Simulating phase transition dynamics on nontrivial domains  Lukasz Bolikowski and Maria Gokieli Interdisciplinary Centre for Mathematical and Computational Modelling University of Warsaw Prosta 69, 00-838 Warsaw, Poland L.Bolikowski@icm.edu.pl, M.Gokieli@icm.edu.pl Abstract. Our goal is to investigate the influence of the geometry and topology of the domain Ω on the solutions of the phase transition and other diffusion-driven phenomena in Ω, modeled e.g. by the Allen-Cahn, Cahn-Hilliard, reaction–diffusion equations. We present FEM numerical schemes for the Allen–Cahn and Cahn–Hilliard equation based on the Eyre’s algorithm and present some numerical results on split and dumb- bell domains. Keywords: diffusion, Cahn-Hilliard, Allen-Cahn, stability, finite ele- ment 1 Introduction and motivation The Allen-Cahn equation: u t - εΔu = u - u 3 on (0, ∞) × Ω (1) where u = u(t, x), u(0, ·)= u 0 given in some domain Ω and with the homogenous Neumann boundary condition ∂ ∂n u =0 on (0, ∞) × ∂Ω, is of reaction-diffusion type. It is also a model of an order–disorder phase tran- sition in alloys, u being then not a concentration, but a non-conserved order parameter [1]. In any interpretation the equation is a dissipative system, driven by the minimization of the free energy F = Z Ω ε 2 |∇u(x)| 2 + 1 4 u 4 - 1 2 u 2 dx in the Hilbert space L 2 (Ω). It takes the so–called gradient form du dt = δF δu (u),