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

HTML articles powered by AMS MathViewer

- by Buyang Li and Zhimin Zhang PDF
- Math. Comp.
**86**(2017), 1579-1608 Request permission

## 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.## References

- Robert A. Adams,
*Sobolev spaces*, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR**0450957** - 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**, DOI 10.1007/s00211-009-0241-7 - 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**, DOI 10.1007/s10440-010-9580-8 - 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**, DOI 10.1016/j.apnum.2009.09.004 - Jöran Bergh and Jörgen Löfström,
*Interpolation spaces. An introduction*, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR**0482275** - 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**, DOI 10.1090/S0025-5718-2011-02540-8 - 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**, DOI 10.1007/978-1-4757-3658-8 - 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**, DOI 10.1007/s10543-006-0087-7 - Zhiming Chen,
*Mixed finite element methods for a dynamical Ginzburg-Landau model in superconductivity*, Numer. Math.**76**(1997), no. 3, 323–353. MR**1452512**, DOI 10.1007/s002110050266 - 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. MR**1856238**, DOI 10.1137/S0036142998349102 - 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** - 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. - 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**, DOI 10.1002/mma.1670161203 - Martin Costabel and Monique Dauge,
*Maxwell and Lamé eigenvalues on polyhedra*, Math. Methods Appl. Sci.**22**(1999), no. 3, 243–258. MR**1672271**, DOI 10.1002/(SICI)1099-1476(199902)22:3 - Monique Dauge,
*Elliptic boundary value problems on corner domains*, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR**961439**, DOI 10.1007/BFb0086682 - P. G. De Gennes,
*Superconductivity of Metal and Alloys*, Advanced Books Classics, Westview Press, 1999. - 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**, DOI 10.1080/00036819408840240 - 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**, DOI 10.1016/0898-1221(94)90091-4 - 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**, DOI 10.1090/S0025-5718-98-00954-5 - Qiang Du,
*Numerical approximations of the Ginzburg-Landau models for superconductivity*, J. Math. Phys.**46**(2005), no. 9, 095109, 22. MR**2171212**, DOI 10.1063/1.2012127 - 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**, DOI 10.1090/S0025-5718-04-01719-3 - H. Frahm, S. Ullah, and A. Dorsey,
*Flux dynamics and the growth of the superconducting phase*, Phys. Rev. Letters,**66**(1991), pp. 3067–3072. - 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**, DOI 10.1137/130918678 - 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**, DOI 10.1016/j.jcp.2015.03.057 - V. Ginzburg and L. Landau,
*Theory of Superconductivity*, Zh. Eksp. Teor. Fiz.,**20**(1950), pp. 1064–1082. - 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**, DOI 10.1137/1.9781611972030.ch1 - 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**, DOI 10.1006/jcph.1996.0022 - 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. - 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** - J.L. Lions,
*Quelques methodes de resolution des problems auxlimites non lineaires*, Dunrod, Paris (1969) (Chinese translation version, Sun Yat-sen University Press, 1992). - 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**, DOI 10.1016/j.jcp.2015.09.049 - F. Liu, M. Mondello, and N. Goldenfeld,
*Kinetics of the superconducting transition*, Phys. Rev. Letters,**66**(1991), pp. 3071–3074. - M. Tinkham,
*Introduction to Superconductivity*, 2nd ed., McGraw-Hill, New York, 1994. - William McLean,
*Strongly elliptic systems and boundary integral equations*, Cambridge University Press, Cambridge, 2000. MR**1742312** - 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**, DOI 10.1137/S1064827595283756 - 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**, DOI 10.1137/S0036142996303092 - 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. - Ian Wood,
*Maximal $L^p$-regularity for the Laplacian on Lipschitz domains*, Math. Z.**255**(2007), no. 4, 855–875. MR**2274539**, DOI 10.1007/s00209-006-0055-6

## Additional Information

**Buyang Li**- Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
- MR Author ID: 910552
- 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
- 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. - © Copyright 2016 American Mathematical Society
- Journal: Math. Comp.
**86**(2017), 1579-1608 - MSC (2010): Primary 65M12, 65M60; Secondary 35Q56, 35K61
- DOI: https://doi.org/10.1090/mcom/3177
- MathSciNet review: 3626529