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Analysis of linearized Galerkin-mixed FEMs for the time-dependent Ginzburg-Landau equations of superconductivity

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Abstract

A linearized backward Euler Galerkin-mixed finite element method is investigated for the time-dependent Ginzburg-Landau (TDGL) equations under the Lorentz gauge. By introducing the induced magnetic field σ = c u r l A as a new variable, the Galerkin-mixed FE scheme offers many advantages over conventional Lagrange type Galerkin FEMs. An optimal error estimate for the linearized Galerkin-mixed FE scheme is established unconditionally. Analysis is given under more general assumptions for the regularity of the solution of the TDGL equations, which includes the problem in two-dimensional nonconvex polygons and certain three dimensional polyhedrons, while the conventional Galerkin FEMs may not converge to a true solution in these cases. Numerical examples in both two and three dimensional spaces are presented to confirm our theoretical analysis. Numerical results show clearly the efficiency of the mixed method, particularly for problems on nonconvex domains.

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References

  1. Alonso, A., Valli, A.: An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comp. 68, 607–631 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alstrom, T., Sorensen, M., Pedersen, N., Madsen, S.: Magnetic flux lines in complex geometry type-II superconductors studied by the time dependent Ginzburg-Landau equation. Acta Appl. Math. 115, 63–74 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc. (N.S.) 47, 281–354 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bergh, J., Lofstrom, J.: Interpolation Spaces: an Introduction. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  6. Braess, D.: Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory, 3rd edn. Cambridge University Press, Cambridge, UK (2007)

    Book  MATH  Google Scholar 

  7. Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, New York (2002)

    Book  MATH  Google Scholar 

  8. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    Book  MATH  Google Scholar 

  9. Chapman, S., Howison, S., Ockendon, J.: Macroscopic models for superconductivity. SIAM Rev. 34, 529–560 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chatzipantelidis, P., Lazarov, R. D., Thomée, V., Wahlbin, L.B.: Parabolic finite element equations in nonconvex polygonal domains. BIT Numer. Math. 46, S113–S143 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, Z.: Mixed finite element methods for a dynamical Ginzburg-Landau model in superconductivity. Numer. Math. 76, 323–353 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, Z., Hoffmann, K.: Numerical studies of a non-stationary Ginzburg-Landau model for superconductivity. Adv. Math. Sci. Appl. 5, 363–389 (1995)

    MathSciNet  MATH  Google Scholar 

  13. Chen, Z., Hoffmann, K., Liang, J.: On a non-stationary Ginzburg-Landau superconductivity model. Math. Methods Appl. Sci. 16, 855–875 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chrysafinos, K., Hou, L. S.: Error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions. SIAM J. Numer. Anal. 40, 282–306 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Rational Mech. Anal. 151, 221–276 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Crabtree, G., Leaf, G., Kaper, H., Vinokur, V., Koshelev, A., Braun, D., Levine, D., Kwok, W., Fendrich, J.: Time-dependent Ginzburg-Landau simulations of vortex guidance by twin boundaries. Phys. C 263, 401–408 (1996)

    Article  Google Scholar 

  17. Dauge, M.: Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Math, p 1341. Springer, Berlin (1988)

    Book  Google Scholar 

  18. Dauge, M.: Singularities of corner problems and problems of corner singularities. ESAIM Proc. 6, 19–40 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Du, Q.: Finite element methods for the time-dependent Ginzburg-Landau model of superconductivity. Comput. Math. Appl. 27, 1–17 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Du, Q.: Discrete gauge invariant approximations of a time-dependent Ginzburg-Landau model of superconductivity. Math. Comp. 67, 965–986 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Du, Q.: Numerical approximations of the Ginzburg-Landau models for superconductivity. J. Math. Phys. 46, 095–109 (2005)

    MathSciNet  Google Scholar 

  22. Du, Q., Gunzburger, M., Peterson, J.: Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34, 54–81 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. Du, Q., Wei, J., Zhao, C.: Vortex solutions of the high- κ high-field Ginzburg-Landau model with an applied current. SIAM J. Math. Anal. 42, 2368–2401 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Feireisl, E., Takac, P.: Long-time stabilization of solutions to the Ginzburg-Landau equations of superconductivity. Monatsh. Math. 133, 197–221 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gao, H., Li, B., Sun, W.: Optimal error estimates of linearized Crank-Nicolson Galerkin FEMs for the time-dependent Ginzburg-Landau equations in superconductivity. SIAM J. Numer. Anal. 52, 1183–1202 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gao, H., Li, B., Sun, W.: Stability and convergence of fully discrete Galerkin FEMs for the nonlinear thermistor equations in a nonconvex polygon. Numer. Math. 136, 383–409 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gao, H., Sun, W.: An efficient fully linearized semi-implicit Galerkin-mixed FEM for the dynamical Ginzburg-Landau equations of superconductivity. J. Comput. Phys. 294, 329–345 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gao, H., Sun, W.: A new mixed formulation and efficient numerical solution of Ginzburg-Landau equations under the temporal gauge. SIAM J. Sci. Comput. 38, A1339–A1357 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gatica, G.: A Simple Introduction to the Mixed Finite Element Method. Theory and Applications, Springer Briefs in Mathematics. Springer, New York (2014)

