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, $\underline u(x, t)\in \Bbb R^2$, such that $$\gamma\frac{\partial\theta}{\partial t}=\nabla\cdot (b(\theta)\nabla [-\gamma\Delta\theta + \gamma^{-1}\Psi'(\theta) + \tfrac 12 c' (\theta){\cal C}\underline{\underline{\cal E}}(\underline u) : \underline{\underline{\cal E}}(\underline u)] ),\quad \nabla\cdot (c(\theta){\cal C} \underline{\underline{\cal E}}(\underline{u})) = \underline 0,$$ subject to an initial condition $\theta^0(\cdot)\in [-1,1]$ on $\theta$ and boundary conditions on both equations. Here $\gamma\in\Bbb R_{>0}$ is the interfacial parameter, $\Psi$ is a nonsmooth double well potential, $ \underline{\underline{\cal E}}$ is the symmetric strain tensor, $\cal C$ is the possibly anisotropic elasticity tensor, $c(s) := c_0+ \frac 12 (1- c_0) (1+s)$ with $c_0(\gamma)\in\Bbb R_{>0}$ and $b(s) := 1 s^2$ is the degenerate diffusion 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.