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Mathematical and numerical analysis of the time-dependent Ginzburg-Landau equations in nonconvex polygons based on Hodge decomposition


Authors: Buyang Li and Zhimin Zhang
Journal: Math. Comp. 86 (2017), 1579-1608
MSC (2010): Primary 65M12, 65M60; Secondary 35Q56, 35K61
DOI: https://doi.org/10.1090/mcom/3177
Published electronically: November 18, 2016
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Abstract: We prove well-posedness of the time-dependent Ginzburg-Landau system in a nonconvex polygonal domain, and decompose the solution as a regular part plus a singular part. We see that the magnetic potential is not in $ H^1(\Omega )$ in general, and so the finite element method (FEM) may give incorrect solutions. To overcome this difficulty, we reformulate the equations into an equivalent system of elliptic and parabolic equations based on the Hodge decomposition, which avoids direct calculation of the magnetic potential. The essential unknowns of the reformulated system admit $ H^1$ solutions and can be solved correctly by the FEMs. We then propose a decoupled and linearized FEM to solve the reformulated equations and present error estimates based on the proved regularity of the solution. Numerical examples are provided to support our theoretical analysis and show the efficiency of the method.


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  • [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • [2] Ana Alonso Rodríguez, Alberto Valli, and Rafael Vázquez Hernández, A formulation of the eddy current problem in the presence of electric ports, Numer. Math. 113 (2009), no. 4, 643-672. MR 2545497, https://doi.org/10.1007/s00211-009-0241-7
  • [3] Tommy Sonne Alstrøm, Mads Peter Sørensen, Niels Falsig Pedersen, and Søren Madsen, Magnetic flux lines in complex geometry type-II superconductors studied by the time dependent Ginzburg-Landau equation, Acta Appl. Math. 115 (2011), no. 1, 63-74. MR 2812976, https://doi.org/10.1007/s10440-010-9580-8
  • [4] Franck Assous and Michael Michaeli, Hodge decomposition to solve singular static Maxwell's equations in a non-convex polygon, Appl. Numer. Math. 60 (2010), no. 4, 432-441. MR 2607801, https://doi.org/10.1016/j.apnum.2009.09.004
  • [5] Jöran Bergh and Jörgen Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275
  • [6] S. C. Brenner, J. Cui, Z. Nan, and L.-Y. Sung, Hodge decomposition for divergence-free vector fields and two-dimensional Maxwell's equations, Math. Comp. 81 (2012), no. 278, 643-659. MR 2869031, https://doi.org/10.1090/S0025-5718-2011-02540-8
  • [7] Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376
  • [8] P. Chatzipantelidis, R. D. Lazarov, V. Thomée, and L. B. Wahlbin, Parabolic finite element equations in nonconvex polygonal domains, BIT 46 (2006), no. suppl., S113-S143. MR 2283311, https://doi.org/10.1007/s10543-006-0087-7
  • [9] Zhiming Chen, Mixed finite element methods for a dynamical Ginzburg-Landau model in superconductivity, Numer. Math. 76 (1997), no. 3, 323-353. MR 1452512, https://doi.org/10.1007/s002110050266
  • [10] Zhiming Chen and Shibin Dai, Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity, SIAM J. Numer. Anal. 38 (2001), no. 6, 1961-1985 (electronic). MR 1856238, https://doi.org/10.1137/S0036142998349102
  • [11] Zhiming Chen and K.-H. Hoffmann, Numerical studies of a non-stationary Ginzburg-Landau model for superconductivity, Adv. Math. Sci. Appl. 5 (1995), no. 2, 363-389. MR 1360996
  • [12] Z. Chen and K. H. Hoffmann, Numerical simulations of dynamical Ginzburg-Landau vortices in superconductivity, in the book ``Numerical Simulation in Science and Engineering'', Notes on Numerical Fluid Mechanics 48 (1994), pp. 31-38. Vieweg, Braun-Schweig/Wiesbaden.
  • [13] Zhi Ming Chen, K.-H. Hoffmann, and Jin Liang, On a nonstationary Ginzburg-Landau superconductivity model, Math. Methods Appl. Sci. 16 (1993), no. 12, 855-875. MR 1247887, https://doi.org/10.1002/mma.1670161203
  • [14] Martin Costabel and Monique Dauge, Maxwell and Lamé eigenvalues on polyhedra, Math. Methods Appl. Sci. 22 (1999), no. 3, 243-258. MR 1672271, https://doi.org/10.1002/(SICI)1099-1476(199902)22:3$ \langle $243::AID-MMA37$ \rangle $3.3.CO;2-S
  • [15] Monique Dauge, Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. MR 961439
  • [16] P. G. De Gennes, Superconductivity of Metal and Alloys, Advanced Books Classics, Westview Press, 1999.
  • [17] Qiang Du, Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity, Appl. Anal. 53 (1994), no. 1-2, 1-17. MR 1379180, https://doi.org/10.1080/00036819408840240
  • [18] Q. Du, Finite element methods for the time-dependent Ginzburg-Landau model of superconductivity, Comput. Math. Appl. 27 (1994), no. 12, 119-133. MR 1284135, https://doi.org/10.1016/0898-1221(94)90091-4
  • [19] Qiang Du, Discrete gauge invariant approximations of a time dependent Ginzburg-Landau model of superconductivity, Math. Comp. 67 (1998), no. 223, 965-986. MR 1464143, https://doi.org/10.1090/S0025-5718-98-00954-5
