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

Anisotropy in wavelet-based phase field models
Pages: 1167 - 1187, Issue 4, June 2016

doi:10.3934/dcdsb.2016.21.1167      Abstract        References        Full text (2812.6K)           Related Articles

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)

1 J. W. Barrett, H. Garcke and R. Nürnberg, Stable phase field approximations of anisotropic solidification, IMA J. Numer. Anal., 34 (2014), 1289-1327.       
2 A. Braides, Gamma-Convergence for Beginners, Oxford University Press, 2002.       
3 E. Burman and J. Rappaz, Existence of solutions to an anisotropic phase-field model, Math. Meth. Appl. Sci., 26 (2003), 1137-1160.       
4 W. K. Burton, N. Cabrera and F. C. Frank, The Growth of Crystals and the Equilibrium Structure of their Surfaces, Phil. Trans. R. Soc. Lond. A, 243 (1951), 299-358.       
5 G. Caginalp, Penrose-Fife modification of solidification equations has no freezing or melting, Appl. Math. Lett., 5 (1992), 93-96.       
6 C. Cattani, Harmonic wavelets towards the solution of nonlinear PDE, Comp. Math. Appl., 50 (2005), 1191-1210.       
7 W. Dahmen, Wavelet and multiscale methods for operator equations, Acta Num., 6 (1997), 55-228.       
8 I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, USA, 1992.       
9 J. A. Dobrosotskaya and A. L. Bertozzi, A wavelet-laplace variational technique for image deconvolution and inpainting, IEEE Trans. Imag. Proc., 17 (2008), 657-663.       
10 J. A. Dobrosotskaya and A. L. Bertozzi, Wavelet analogue of the Ginzburg-Landau energy and its Gamma-convergence, Interf. Free Boundaries, 12 (2010), 497-525.       
11 J. A. Dobrosotskaya and A. L. Bertozzi, Analysis of the wavelet Ginzburg-Landau energy in image applications with edges, SIAM J. Imaging Sci., 6 (2013), 698-729.       
12 M. E. Glicksman, Principles of Solidification, Springer, 2011.
13 C. Herring, Some theorems on the free energies of crystal surfaces, Phys. Rev., 82 (1951), 87-93.
14 M. Holmström, Solving hyperbolic PDEs using interpolating wavelets, SIAM J. Sci. Comput., 21 (1999), 405-420.       
15 M. Holmström and J. Waldén, Adaptive wavelet methods for hyperbolic PDEs, J Sci. Comp., 13 (1998), 19-49.       
16 L. Jameson, A wavelet-optimized, very high order adaptive grid and order numerical method, SIAM J. Sci. Comput., 19 (1998), 1980-2013.       
17 A. Karma and W.-J. Rappel, Numerical simulation of three-dimensional dendritic growth, Phys. Rev. Lett., 77 (1996), p4050.
18 R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Physica D, 63 (1993), 410-423.
19 B. Li, J. Lowengrub, A. Rätz and A. Voigt, Geometric evolution laws for thin crystalline films: Modeling and numerics, Commun. Comput. Phys., 6 (2009), 433-482.       
20 S. Mallat, A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way, Academic Press, 2009.       
21 G. B. McFadden, Phase-field models of solidification, in Recent Advances in Numerical Methods for Partial Differential Equations and Applications, Contemporary Mathematics, American Mathematical Society, 306 (2002), 107-145.       
22 G. B. McFadden, A. A. Wheeler, R. J. Braun, S. R. Coriell and R. F. Sekerka, Phase-field models for anisotropic interfaces, Phys. Rev. E, 48 (1993), 2016-2024.       
23 A. Miranville, Some mathematical models in phase transitions, DCDS-S, 7 (2014), 271-306.       
24 L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 123-142.       
25 O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990), 44-62.       
26 O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a ''thermodynamically consistent'' phase field model, Physica D, 69 (1993), 107-113.       
27 K. Schneider and O. V. Vasilyev, Wavelet methods in computational fluid dynamics, Ann. Rev. Fluid Mech., 42 (2010), 473-503.       
28 I. Steinbach, Phase-field models in materials science, Mod. Sim. Mater. Sci. Eng., 17 (2009), 073001.
29 O. V. Vasilyev and S. Paolucci, A fast adaptive wavelet collocation algorithm for multidimensional PDEs, J. Comp. Phys., 138 (1997), 16-56.       
30 O. V. Vasilyev, S. Paolucci and M. Sen, A multilevel wavelet collocation method for solving partial differential equations in a finite domain, J. Comp. Phys., 120 (1995), 33-47.       
31 S.-L. Wang, R. F. Sekerka, A. A. Wheeler, B. T. Murray, S. R. Coriell, R. J. Braun and G. B. McFadden, Thermodynamically-consistent phase-field models for solidification, Physica D, 69 (1993), 189-200.       
32 A. A. Wheeler, B. T. Murray and R. J. Schaefer, Computation of dendrites using a phase field model, Physica D, 66 (1993), 243-262.
33 G. Wulff, Zur frage der geschwindigkeit des wachstums und der auflösung der krystallflächen, Zeitschrift f. Krystall. Mineral., 34 (1901), 449-530.
34 S.-M. Zheng, Nonlinear Evolution Equations, Pitman series Monographs and Survey in Pure and Applied Mathematics 133, Chapman Hall/CRC, Boca Raton, Florida, 2004.       

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