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Analysis and simulation for an isotropic phase-field model describing grain growth. (English) Zbl 1304.35094
Summary: 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.
MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K57 Reaction-diffusion equations
74H40 Long-time behavior of solutions for dynamical problems in solid mechanics
74N20 Dynamics of phase boundaries in solids
35B41 Attractors
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