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An upwinding mixed finite element method for a mean field model of superconducting vortices

Published online by Cambridge University Press:  15 April 2002

Zhiming Chen  and
Qiang Du
Zhiming Chen
Affiliation:
Institute of Mathematics, Academia Sinica, Beijing 100080, P.R. China. e-mail: zmchen@math03.math.ac.cn
Qiang Du
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, and Department of Mathematics, Iowa State University, Ames, IA 50011, USA. e-mail: madu@uxmail.ust.hk
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Abstract

In this paper, we construct a combined upwinding and mixed finite element method for the numerical solution of a two-dimensional mean field model of superconducting vortices. An advantage of our method is that it works for any unstructured regular triangulation. A simple convergence analysis is given without resorting to the discrete maximum principle. Numerical examples are also presented.

Keywords

Mean field modelsuperconductivityvorticesmixed finite elementunstructured gridconvergence analysis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 3 , May 2000 , pp. 687 - 706
Copyright
© EDP Sciences, SMAI, 2000

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References

S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer-Verlag, New York (1994).
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer, New York (1991).
Chapman, S.J., A mean-field model of superconducting vortices in three dimensions. SIAM J. Appl. Math. 55 (1995) 1259-1274. CrossRef
Chapman, S.J. and Richardson, G., Motion of vortices in type-II superconductors. SIAM J. Appl. Math. 55 (1995) 1275-1296. CrossRef
Chapman, S.J., Rubenstein, J., and Schatzman, M., A mean-field model of superconducting vortices. Euro. J. Appl. Math. 7 (1996) 97-111. CrossRef
Z. Chen and S. Dai, Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity. (Preprint, 1998).
Cockburn, B., Hou, S. and Shu, C.-W., The Runge-Kutta local projection discontinuos galerkin finite element method for conservation laws IV: The multidimensional case. Math. Com. 54 (1990) 545-581.
Q. Du, Convergence analysis of a hybrid numerical method for a mean field model of superconducting vortices. SIAM Numer. Analysis, (1998).
Du, Q., Gunzburger, M., and Peterson, J., Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Review 34 (1992) 54-81. CrossRef
Du, Q., Gunzburger, M., and Peterson, J., Computational simulations of type-II superconductivity including pinning mechanisms. Phys. Rev. B 51 (1995) 16194-16203. CrossRefPubMed
Q. Du, M. Gunzburger and H. Lee, Analysis and computation of a mean field model for superconductivity. Numer. Math. 81 539-560 (1999).
Gray, Q. Du , High-kappa limit of the time dependent Ginzburg-Landau model for superconductivity. SIAM J. Appl. Math. 56 (1996) 1060-1093.
Dynamics, W. E of vortices in Ginzburg-Landau theories with applications to superconductivity. Phys. D 77 (1994) 383-404.
C. Elliott and V. Styles, Numerical analysis of a mean field model of superconductivity. preprint.
V. Girault and -A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer, Berlin (1986).
Grisvard, Elliptic Problems on Non-smooth Domains. Pitman, Boston (1985).
Huang, C. and Svobodny, T., Evolution of Mixed-state Regions in type-II superconductors. SIAM J. Math. Anal. 29 (1998) 1002-1021. CrossRef
Lesaint and P.A. Raviart, On a Finite Element Method for Solving the Neutron Transport equation, in: Mathematical Aspects of the Finite Element Method in Partial Differential Equations, C. de Boor Ed., Academic Press, New York (1974).
Prigozhin, L., On the Bean critical-state model of superconductivity. Euro. J. Appl. Math. 7 (1996) 237-247.
Prigozhin, L., The Bean model in superconductivity: variational formulation and numerical solution. J. Com Phys. 129 (1996) 190-200. CrossRef
Raviart and J. Thomas, A mixed element method for 2nd order elliptic problems, in: Mathematical Aspects of the Finite Element Method, Lecture Notes on Mathematics, Springer, Berlin 606 (1977).
R. Schatale and V. Styles, Analysis of a mean field model of superconducting vortices (preprint).
R. Temam, Navier-Stokes equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1984).