Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-20T19:19:26.360Z Has data issue: false hasContentIssue false
  • English
  • Français

Robust a priori error analysis for the approximation of degree-one Ginzburg-Landau vortices

Published online by Cambridge University Press:  15 September 2005

Sören Bartels
Sören Bartels*
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA. sba@math.umd.edu
Get access

Abstract

This article discusses the numerical approximation of time dependent Ginzburg-Landau equations. Optimal error estimates which are robust with respect to a large Ginzburg-Landau parameter are established for a semi-discrete in time and a fully discrete approximation scheme. The proofs rely on an asymptotic expansion of the exact solution and a stability result for degree-one Ginzburg-Landau vortices. The error bounds prove that degree-one vortices can be approximated robustly while unstable higher degree vortices are critical.

Keywords

Ginzburg-Landau equationsnumerical approximationerror analysisspectral estimatefinite element method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 5 , September 2005 , pp. 863 - 882
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

H.W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983), 311–341.
S. Bartels, A posteriori error analysis for Ginzburg-Landau type equations. In preparation (2004).
Beaulieu, A., Some remarks on the linearized operator about the radial solution for the Ginzburg-Landau equation. Nonlinear Anal. 54 (2003) 1079–1119. CrossRef
F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston, MA (1994).
S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Texts in Applied Mathematics, Springer-Verlag, New York (2002).
Chen, X., Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces. Comm. Partial Differential Equations 19 (1994) 1371–1395. CrossRef
Chen, Z. and Hoffmann, K.-H., Numerical studies of a non-stationary Ginzburg-Landau model for superconductivity. Adv. Math. Sci. Appl. 5 (1995) 363–389.
Chen, X., Elliott, C.M. and Shooting, T. Qi method for vortex solutions of a complex-valued Ginzburg-Landau equation. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 1075–1088. CrossRef
de Mottoni, P. and Schatzman, M., Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347 (1995) 1533–1589. CrossRef
Ding, S. and Liu, Z., Hölder convergence of Ginzburg-Landau approximations to the harmonic map heat flow. Nonlinear Anal. 46 (2001) 807–816. CrossRef
Q. Du, M. Gunzburger and J. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34 (1992), 54–81
Du, Q., Gunzburger, M. and Peterson, J., Finite element approximation of a periodic Ginzburg-Landau model for type- ${\rm II}$ superconductors. Numer. Math. 64 (1993) 85–114. CrossRef
Dynamics, W. E of vortices in Ginzburg-Landau theories with applications to superconductivity. Phys. D 77 (1994) 383–404.
L.C. Evans, Partial differential equations. Graduate Studies in Mathematics, American Mathematical Society, Providence, RI (1998).
Feng, X. and Prohl, A., Numerical analysis of the Cahn-Hilliard equation and approximation of the Hele-Shaw problem. Interfaces Free Bound. 7 (2005) 1–28. CrossRef
Feng, X. and Prohl, A., Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94 (2003) 33–65. CrossRef
Feng, X. and Prohl, A., 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
Ginzburg, V. and Landau, L., 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.
Hervé, R.-M. and Hervé, M., 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. CrossRef
Hoffmann, K.-H., Zou, J., 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. CrossRef
A. Jaffe and C. Taubes, Vortices and monopoles. Progress in Physics, Birkhäuser Boston, Inc., Boston, MA (1994).
D. Kessler, R.H. Nochetto and A. Schmidt, A posteriori error control for the Allen-Cahn problem: circumventing Gronwall's inequality. Preprint (2003).
Lieb, E.H. and Loss, M., Symmetry of the Ginzburg-Landau minimizer in a disc. Math. Res. Lett. 1 (1994) 701–715. CrossRef
Lin, F.H., Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds. Comm. Pure Appl. Math. 51 (1998) 385–441. 3.0.CO;2-5>CrossRef
Lin, F.H., Some dynamical properties of Ginzburg-Landau vortices. Comm. Pure Appl. Math. 49 (1996) 323–359. 3.0.CO;2-E>CrossRef
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.
Lin, F.H. and Du, Ginzburg-Landau, Q. vortices: dynamics, pinning, and hysteresis. SIAM J. Math. Anal. 28 (1997) 1265–1293. CrossRef
Lin, T.C., The stability of the radial solution to the Ginzburg-Landau equation. Comm. Partial Differential Equations 22 (1997) 619–632.
T.C. Lin, Spectrum of the linearized operator for the Ginzburg-Landau equation. Electron. J. Differential Equations 42 (2000), 25 (electronic).
Mironescu, P., On the stability of radial solutions of the Ginzburg-Landau equation. J. Funct. Anal. 130 (1995) 334–344. CrossRef
Mironescu, P., 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.
Mu, M., Deng, Y. and Chou, C.-C., Numerical methods for simulating Ginzburg-Landau vortices. SIAM J. Sci. Comput. 19 (1998) 1333–1339. CrossRef
Neu, J.C., Vortices in complex scalar fields. Phys. D 43 (1990) 385–406. CrossRef
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).
V. Thomée, Galerkin finite element methods for parabolic problems. Springer Series in Computational Mathematics, Springer-Verlag, Berlin (1997).
Wheeler, M.F., A priori $L\sb{2}$ error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10 (1973) 723–759. CrossRef