In the present paper, a careful numerical study is undertaken, utilizing a variety of benchmarks as well as new upper bounds for coarsening, in order to clarify evolutionary properties and to explore the differences in the two variant models.