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

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  • The deep quench obstacle problem $$ {\rm{\bf{(DQ)}}} \begin{equation}\left\{ \begin{array}{l} \frac{\partial u}{\partial t}=\nabla \cdot M(u) \nabla w, \\ w + \epsilon^2 \triangle u + u \in \partial \Gamma(u), \end{array} \right. \end{equation}$$ for $(x,t) \in \Omega \times (0,T)$, models phase separation at low temperatures. In (DQ), $\epsilon>0,$ $\partial \Gamma(\cdot)$ is the sub-differential of the indicator function $I_{[-1,1]}(\cdot),$ and $u(x,t)$ should satisfy $\nu \cdot \nabla u=0$ on the ``free boundary'' where $u=\pm 1$. 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.
    Mathematics Subject Classification: 35K86, 74N20, 35K35, 35K65.

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