[1] 
H. Garcke, M. Rumpf, and U. Weikard.
The CahnHilliard equation with elasticity, finite element
approximation and qualitative analysis.
Interfaces and Free Boundaries, 3(1):101118, 2001. [ bib  .pdf 1 ] We consider the CahnHilliard equation  a fourthorder, 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 CahnHilliard 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 Astable Θscheme 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.