  • [20] Qiang Du, Numerical approximations of the Ginzburg-Landau models for superconductivity, J. Math. Phys. 46 (2005), no. 9, 095109, 22. MR 2171212, https://doi.org/10.1063/1.2012127
  • [21] Qiang Du and Lili Ju, Approximations of a Ginzburg-Landau model for superconducting hollow spheres based on spherical centroidal Voronoi tessellations, Math. Comp. 74 (2005), no. 251, 1257-1280. MR 2137002, https://doi.org/10.1090/S0025-5718-04-01719-3
  • [22] H. Frahm, S. Ullah, and A. Dorsey, Flux dynamics and the growth of the superconducting phase, Phys. Rev. Letters, 66 (1991), pp. 3067-3072.
  • [23] Huadong Gao, Buyang Li, and Weiwei Sun, Optimal error estimates of linearized Crank-Nicolson Galerkin FEMs for the time-dependent Ginzburg-Landau equations in superconductivity, SIAM J. Numer. Anal. 52 (2014), no. 3, 1183-1202. MR 3201193, https://doi.org/10.1137/130918678
  • [24] Huadong Gao and Weiwei Sun, An efficient fully linearized semi-implicit Galerkin-mixed FEM for the dynamical Ginzburg-Landau equations of superconductivity, J. Comput. Phys. 294 (2015), 329-345. MR 3343730, https://doi.org/10.1016/j.jcp.2015.03.057
  • [25] V. Ginzburg and L. Landau, Theory of Superconductivity, Zh. Eksp. Teor. Fiz., 20 (1950), pp. 1064-1082.
  • [26] Pierre Grisvard, Elliptic problems in nonsmooth domains, Classics in Applied Mathematics, vol. 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. Reprint of the 1985 original [ MR0775683]; With a foreword by Susanne C. Brenner. MR 3396210
  • [27] William D. Gropp, Hans G. Kaper, Gary K. Leaf, David M. Levine, Mario Palumbo, and Valerii M. Vinokur, Numerical simulation of vortex dynamics in type-II superconductors, J. Comput. Phys. 123 (1996), no. 2, 254-266. MR 1372372, https://doi.org/10.1006/jcph.1996.0022
  • [28] L. P. Gor'kov and G. M. Eliashberg, Generalization of the Ginzburg-Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities, Soviet Phys. JETP, 27 (1968), pp. 328-334.
  • [29] R. Bruce Kellogg, Corner singularities and singular perturbations, Ann. Univ. Ferrara Sez. VII (N.S.) 47 (2001), 177-206 (English, with English and Italian summaries). MR 1897566
  • [30] J.L. Lions, Quelques methodes de resolution des problems auxlimites non lineaires, Dunrod, Paris (1969) (Chinese translation version, Sun Yat-sen University Press, 1992).
  • [31] Buyang Li and Zhimin Zhang, A new approach for numerical simulation of the time-dependent Ginzburg-Landau equations, J. Comput. Phys. 303 (2015), 238-250. MR 3422711, https://doi.org/10.1016/j.jcp.2015.09.049
  • [32] F. Liu, M. Mondello, and N. Goldenfeld, Kinetics of the superconducting transition, Phys. Rev. Letters, 66 (1991), pp. 3071-3074.
  • [33] M. Tinkham, Introduction to Superconductivity, 2nd ed., McGraw-Hill, New York, 1994.
  • [34] William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
  • [35] Mo Mu, A linearized Crank-Nicolson-Galerkin method for the Ginzburg-Landau model, SIAM J. Sci. Comput. 18 (1997), no. 4, 1028-1039. MR 1453555, https://doi.org/10.1137/S1064827595283756
  • [36] Mo Mu and Yunqing Huang, An alternating Crank-Nicolson method for decoupling the Ginzburg-Landau equations, SIAM J. Numer. Anal. 35 (1998), no. 5, 1740-1761. MR 1640013, https://doi.org/10.1137/S0036142996303092
  • [37] D. Y. Vodolazov, I. L. Maksimov, and E. H. Brandt, Vortex entry conditions in type-II superconductors. Effect of surface defects, Physica C, 384 (2003), pp. 211-226.
  • [38] Ian Wood, Maximal $ L^p$-regularity for the Laplacian on Lipschitz domains, Math. Z. 255 (2007), no. 4, 855-875. MR 2274539, https://doi.org/10.1007/s00209-006-0055-6

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Additional Information

Buyang Li
Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
Email: buyang.li@polyu.edu.hk, libuyang@gmail.com

Zhimin Zhang
Affiliation: Beijing Computational Science Research Center, Beijing, 100193, People’s Republic of China — and — Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: zmzhang@csrc.ac.cn, zzhang@math.wayne.edu

DOI: https://doi.org/10.1090/mcom/3177
Keywords: Superconductivity, reentrant corner, singularity, well-posedness, finite element method, convergence, Hodge decomposition
Received by editor(s): April 17, 2015
Received by editor(s) in revised form: February 8, 2016
Published electronically: November 18, 2016
Additional Notes: The work of the first author was supported in part by National Natural Science Foundation of China (NSFC) under grant 11301262. The research stay of this author at Universität Tübingen was funded by the Alexander von Humboldt Foundation
The second author is the corresponding author, whose work was supported in part by the National Natural Science Foundation of China (NSFC) under grants 11471031, 91430216, and U1530401, and by the US National Science Foundation (NSF) through grant DMS-1419040.
Article copyright: © Copyright 2016 American Mathematical Society

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