Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T17:08:08.666Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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).