In this paper we consider the numerical approximation of phase field and sharp interface models for diffusion-induced grain boundary motion. The phase field model consists of a double-obstacle Allen-Cahn equation with a forcing obtained from the solution of a degenerate diffusion equation. On the other hand the sharp interface model consists of forced mean curvature flow coupled to a diffusion equation holding on the interface itself. Formal asymptotics yield the sharp interface model as the limit of the phase field equations as the width of the associated diffuse interface tends to zero. A finite-element approximation of the phase field model is presented and is shown to be convergent to a weak solution. Numerical simulations of both models are described and compared. It is shown that the two models are consistent.
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Charles M. Elliott, Klaus Deckelnick, Vanessa Styles, Numerical diffusion-induced grain boundary motion. Interfaces Free Bound. 3 (2001), no. 4, pp. 393–414