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

Lubomir Banas, Amy Novick-Cohen, Robert Nürnberg

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)37-64
Number of pages28
JournalNetworks and Heterogeneous Media
Volume8
Issue number1
DOIs
Publication statusPublished - Mar 2013

Keywords

  • Cahn-Hilliard equation
  • degenerate deep quench obstacle problem
  • coarsening
  • phase separation
  • CAHN-HILLIARD EQUATION
  • FINITE-ELEMENT APPROXIMATION
  • FE-CR ALLOYS
  • LOGARITHMIC FREE-ENERGY
  • PHASE FIELD MODEL
  • SPINODAL DECOMPOSITION
  • INTERPENETRATING MICROSTRUCTURES
  • VOID ELECTROMIGRATION
  • COARSENING RATES
  • COMPUTER-MODELS

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