    Google Scholar 

  30. Girault, V., Raviart, P.: Finite Element Methods for Navier-Stokes Equations. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  31. Gor’kov, L., Eliashberg, G.: Generalization of the Ginzburg-Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities. Soviet Phys.-JETP 27, 328–334 (1968)

    Google Scholar 

  32. Gropp, W., Kaper, H., Leaf, G., Levine, D., Palumbo, M., Vinokur, V.: Numerical simulation of vortex dynamics in type-II superconductors. J. Comput. Phys. 123, 254–266 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Gunter, D., Kaper, H., Leaf, G.: Implicit integration of the time-dependent Ginzburg-Landau equations of superconductivity. SIAM J. Sci. Comput. 23, 1943–1958 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Levermore, C., Oliver, M.: The complex Ginzburg-Landau equation as a model problem. Lectures Appl. Math. 31, 141–190 (1996)

    MathSciNet  MATH  Google Scholar 

  35. Li, B.: Convergence of a decoupled mixed FEM for the dynamic Ginzburg-Landau equations in nonsmooth domains with incompatible initial data, Calcolo online. https://doi.org/10.1007/s10092-017-0237-0 (2017)

  36. Li, B., Yang, C.: Global well-posedness of the time-dependent Ginzburg-Landau superconductivity model in curved polyhedra, arXiv:1411.4235

  37. Li, B., Zhang, Z.: Mathematical and numerical analysis of time-dependent Ginzburg-Landau equations in nonconvex polygons based on Hodge decomposition. Math. Comp. 86, 1579–1608 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Li, B., Zhang, Z.: A new approach for numerical simulation of the time-dependent Ginzburg-Landau equations. J. Comput. Phys. 303, 238–250 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Li, B., Sun, W.: Maximal L p analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra. Math. Comput. 86, 1071–1102 (2017)

    Article  MATH  Google Scholar 

  40. Logg, A., Mardal, K., Wells, G. (eds.): Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012). https://doi.org/10.1007/978-3-642-23099-8

  41. Mu, M.: A linearized Crank-Nicolson-Galerkin method for the Ginzburg-Landau model. SIAM J. Sci. Comput. 18, 1028–1039 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  42. Mu, M., Huang, Y.: An alternating Crank-Nicolson method for decoupling the Ginzburg-Landau equations. SIAM J. Numer. Anal. 35, 1740–1761 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  43. Nirenberg, L.: An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa (3) 20, 733–737 (1966)

    MathSciNet  MATH  Google Scholar 

  44. Peres-Hari, L., Rubinstein, J., Sternberg, P.: Kinematic and dynamic vortices in a thin film driven by an applied current and magnetic field. Physica D 261, 31–41 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Richardson, W., Pardhanani, A., Carey, G., Ardelea, A.: Numerical effects in the simulation of Ginzburg-Landau models for superconductivity. Int. J. Numer. Meth. Engng. 59, 1251–1272 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  46. Rodriguez-Bernal, A., Wang, B., Willie, R.: Asymptotic behavior of the time-dependent Ginzburg-Landau equations of superconductivity. Math. Methods Appl. Sci. 22, 1647–1669 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  47. Tang, Q., Wang, S.: Time dependent Ginzburg-Landau equations of superconductivity. Physica D 88, 139–166 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  48. Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)

    MATH  Google Scholar 

  49. Winiecki, T., Adams, C.: A fast semi-implicit finite difference method for the TDGL equation. J. Comput. Phys. 179, 127–139 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The first author would like to thank Prof. Douglas Arnold for useful discussions on mixed methods for vector Poisson equations. The first author also acknowledges helpful discussions with Dr. Buyang Li. The discrete Sobolev embedding inequality in the ?? was proved after a discussion with Buyang Li. The authors are grateful to the referee for many valuable comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Huadong Gao

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Huadong Gao

  3. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    Weiwei Sun

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  1. Huadong Gao
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  2. Weiwei Sun
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Correspondence to Huadong Gao.

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Communicated by: Francesca Rapetti

The work of the author Huadong Gao was supported in part by the National Science Foundation of China No. 11501227 and Fundamental Research Funds for the Central Universities, HUST, P.R. China, under Grant No. 2014QNRC025, No. 2015QN13, and No. 2017KFYXJJ089.

The work of the author Weiwei Sun was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China. (Project No. CityU 11300517).

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Gao, H., Sun, W. Analysis of linearized Galerkin-mixed FEMs for the time-dependent Ginzburg-Landau equations of superconductivity. Adv Comput Math 44, 923–949 (2018). https://doi.org/10.1007/s10444-017-9568-2

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  • DOI: https://doi.org/10.1007/s10444-017-9568-2

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