Mathematics of Computation

Approximations of a Ginzburg-Landau model for superconducting hollow spheres based on spherical centroidal Voronoi tessellations

Author(s): Qiang Du; Lili Ju.
Journal: Math. Comp. 74 (2005), 1257-1280.
MSC (2000): Primary 65N15, 65N99; Secondary 82D55
Posted: December 8, 2004
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Abstract: In this paper the numerical approximations of the Ginzburg- Landau model for a superconducting hollow spheres are constructed using a gauge invariant discretization on spherical centroidal Voronoi tessellations. A reduced model equation is used on the surface of the sphere which is valid in the thin spherical shell limit. We present the numerical algorithms and their theoretical convergence as well as interesting numerical results on the vortex configurations. Properties of the spherical centroidal Voronoi tessellations are also utilized to provide a high resolution scheme for computing the supercurrent and the induced magnetic field.

1.: S. ADLER AND T. PIRAN, Relaxation methods for gauge field equilibrium equations, Rev. Mod. Phys., 56 , pp. 1-40, 1984. MR 85c:81022
2.: A. AFTALION AND Q. DU, The bifurcation diagram for the Ginzburg-Landau system for superconductivity, Physica D, 163, pp. 94-105, 2001. MR 2003d:82125
3.: B. BAELUS AND F. PEETERS, Dependence of the vortex configuration on the geometry of mesoscopic flat samples, Phys. Rev. B, 65, 104515, 2002.
4.: B. BAELUS, F. PEETERS AND V. SCHWEIGERT, Saddle-point states and energy barriers for vortex entrance and exit in superconducting disks and rings, Phys. Rev. B, 63, 144517, 2001.
5.: S. CHAPMAN, Q. DU AND M. GUNZBURGER, A variable thickness thin film model for superconductivity, Z. Angew. Math. Phys., 47, pp. 410-431 1995. MR 2003d:82125
6.: S. CHAPMAN, Q. DU AND M. GUNZBURGER, A Ginzburg-Landau model for superconducting shells, preprint.
7.: S. CHAPMAN, D. HERON, The Motion of Superconducting Vortices in Thin Films of Varying Thickness, SIAM J. Applied Mathematics, 58, pp. 1808-1825, 1998. MR 99k:82077
8.: Z. CHEN AND S. DAI, Adaptive Galerkin method with error control for a dynamical Ginzburg-Landau model in superconductivity SIAM J. Numer. Anal., 38, pp. 1961-1985, 2001. MR 2002g:65114
9.: M. COFFEY, London model for the levitation force between a horizontally oriented point magnetic dipole and superconducting sphere, Phys. Rev. B, 65, 214524, 2002.
10.: S. DING AND Q. DU, Critical magnetic field and asymptotic behavior for superconducting thin films, SIAM J. Mathematical Analysis, 34, pp. 239-256, 2002. MR 2004c:35398
11.: M. DODGSON AND M.A. MOORE, Vortices in a thin-film superconductor with a spherical geometry, Phys. Rev. B, 55, pp. 3816-3831, 1997.
12.: Q. DU, Discrete gauge invariant approximations of a time-dependent Ginzburg-Landau model of superconductivity, Math. Comp., 67, pp. 965-986, 1998. MR 98m:820774
13.: Q. DU, M. GUNZBURGER, AND L. JU, Constrained centroidal Voronoi tessellations on general surfaces, SIAM J. Sci. Comput., 24, pp.1488-1506, 2003. MR 2004d:65155
14.: Q. DU, M. GUNZBURGER AND L. JU, Voronoi-based finite volume methods, optimal Voronoi meshes, and PDEs on the sphere, Comp. Meth. Appl. Mech. Engrg., 192, pp. 779-802, 2003. MR 2004g:65142
15.: Q. DU, M. GUNZBURGER M. AND J. PETERSON, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Review, 34, pp. 54-81, 1992. MR 93g:82109
16.: Q. DU, M. GUNZBURGER AND J. PETERSON, Computational simulations of type-II superconductivity including pinning mechanisms, Phys. Rev. B, 51, pp. 16194-16203, 1995.
