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

Maciek D. Korzec; Hao Wu

doi:10.3934/dcdsb.2014.19.2227

Article Contents

2014, Volume 19, Issue 7: 2227-2246. Doi: 10.3934/dcdsb.2014.19.2227

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

Maciek D. Korzec^1, and
Hao Wu^2,

1.
Institute of Mathematics, Technical University Berlin, Straße des 17. Juni 136, 10623 Berlin
2.
School of Mathematical Sciences, Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Han Dan Road 220, 200433 Shanghai

Received: February 28, 2013

Revised: May 31, 2013

Published: August 2014

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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 consistent discrete free energy dissipation.

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Mathematics Subject Classification: Primary: 35K20, 35K57; Secondary: 74H40, 74N20.

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