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

The deep quench obstacle problem for (x, t) ∈Ω× (0, T ), models phase separation at low temperatures. In (DQ), ε > 0, ∂(·) is the sub-differential of the indicator function I[-1,1](·), and u(x, t) should satisfy ν · ∇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-u2, 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. ©American Institute of Mathematical Sciences.