17.: Q. DU AND L. JU, Finite volume methods on spheres and spherical centroidal Voronoi tessellations, IMA preprint, No. 1918, 2003.
18.: Q. DU AND L. JU, Numerical simulation of the quantized vortices on a thin superconducting hollow sphere, to appear in J. Computational Phys., 2004.
19.: Q. DU AND F. LIN, Ginzburg-Landau vortices: dynamics, pinning and hysteresis, SIAM J. Math. Anal., 28, pp. 1265-1293, 1999. MR 99c:82075
20.: Q. DU, R. NICOLAIDES AND X. WU, Analysis and convergence of a covolume approximation of the Ginzburg-Landau models of superconductivity, SIAM J. Numer. Anal., 35, pp. 1049-1072, 1998. MR 99e:65178
21.: W. E., Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, Phys. D, 77, pp. 383-404, 1994. MR 95g:82097
22.: V. FOMIN, V. MISKO, J. DEVREESE AND V. MOSHCHALKOV, Superconducting mesoscopic square loop, Phys. Rev. B, 58, pp. 11703-11715, 1998. MR
23.: W. GROPP, H. KAPER, G. LEAF, D. LEVINE, M. PALUMBO, AND V. VINOKUR, Numerical simulation of vortex dynamics in type-II superconductors, J. Comp. Phys., 123, pp. 254-266, 1996.
24.: E. HEBEY, Sobolev Spaces on Riemannian Manifolds, Springer, Berlin, 1991. MR 98k:46049
25.: B. IANOTTA, Music of the Spheres, New Scientist, 31, pp.28-31, 1996.
26.: H. JADALLAH, J. RUBINSTEIN, AND P. STERNBERG, Phase Transition Curves for Mesoscopic Superconducting Samples, Phys. Rev. Lett, 82, pp. 2935, 1999.
27.: E. POLAK, Computational Methods in Optimization, Academic Press, New York, 1971. MR 43:8222
28.: R. RENKA; Algorithm 772: STRIPACK: Delaunay triangulation and Voronoi diagrams on the surface of a sphere, ACM Trans. Math. Soft., 23, pp. 416-434, 1997.
29.: B. RICHTER AND R. WARBURTON, A new generation of superconducting gravimeters, in Proceedings of the 13th Intern. Sym. on Earth Tides, Brussel, pp. 545-556, 1998, Série Géophysique, Royal Observatory of Belgium.
30.: V. SCHWEIGERT, F. PEETERS AND P. DEO, Vortex Phase Diagram for Mesoscopic Superconducting Disks, Phys. Rev. Lett, 81, pp. 2783-2786, 1998.
31.: R. TAO, X. ZHANG, X. TANG, AND P. ANDERSON, Formation of High Temperature Superconducting Balls, Phys. Rev. Lett, 83, pp. 5575-5578, 1999.
32.: B. TENT, D. QU, D. SHI, W. BRESSER, P. BOOLCHAND AND Z. CAI, Angle dependence of magnetization in a single-domain sphere, Phys. Rev. B, 58, 11761, 1998.
33.: M. TINKHAM, Introduction to Superconductivity, 2nd ed., McGraw-Hill, New York, 1994.
34.: J. YEO AND M. MOORE, Non-integer flux quanta for a spherical superconductor, Phys. Rev. B, 58, pp. 10785-10789, 1998.
35.: Y. XIAO, S. BUCHMAN, G KEISER, B. MUHLFELDER, J. TURNEAURE, AND C. WU, Magnetic flux distribution on a spherical superconducting shell, Physica B, 194, pp.65-66, 1994.

Retrieve articles in Mathematics of Computation with MSC (2000): 65N15, 65N99, 82D55

Qiang Du
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: qdu@math.psu.edu

Lili Ju
Affiliation: Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455
Email: ju@ima.umn.edu

DOI: 10.1090/S0025-5718-04-01719-3
PII: S 0025-5718(04)01719-3
Keywords: Ginzburg-Landau model of superconductivity, finite volume, gauge invariance, convergence, spherical centroidal Voronoi tessellations
Received by editor(s): July 13, 2003
Received by editor(s) in revised form: January 5, 2004
Posted: December 8, 2004
Additional Notes: This work is supported in part by NSF-DMS 0196522 and NSF-ITR 0205232
Copyright of article: Copyright 2004, American Mathematical Society

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