Preprint 06-2013

Analysis and simulation for an isotropic phase-field model describing grain growth

Author(s) : Maciek Korzec , Hao Wu

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 06-2013

MSC 2000

35K20 Boundary value problems for second-order, parabolic equations

74H40 Long-time behavior of solutions

Abstract :
A phase-field system of coupled Allen--Cahn type PDEs describing grain growth is analyzed and simulated. In the periodic setting, we prove the existence and uniqueness of global weak solutions to the problem. Then we investigate the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. Namely, the problem possesses a global attractor as well as an exponential attractor, which entails that the global attractor has finite fractal dimension. Moreover, we show that each trajectory converges to a single equilibrium. A time-adaptive numerical scheme based on trigonometric interpolation is presented. It allows to track the approximated long-time behavior accurately and leads to a convergence rate. The scheme exhibits a physically aspired discrete free energy dissipation.

Keywords : Grain growth, phase-field system, well-posedness, long-time behavior, numerical simulation.