Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid

We consider a fully practical finite element approximation of the degenerate Cahn-Hilliard equation with elasticity: Find the conserved order parameter, $\theta(x,t)\in[-1,1]$, and the displacement field, $\underline{u}(x,t) \in \mathbb{R}^2$, such that

$\gamma\,\textstyle{\frac{\partial \theta}{\partial t}}$	$= \nabla\,.\, (b(\theta) \,\nabla \,[- \gamma \, \Delta \theta + ... ...line{u}): \underline{\underline{\mathcal{E}}} (\underline{u})]\,) \,, \nonumber$
$\nabla \,.\, (c(\theta) \,\mathcal{C}\, \underline{\underline{\mathcal{E}}} (\underline{u}) )$	$= \underline{0}\,,$

subject to an initial condition $\theta^0(\cdot) \in [-1,1]$ on $\theta$ and boundary conditions on both equations. Here $\gamma \in {\mathbb{R}}_{>0}$ is the interfacial parameter, $\Psi$ is a non-smooth double well potential, $\underline{\underline{\mathcal{E}}}$ is the symmetric strain tensor, $\mathcal{C}$ is the possibly anisotropic elasticity tensor, $c(s):=c_0+\textstyle\frac12\,(1-c_0)\,(1+s)$ with $c_0(\gamma)\in {\mathbb{R}}_{>0}$ and $b(s):=1-s^2$ is the degenerate diffusional mobility. In addition to showing stability bounds for our approximation, we prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system in two space dimensions. Finally, some numerical experiments are presented.