Finite Element Approximation of a Phase Field Model for Void Electromigration

Abstract

We consider a fully practical finite element approximation of the nonlinear degenerate parabolic system \begin{align} \gamma\,\textstyle{\frac{\partial u}{\partial t}} - \nabla . ( \,b(u) \,\nabla [ w +\alpha\,\phi]\, ) = 0\,, \quad w = - \gamma\, \Delta u + \gamma^{-1}\,\Psi'(u)\,, %\nonumber \\ \quad \nabla . (\, c(u) \,\nabla \phi ) = 0\, \nonumber \end{align} subject to an initial condition $u^0(\cdot) \in [-1,1]$ on u and flux boundary conditions on all three equations. Here $\gamma \in {\mathbb R}_{>0}$, $\alpha \in {\mathbb R}_{\geq 0}$, $\Psi$ is a nonsmooth double well potential, and $c(u) :=1+u$, $b(u):=1-u^2$ are degenerate coefficients. %diffusional mobility. The degeneracy in b restricts $u(\cdot,\cdot) \in [-1,1]$. The above, in the limit $\gamma \rightarrow 0$, models the evolution of voids by surface diffusion in an electrically conducting solid. 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. Furthermore, an iterative scheme for solving the resulting nonlinear discrete system is introduced and analyzed. Finally, some numerical experiments are presented.