Skip to main content
SpringerLink
Log in

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

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary.

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. Babuška, I., Osborn, J.: Eigenvalue problems. In: Handbook of numerical analysis II, North-Holland, 1991, pp. 641–787

  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. Beaulieu, A.: Some remarks on the linearized operator about the radial solution for the Ginzburg-Landau equation. Nonlinear Anal. 54, 1079–1119 (2003)

    Article  Google Scholar 

  5. Carstensen, C.: Quasi interpolation and a posteriori error analysis in finite element method. Math. Modelling Numer. Anal. 33, 1187–1202 (1999)

    Article  Google Scholar 

  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. Clément, P.: Approximation by finite element functions using local regularization. RAIRO Sér. Rouge Anal. Numér. R–2, 77–84 (1975)

  8. de Mottoni, P., Schatzman, M.: Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347, 1533–1589 (1995)

    Google Scholar 

  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. 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. 407–440

  11. Feng, X., Prohl, A.: Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94, 33–65 (2003)

    Article  Google Scholar 

  12. Feng, X., Prohl, A.: Error analysis of a mixed finite element method for the Cahn–Hilliard equation. Numer. Math. 99, 47–84 (2004)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  14. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer, Berlin, 2001

  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. 138–167

    Google Scholar 

  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. 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)

    Article  Google Scholar 

  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. 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. Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner Skripten zur Numerik, Teubner, Stuttgart, 1996

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Sören Bartels

Authors
  1. Sören Bartels
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sören Bartels.

Additional information

Mathematics Subject Classification(2000): 65M15, 65M60, 65M50.

AcknowledgmentS.B. is thankful to G. Dolzmann and R.H. Nochetto for stimulating discussions. This work was supported by a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bartels, S. A posteriori error analysis for time-dependent Ginzburg-Landau type equations. Numer. Math. 99, 557–583 (2005). https://doi.org/10.1007/s00211-004-0560-7

Download citation

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-004-0560-7

Keywords

Search

Navigation