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

Barrett, John W. and Garcke, Harald and Nürnberg, Robert (2006) Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid. Mathematics of Computation (MCOM) 75 (253), pp. 7-41.

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Abstract

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.