Finite element approximation of a three dimensional phase field model for void electromigration

We consider a finite element approximation of a phase field model for the evolution of voids by surface diffusion in an electrically conducting solid. The phase field equations are given by the nonlinear degenerate parabolic system γ∂u/∂t-∇.(b(u)∇[w+αφ])=0, w=-γΔu+γ -1Ψ́(u), ∇.(c(u) ∇φ)=0 subject to an initial condition u 0(·)ε [-1,1] on u and flux boundary conditions on all three equations. Here γ ε ℝ >0, α ε ℝ ≥0, Ψ is a non-smooth double well potential, and c(u):=1+u, b(u):=1-u 2 are degenerate coefficients. On extending existing results for the simplified two dimensional phase field model, we show stability bounds for our approximation and prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system in three space dimensions. Furthermore, a new iterative scheme for solving the resulting nonlinear discrete system is introduced and some numerical experiments are presented. © 2008 Springer Science+Business Media, LLC.