Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 058, 10 pages      math.AP/0606045      https://doi.org/10.3842/SIGMA.2006.058

A Dual Mesh Method for a Non-Local Thermistor Problem

Abderrahmane El Hachimi a, Moulay Rchid Sidi Ammi b and Delfim F.M. Torres b
a) UFR: Applied and Industrial Mathematics, University of Chouaib Doukkali, El Jadida, Maroc
b) Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received December 20, 2005, in final form May 08, 2006; Published online June 02, 2006

Abstract
We use a dual mesh numerical method to study a non-local parabolic problem arising from the well-known thermistor problem.

Key words: non-local thermistor problem; joule heating; box scheme method.

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References

  1. Allegretto W., Lin Y., Zhou A., A box scheme for coupled systems resulting from microsensor thermistor problems, Dynam. Contin. Discrete Impuls. Systems, 1999, V.5, N 1-4, 209-223.
  2. Antontsev S.N., Chipot M., The thermistor problem: existence, smoothness uniqueness, blowup, SIAM J. Math. Anal., 1994, V.25, N 4, 1128-1156.
  3. Cai Z.Q., On the finite volume element method, Numer. Math., 1991, V.58, N 7, 713-735.
  4. Ciarlet P.G., The finite element method for elliptic problems, Amsterdam, North-Holland, 1978.
  5. Chatzipantelidis P., Lazarov R.D., Thomée V., Error estimates for a finite volume element method for parabolic equations in convex polygonal domains, Numer. Methods Partial Differential Equations, 2004, V.20, Issue 5, 650-674.
  6. Cimatti G., On the stability of the solution of the thermistor problem, Appl. Anal., 1999, V.73, N 3-4, 407-423.
  7. Cimatti G., Stability and multiplicity of solutions for the thermistor problem, Ann. Mat. Pura Appl. (4), 2002, V.181, N 2, 181-212.
  8. El Hachimi A., Sidi Ammi M.R., Existence of weak solutions for the thermistor problem with degeneracy, in Proceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf., 2002, V.9, 127-137.
  9. El Hachimi A., Sidi Ammi M.R., Thermistor problem: a nonlocal parabolic problem, in Proceedings of the 2004-Fez Conference on Differential Equations and Mechanics, Electron. J. Differ. Equ. Conf., 2004, V.11, 117-128.
  10. El Hachimi A., Sidi Ammi M.R., Existence of global solution for a nonlocal parabolic problem, Electron. J. Qual. Theory Differ. Equ., 2005, N 1, 9 pages.
  11. El Hachimi A., Sidi Ammi M.R., Semi-discretization for a non local parabolic problem, Int. J. Math. Math. Sci., 2005, N 10, 1655-1664.
  12. Elliott C.M., Larsson S., A finite element model for the time-dependent Joule heating problem, Math. Comp., 1995, V.64, N 212, 1433-1453.
  13. Lacey A.A., Thermal runaway in a non-local problem modelling Ohmic heating. I. Model derivation and some special cases, European J. Appl. Math., 1995, V.6, N 2, 127-144.
  14. Lacey A.A., Thermal runaway in a non-local problem modelling Ohmic heating. II. General proof of blow-up and asymptotics of runaway, European J. Appl. Math., 1995, V.6, N 3, 201-224.
  15. Tzanetis D.E., Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating, Electron. J. Differential Equations, 2002, N 11, 26 pages.
  16. Xu X., Local and global existence of continuous temperature in the electrical heating of conductors, Houston J. Math., 1996, V.22, N 2, 435-455.
  17. Xu X., Existence and uniqueness for the nonstationary problem of the electrical heating of a conductor due to the Joule-Thomson effect, Int. J. Math. Math. Sci., 1993, V.16, N 1, 125-138.
  18. Yue X.Y., Numerical analysis of nonstationary thermistor problem, J. Comput. Math., 2004, V.12, N 3, 213-223.


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