Finite Element Approximation of a Three Dimensional Phase Field Model for Void Electromigration

• Published in: Journal of scientific computing Vol. 37; no. 2; pp. 202 - 232
• Main Authors: Banas, Lubomir, Nurnberg, Robert
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.11.2008; Springer Nature B.V
• Subjects: Algorithms; Approximation; Boundary conditions; Computational Mathematics and Numerical Analysis; Conduction; Discrete systems; Electromigration; Evolution; Finite element method; Iterative methods; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Surface diffusion; Theoretical; Three dimensional models; Voids; Phase field model; Degenerate Cahn–Hilliard equation; Finite elements; Multigrid methods; Void electromigration; Fourth order degenerate parabolic system; Surface diffusion; Convergence analysis
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-008-9203-y

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 subject to an initial condition u 0 (⋅)∈[−1,1] on u and flux boundary conditions on all three equations. Here γ ∈ℝ >0 , α ∈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.