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

gamma partial derivative u/partial derivative t - del.(b(u)del[w + alpha phi]) = 0, w = -gamma Delta u + gamma(-1)Psi'(u), del.(c(u)del phi) = 0

subject to an initial condition u(0)(.) is an element of [-1, 1] on u and flux boundary conditions on all three equations. Here gamma is an element of R->0, alpha is an element of R->= 0, Psi is a non-smooth double well potential, and c(u) := 1 + u, b(u) := 1 - u(2) are degenerate coefficients. 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.