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Efficient Numerical Solution of Dynamical Ginzburg-Landau Equations under the Lorentz Gauge

  • Huadong Gao (a1)

In this paper, a new numerical scheme for the time dependent Ginzburg-Landau (GL) equations under the Lorentz gauge is proposed. We first rewrite the original GL equations into a new mixed formulation, which consists of three parabolic equations for the order parameter ψ, the magnetic field σ=curlA, the electric potential θ=divA and a vector ordinary differential equation for the magnetic potential A, respectively. Then, an efficient fully linearized backward Euler finite element method (FEM) is proposed for the mixed GL system, where conventional Lagrange element method is used in spatial discretization. The new approach offers many advantages on both accuracy and efficiency over existing methods for the GL equations under the Lorentz gauge. Three physical variables ψ, σ and θ can be solved accurately and directly. More importantly, the new approach is well suitable for non-convex superconductors. We present a set of numerical examples to confirm these advantages.

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*Corresponding author. Email address: (H. Gao)
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[1] Alstrom, T., Sorensen, M., Pedersen, N. and Madsen, S., Magnetic flux lines in complex geometry type-II superconductors studied by the time dependent Ginzburg-Landau equation, Acta Appl. Math., 115(2011), 6374.
[2] Benzi, M., Golub, G. and Liesen, J., Numerical solution of saddle point problems, Acta Numerica, 14(2005), 1137.
[3] Bethuel, F., Brezis, H. and Hélein, F., Ginzburg-Landau Vortices, Progress in Nonlinear Partial Differential Equations and Their Applications 13, Birkhäuser Boston, Boston, 1994.
[4] Brenner, S. and Scott, L., The Mathematical Theory of Finite Element Methods, Springer, New York, 2002.
[5] Chen, L., Wu, Y., Zhong, L. and Zhou, J., Multigrid preconditioners for mixed finite element methods of vector Laplacian, arXiv:1601.04095.
[6] Chen, Z., Mixed finite element methods for a dynamical Ginzburg-Landau model in superconductivity, Numer. Math., 76(1997), 323353.
[7] Chen, Z. and Hoffmann, K., Numerical studies of a non-stationary Ginzburg-Landau model for superconductivity, Adv. Math. Sci. Appl., 5(1995), 363389.
[8] Chen, Z., Hoffmann, K. and Liang, J., On a non-stationary Ginzburg-Landau superconductivity model, Math. Methods Appl. Sci., 16(1993), 855875.
[9] Chrysafinos, K. and Hou, L., Error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions, SIAM J. Numer. Anal., 40(2002), 282306.
[10] Du, Q., Finite element methods for the time-dependent Ginzburg-Landau model of superconductivity, Comput. Math. Appl., 27(1994), 119133.
[11] Du, Q., Gunzburger, M. and Peterson, J., Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Rev., 34(1992), 5481.
[12] Fleckinger-Pelle, J. and Kaper, H., Gauges for the Ginzburg-Landau equations of superconductivity, Z. Angew. Math. Mech., 76(S2)(1996), 345348.
[13] Frahm, H., Ullah, S. and Dorsey, A., Flux dynamics and the growth of the superconducting phase, Phys. Rev. Lett., 66(1991), 30673070.
[14] Gao, H., Li, B. and 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(2014), 11831202.
[15] Gao, H. and Sun, W., An efficient fully linearized semi-implicit Galerkin-mixed FEM for the dynamical Ginzburg-Landau equations of superconductivity, J. Comput. Phys., 294(2015), 329345.
[16] Gao, H. and Sun, W., A new mixed formulation and efficient numerical solution of Ginzburg-Landau equations under the temporal gauge. SIAM J. Sci. Comput., 38(2016), A1339A1357.
[17] Gropp, W., Kaper, H., Leaf, G., Levine, D., Palumbo, M. and Vinokur, V., Numerical simulation of vortex dynamics in type-II superconductors, J. Comput. Phys., 123(1996), 254266.
[18] Gunter, D., Kaper, H. and Leaf, G., Implicit integration of the time-dependent Ginzburg-Landau equations of superconductivity, SIAM J. Sci. Comput., 23(2002), 19431958.
[19] Kim, S., Burkardt, J., Gunzburger, M., Peterson, J. and Hu, C., Effects of sample geometry on the dynamics and configurations of vortices in mesoscopic superconductors, Phys. Rev. B, 76(2007), 024509.
[20] Li, B. and Zhang, Z., Mathematical and numerical analysis of time-dependent Ginzburg-Landau equations in nonconvex polygons based on Hodge decomposition, Math. Comp., 86(2017), 15791608.
[21] Li, B. and Zhang, Z., A new approach for numerical simulation of the time-dependent Ginzburg-Landau equations, J. Comput. Phys., 303(2015), 238250.
[22] Logg, A., Mardal, K. and Wells, G. (Eds.), Automated Solution of Differential Equations by the Finite Element Method, Springer, Berlin, 2012.
[23] Mu, M., A linearized Crank-Nicolson-Galerkin method for the Ginzburg-Landau model, SIAM J. Sci. Comput., 18(1997), 10281039.
[24] Mu, M. and Huang, Y., An alternating Crank-Nicolson method for decoupling the Ginzburg-Landau equations, SIAM J. Numer. Anal., 35(1998), 17401761.
[25] Peng, L., Wei, Z. and Xu, D., Vortex states in mesoscopic superconductors with a complex geometry: A finite element analysis, Int. J. Mod. Phys. B, 28(2014), 1450127,
[26] Raza, N., Sial, S. and Siddiqi, S., Approximating time evolution related to Ginzburg-Landau functionals via Sobolev gradient methods in a finite-element setting, J. Comput. Phys., 229(2010), 16211625.
[27] Richardson, W., Pardhanani, A., Carey, G. and Ardelea, A., Numerical effects in the simulation of Ginzburg-Landau models for superconductivity, Int. J. Numer. Meth. Engng., 59(2004), 12511272.
[28] Yang, C., A linearized Crank-Nicolson-Galerkin FEM for the time-dependent Ginzburg-Landau equations under the temporal gauge, Numer. Methods Partial Differential Equations, 30(2014), 12791290.
[29] Zhang, Y., Sun, Z. and Wang, T., Convergence analysis of a linearized Crank-Nicolson scheme for the two-dimensional complex Ginzburg-Landau equation, Numer. Methods Partial Differential Equations, 29(2013), 14871503.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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