Numerische Mathematik

, Volume 64, Issue 1, pp 85–114 | Cite as

Finite element approximation of a periodic Ginzburg-Landau model for type-II superconductors

  • Qiang Du
  • Max Gunzburger
  • Janet Peterson
Article

Summary

We consider efficient finite element algorithms for the computational simulation of type-II superconductors. The algorithms are based on discretizations of a periodic Ginzburg-Landau model. Periodicity is defined with respect to a non-orthogonal lattice that is not necessarily aligned with the coordinate axes; also, the primary dependent variables employed in the model satisfy non-standard “quasi”-periodic boundary conditions. After introducing the model, we define finite element schemes, derive error estimates of optimal order, and present the results of some numerical calculations. For a similar quality of simulation, the resulting algorithms seem to be significantly less costly than are previously used numerical approximation methods.

Mathematics Subject Classification (1991)

65N30 35J60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abrikosov, A. (1957): On the magnetic properties of superconductors of the second type. Zb. Eksperim. i Teor. Fiz.32, 1442–1452 [English translation: (1957) Soviet Phys.-JETP5, 1174–1182]Google Scholar
  2. 2.
    Adams, A. (1975): Sobolev spaces. Academic Press, New YorkGoogle Scholar
  3. 3.
    Babuška, I., Aziz, A. (1972): Survey lectures on the mathematical foundations of the finite element method. In: A. Aziz, ed., The mathematical foundations of the finite element method with application to partial differential equations, pp. 3–359. Academic Press, New YorkGoogle Scholar
  4. 4.
    Bardeen, J. (1956): Theory of superconductivity. In: S. Flŭgge, ed., Encyclopedia of Physics XV, pp. 17–369. Springer, Berlin Heidelberg New York.Google Scholar
  5. 5.
    Brandt, E. (1972): Ginsburg-Landau theory of the vortex lattice in type-II superconductors for all values ofk andB. Phys. Stat. Sol. (b)51, 345–358Google Scholar
  6. 6.
    Brezzi, F., Rappaz, J., Raviart, P.-A. (1980): Finite-dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions, Numer. Math.36, 1–25Google Scholar
  7. 7.
    Chapman, S., Howison, S., Ockendon, J. (1992): Macroscopic models for superconductivity. SIAM Review (to appear)Google Scholar
  8. 8.
    Ciarlet, P. (1978): The finite element method for elliptic problems. North-Holland, AmsterdamGoogle Scholar
  9. 9.
    Crouziex, M., Rappaz, J. (1989): On numerical, approximation in bifurcation theory. Masson, ParisGoogle Scholar
  10. 10.
    DeGennes, P. (1966): Superconductivity in metals and alloys. Benjamin, New YorkGoogle Scholar
  11. 11.
    Doria, M., Gubernatis, J., Rainer, D. (1989): Virial theorem for Ginzburg-Landau theories with potential application to numerical studies of type II superconductors. Phys. Rev. B39, 9573–9575Google Scholar
  12. 12.
    Doria, M., Gubernatis, J., Rainer, D. (1990): Solving the Ginzburg-Landau equations by simulated annealing. Phys. Rev. B41, 6335–6340Google Scholar
  13. 13.
    Du, Q., Gunzburger, M., Peterson, J. (1992): Analysis and approximation of Ginzburg-Landau models for superconductivity. SIAM Review34, 54–81Google Scholar
  14. 14.
    Du, Q., Gunzburger, M., Peterson, J. (1992): Modeling and analysis of a periodic Ginzburg-Landau model for type-II superconductors. SIAM J. Appl. Math. (to appear)Google Scholar
  15. 15.
    Eilenberger, G. (1964): Zu Abrikosovs Theorie der periodischen Lösungen der GL-Gleichungen für Suparaleiter 2. Art Z. Phys.180, 32–42Google Scholar
  16. 16.
    Girault, V., Raviart, P.-A. (1986): Finite element methods for Navier-Stokes equations. Springer, Berlin Heidelberg New YorkGoogle Scholar
  17. 17.
    Kleiner, W., Roth, L., Autler, S. (1964): Bulk solution of Ginzburg-Landau equations for type II superconductors: upper critical field region. Phys. Rev.133, A1226-A1227Google Scholar
  18. 18.
    Koppe, H., Willebrand, J. (1970): Approximate calculation of the reversible magnetization curves of type II superconductors. Low Temp. Phys.2, 499–506Google Scholar
  19. 19.
    Kuper, C. (1968): An introduction of the theory of super conductivity. Clarendon, OxfordGoogle Scholar
  20. 20.
    Lasher, G. (1965): Series solution of the Ginzburg-Landau equations for the Abrisokov mixed state. Phys. Rev. A,140, 523–528Google Scholar
  21. 21.
    Odeh, F. (1967): Existence and bifurcation theorems for the Ginzburg-Landau equations. J. Math. Phys.8, 2351–2356Google Scholar
  22. 22.
    St. James, D., Sarma, G., Thomas, E. (1969): Type II superconductivity. Pergamon, OxfordGoogle Scholar
  23. 23.
    Temam, R. (1983): Navier-Stokes equations and nonlinear functional analysis. SIAM, PhiladelphiaGoogle Scholar
  24. 24.
    Tinkham, M. (1975): Introduction to superconductivity. McGraw-Hill, New YorkGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Qiang Du
    • 1
  • Max Gunzburger
    • 2
  • Janet Peterson
    • 2
  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Department of MathematicsVirginia TechBlacksburgUSA

Personalised recommendations