Free Access
Issue
ESAIM: M2AN
Volume 39, Number 5, September-October 2005
Page(s) 863 - 882
DOI https://doi.org/10.1051/m2an:2005038
Published online 15 September 2005
  1. H.W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983), 311–341. [Google Scholar]
  2. S. Bartels, A posteriori error analysis for Ginzburg-Landau type equations. In preparation (2004). [Google Scholar]
  3. A. Beaulieu, Some remarks on the linearized operator about the radial solution for the Ginzburg-Landau equation. Nonlinear Anal. 54 (2003) 1079–1119. [CrossRef] [MathSciNet] [Google Scholar]
  4. F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston, MA (1994). [Google Scholar]
  5. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Texts in Applied Mathematics, Springer-Verlag, New York (2002). [Google Scholar]
  6. X. Chen, Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces. Comm. Partial Differential Equations 19 (1994) 1371–1395. [Google Scholar]
  7. Z. Chen and K.-H. Hoffmann, Numerical studies of a non-stationary Ginzburg-Landau model for superconductivity. Adv. Math. Sci. Appl. 5 (1995) 363–389. [MathSciNet] [Google Scholar]
  8. X. Chen, C.M. Elliott and T. Qi, Shooting method for vortex solutions of a complex-valued Ginzburg-Landau equation. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 1075–1088. [MathSciNet] [Google Scholar]
  9. P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347 (1995) 1533–1589. [CrossRef] [MathSciNet] [Google Scholar]
  10. S. Ding and Z. Liu, Hölder convergence of Ginzburg-Landau approximations to the harmonic map heat flow. Nonlinear Anal. 46 (2001) 807–816. [CrossRef] [MathSciNet] [Google Scholar]
  11. Q. Du, M. Gunzburger and J. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34 (1992), 54–81 [Google Scholar]
  12. Q. Du, M. Gunzburger and J. Peterson, Finite element approximation of a periodic Ginzburg-Landau model for type-Formula superconductors. Numer. Math. 64 (1993) 85–114. [CrossRef] [MathSciNet] [Google Scholar]
  13. W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity. Phys. D 77 (1994) 383–404. [CrossRef] [MathSciNet] [Google Scholar]
  14. L.C. Evans, Partial differential equations. Graduate Studies in Mathematics, American Mathematical Society, Providence, RI (1998). [Google Scholar]
  15. X. Feng and A. Prohl, Numerical analysis of the Cahn-Hilliard equation and approximation of the Hele-Shaw problem. Interfaces Free Bound. 7 (2005) 1–28. [Google Scholar]
  16. X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94 (2003) 33–65. [CrossRef] [MathSciNet] [Google Scholar]
  17. X. Feng and A. Prohl, Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Math. Comp. 73 (2004) 541-567. [CrossRef] [MathSciNet] [Google Scholar]
  18. V. Ginzburg and L. Landau, On the theory of superconductivity. Zh. Èksper. Teoret. Fiz. 20 (1950) 1064–1082, in Men of Physics, L.D. Landau, D. ter Haar, Eds., Pergamon, Oxford (1965) 138–167. [Google Scholar]
  19. R.-M. Hervé and M. Hervé, Étude qualitative des solutions réelles d'une équation différentielle liée à l'équation de Ginzburg-Landau. Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994) 427–440. [Google Scholar]
  20. K.-H. Hoffmann, J. Zou, Finite element approximations of Landau-Ginzburg's equation model for structural phase transitions in shape memory alloys. RAIRO Modél. Math. Anal. Numér. 29 (1995) 629–655. [MathSciNet] [Google Scholar]
  21. A. Jaffe and C. Taubes, Vortices and monopoles. Progress in Physics, Birkhäuser Boston, Inc., Boston, MA (1994). [Google Scholar]
  22. D. Kessler, R.H. Nochetto and A. Schmidt, A posteriori error control for the Allen-Cahn problem: circumventing Gronwall's inequality. Preprint (2003). [Google Scholar]
  23. E.H. Lieb and M. Loss, Symmetry of the Ginzburg-Landau minimizer in a disc. Math. Res. Lett. 1 (1994) 701–715. [MathSciNet] [Google Scholar]
  24. F.H. Lin, Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds. Comm. Pure Appl. Math. 51 (1998) 385–441. [CrossRef] [MathSciNet] [Google Scholar]
  25. F.H. Lin, Some dynamical properties of Ginzburg-Landau vortices. Comm. Pure Appl. Math. 49 (1996) 323–359. [CrossRef] [MathSciNet] [Google Scholar]
  26. F.H. Lin, The dynamical law of Ginzburg-Landau vortices. Proc. of the Conference on Nonlinear Evolution Equations and Infinite-dimensional Dynamical Systems (Shanghai, 1995), World Sci. Publishing, River Edge, NJ (1997) 101–110. [Google Scholar]
  27. F.H. Lin and Q. Du, Ginzburg-Landau vortices: dynamics, pinning, and hysteresis. SIAM J. Math. Anal. 28 (1997) 1265–1293. [CrossRef] [MathSciNet] [Google Scholar]
  28. T.C. Lin, The stability of the radial solution to the Ginzburg-Landau equation. Comm. Partial Differential Equations 22 (1997) 619–632. [CrossRef] [MathSciNet] [Google Scholar]
  29. T.C. Lin, Spectrum of the linearized operator for the Ginzburg-Landau equation. Electron. J. Differential Equations 42 (2000), 25 (electronic). [Google Scholar]
  30. P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation. J. Funct. Anal. 130 (1995) 334–344. [CrossRef] [MathSciNet] [Google Scholar]
  31. P. Mironescu, Les minimiseurs locaux pour l'équation de Ginzburg-Landau sont à symétrie radiale. C. R. Acad. Sci. Paris Sér. I Math. 323 (1996) 593–598. [Google Scholar]
  32. M. Mu, Y. Deng and C.-C. Chou, Numerical methods for simulating Ginzburg-Landau vortices. SIAM J. Sci. Comput. 19 (1998) 1333–1339. [CrossRef] [MathSciNet] [Google Scholar]
  33. J.C. Neu, Vortices in complex scalar fields. Phys. D 43 (1990) 385–406. [CrossRef] [MathSciNet] [Google Scholar]
  34. F. Pacard and T. Riviere, Linear and nonlinear aspects of vortices. The Ginzburg-Landau model. Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston, MA (2000). [Google Scholar]
  35. V. Thomée, Galerkin finite element methods for parabolic problems. Springer Series in Computational Mathematics, Springer-Verlag, Berlin (1997). [Google Scholar]
  36. M.F. Wheeler, A priori Formula error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10 (1973) 723–759. [CrossRef] [MathSciNet] [Google Scholar]

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