Numerische Mathematik

, Volume 99, Issue 4, pp 557–583 | Cite as

A posteriori error analysis for time-dependent Ginzburg-Landau type equations

  • Sören Bartels


This work presents an a posteriori error analysis for the finite element approximation of time-dependent Ginzburg-Landau type equations in two and three space dimensions. The solution of an elliptic, self-adjoint eigenvalue problem as a post-processing procedure in each time step of a finite element simulation leads to a fully computable upper bound for the error. Theoretical results for the stability of degree one vortices in Ginzburg-Landau equations and of generic interfaces in Allen-Cahn equations indicate that the error estimate only depends on the inverse of a small parameter in a low order polynomial. The actual dependence of the error estimate upon this parameter is explicitly determined by the computed eigenvalues and can therefore be monitored within an approximation scheme. The error bound allows for the introduction of local refinement indicators which may be used for adaptive mesh and time step size refinement and coarsening. Numerical experiments underline the reliability of this approach.


Vortex Error Estimate Finite Element Simulation Order Polynomial Element Approximation 
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  1. 1.
    Allen, S., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta. Metall. 27, 1084–1095 (1979)Google Scholar
  2. 2.
    Babuška, I., Osborn, J.: Eigenvalue problems. In: Handbook of numerical analysis II, North-Holland, 1991, pp. 641–787Google Scholar
  3. 3.
    Bartels, S., Carstensen, C., Dolzmann, G.: Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis. Numer. Math. 99, 1–24 (2004)Google Scholar
  4. 4.
    Beaulieu, A.: Some remarks on the linearized operator about the radial solution for the Ginzburg-Landau equation. Nonlinear Anal. 54, 1079–1119 (2003)CrossRefGoogle Scholar
  5. 5.
    Carstensen, C.: Quasi interpolation and a posteriori error analysis in finite element method. Math. Modelling Numer. Anal. 33, 1187–1202 (1999)CrossRefGoogle Scholar
  6. 6.
    Chen, X.: Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces. Commun. Partial Differential Equations 19, 1371–1395 (1994)Google Scholar
  7. 7.
    Clément, P.: Approximation by finite element functions using local regularization. RAIRO Sér. Rouge Anal. Numér. R–2, 77–84 (1975)Google Scholar
  8. 8.
    de Mottoni, P., Schatzman, M.: Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347, 1533–1589 (1995)Google Scholar
  9. 9.
    Du, Q., Gunzburger, M., Peterson, J.: Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34, 54–81 (1992)Google Scholar
  10. 10.
    Elliott, C.M.: Approximation of curvature dependent interface motion. The state of the art in numerical analysis, Inst. Math. Appl. Conf. Ser. New Ser. 63, Oxford Univ. Press, 1997, pp. 407–440Google Scholar
  11. 11.
    Feng, X., Prohl, A.: Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94, 33–65 (2003)CrossRefGoogle Scholar
  12. 12.
    Feng, X., Prohl, A.: Error analysis of a mixed finite element method for the Cahn–Hilliard equation. Numer. Math. 99, 47–84 (2004)CrossRefGoogle Scholar
  13. 13.
    Feng, X., 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, 541–567 (2004)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer, Berlin, 2001Google Scholar
  15. 15.
    Ginzburg, V., Landau, L.: On the theory of superconductivity. Zh. Èksper. Teoret. Fiz. 20, 1064–1082, (1950); In: L.D. Landau, D. ter Haar, (ed.), Men of Physics: Pergamon, Oxford, 1965, pp. 138–167Google Scholar
  16. 16.
    Kessler, D., Nochetto, R.H., Schmidt, A.: A posteriori error control for the Allen-Cahn problem: circumventing Gronwall’s inequality. M2AN Math. Model. Numer. Anal. 38, 129–142 (2004)Google Scholar
  17. 17.
    Larson, M.G.: A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems. SIAM J. Numer. Anal. 38, 608–625 (2000)CrossRefGoogle Scholar
  18. 18.
    Lieb, E.H., Loss, M.: Symmetry of the Ginzburg-Landau minimizer in a disc. Math. Res. Lett. 1, 701–715 (1994)Google Scholar
  19. 19.
    Nochetto, R.H., Schmidt, A., Verdi, C.: A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comp. 69, 1–24 (1999)Google Scholar
  20. 20.
    Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner Skripten zur Numerik, Teubner, Stuttgart, 1996Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandUSA

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