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Structure Preserving Discretization of Allen–Cahn Type Problems Modeling the Motion of Phase Boundaries

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Abstract

We study the systematic numerical approximation of a class of Allen–Cahn type problems modeling the motion of phase interfaces. The common feature of these models is an underlying gradient flow structure which gives rise to a decay of an associated energy functional along solution trajectories. We first study the discretization in space by a conforming Galerkin approximation of a variational principle which characterizes smooth solutions of the problem. Well-posedness of the resulting semi-discretization is established and the energy decay along discrete solution trajectories is proven. A problem adapted implicit time-stepping scheme is then proposed and we establish its well-posed and decay of the free energy for the fully discrete scheme. Some details about the numerical realization by finite elements are discussed, in particular the iterative solution of the nonlinear problems arising in every time-step. The theoretical results are illustrated by numerical tests which also provide further evidence for asymptotic expansions of the interface velocities derived by Alber et al. and support the observation that their hybrid Allen–Cahn model avoids the problem of mesh-locking to a large extent.

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Acknowledgements

This work was supported by the German Research Foundation (DFG) via grants IRTG 1529, TRR 146 projection C3, TRR 154 project C4, and Eg-331/1-1 and by the German Excellence Initiative via grant GSC 233.

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  1. Department of Mathematics, TU Darmstadt, Darmstadt, Germany

    Anke Böttcher & Herbert Egger

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  1. Anke Böttcher
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  2. Herbert Egger
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Correspondence to Herbert Egger.

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Böttcher, A., Egger, H. Structure Preserving Discretization of Allen–Cahn Type Problems Modeling the Motion of Phase Boundaries. Vietnam J. Math. 48, 847–863 (2020). https://doi.org/10.1007/s10013-020-00428-w

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