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

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Abstract

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

$$\gamma\frac{\partial u}{\partial t}-\nabla.(b(u)\nabla[w+\alpha\phi])=0,\qquad w=-\gamma\Delta u+\gamma^{-1}\Psi'(u),\qquad\nabla.(c(u)\nabla\phi)=0$$

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.

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Author notes

  1. Ľubomír Baňas

    Present address: Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, UK

Authors and Affiliations

  1. Department of Mathematics, Imperial College London, London, SW7 2AZ, UK

    Ľubomír Baňas & Robert Nürnberg

Authors
  1. Ľubomír Baňas
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  2. Robert Nürnberg
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Correspondence to Robert Nürnberg.

Additional information

L. Baňas was supported by the EPSRC grant EP/C548973/1.

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Baňas, Ľ., Nürnberg, R. Finite Element Approximation of a Three Dimensional Phase Field Model for Void Electromigration. J Sci Comput 37, 202–232 (2008). https://doi.org/10.1007/s10915-008-9203-y

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  • DOI: https://doi.org/10.1007/s10915-008-9203-y

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