A Fourier Spectral Moving Mesh Method for the Cahn-Hilliard Equation with Elasticity
W. M. Feng 1, P. Yu 2, S. Y. Hu 3, Z. K. Liu 1, Q. Du 2*, L. Q. Chen 11 Department of Materials Science and Engineering, Pennsylvania State University, University Park, PA 16802, USA.
2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA.
3 Pacific Northwest National Laboratory, Richland, WA 99354, USA.
Received 24 September 2007; Accepted (in revised version) 27 December 2007
Available online 1 August 2008
In recent years, Fourier spectral methods have emerged as competitive numerical methods for large-scale phase field simulations of microstructures in computational materials sciences. To further improve their effectiveness, we recently developed a new adaptive Fourier-spectral semi-implicit method (AFSIM) for solving the phase field equation by combining an adaptive moving mesh method and the semi-implicit Fourier spectral algorithm. In this paper, we present the application of AFSIM to the Cahn-Hilliard equation with inhomogeneous, anisotropic elasticity. Numerical implementations and test examples in both two and three dimensions are considered with a particular illustration using the well-studied example of mis-fitting particles in a solid as they approach to their equilibrium shapes. It is shown that significant savings in memory and computational time is achieved while accurate solutions are preserved.AMS subject classifications: 65M70, 65M50, 74B20
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Key words: Phase field, diffuse interface, moving mesh, adaptive mesh, Fourier-spectral method, adaptive spectral method, Cahn-Hilliard equation, elasticity.
Email: email@example.com (Q. Du)