@ARTICLE{GaRuWe01,
author = {Garcke, H. and Rumpf, M. and Weikard, U.},
title = {The {C}ahn-{H}illiard Equation with Elasticity, Finite Element Approximation
and Qualitative Analysis},
journal = {Interfaces and Free Boundaries},
year = {2001},
volume = {3},
pages = {101--118},
number = {1},
abstract = {We consider the Cahn-Hilliard equation - a fourth-order, nonlinear
parabolic diffusion equation describing phase separation of a binary
alloy which is quenched below a critical temperature. The occurrence
of two phases is due to a nonconvex double well free energy. The
evolution initially leads to a very fine microstructure of regions
with different phases which tend to become coarser at later times.
The resulting phases might have different elastic properties caused
by a different lattice spacing. This effect is not reflected by the
standard Cahn--Hilliard model. Here, we discuss an approach which
contains anisotropic elastic stresses by coupling the expanded diffusion
equation with a corresponding quasistationary linear elasticity problem
for the displacements on the microstructure. Convergence and a discrete
energy decay property are stated for a finite element discretization.
An appropriate timestep scheme based on the strongly A-stable $\Theta$-scheme
and a spatial grid adaptation by refining and coarsening improve
the algorithms efficiency significantly. Various numerical simulations
outline different qualitative effects of the generalized model. Finally,
a surprising stabilizing effect of the anisotropic elasticity is
observed in the limit case of a vanishing fourth order term, originally
representing interfacial energy.},
pdf = {http://numod.ins.uni-bonn.de/research/papers/public/GaRuWe01.pdf}
}