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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Anisotropy in wavelet-based phase field models

Pages: 1167 - 1187, Volume 21, Issue 4, June 2016      doi:10.3934/dcdsb.2016.21.1167

 
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Maciek Korzec - Technische Universität Berlin, Institute of Mathematics, Straße des 17. Juni 136, 10623 Berlin, Germany (email)
Andreas Münch - Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom (email)
Endre Süli - Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliff e Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom (email)
Barbara Wagner - Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany (email)

Abstract: When describing the anisotropic evolution of microstructures in solids using phase-field models, the anisotropy of the crystalline phases is usually introduced into the interfacial energy by directional dependencies of the gradient energy coefficients. We consider an alternative approach based on a wavelet analogue of the Laplace operator that is intrinsically anisotropic and linear. The paper focuses on the classical coupled temperature/Ginzburg--Landau type phase-field model for dendritic growth. For the model based on the wavelet analogue, existence, uniqueness and continuous dependence on initial data are proved for weak solutions. Numerical studies of the wavelet based phase-field model show dendritic growth similar to the results obtained for classical phase-field models.

Keywords:  Phase-field model, wavelets, sharp interface model, free boundaries.
Mathematics Subject Classification:  Primary: 34E13, 74N20; Secondary: 74E10.

Received: May 2015;      Revised: January 2016;      Available Online: March 2016.

 References