# Numerical analysis of the quasistatic thermoviscoelastic thermistor problem

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

- Volume: 40, Issue: 2, page 353-366
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topFernández, José R.. "Numerical analysis of the quasistatic thermoviscoelastic thermistor problem." ESAIM: Mathematical Modelling and Numerical Analysis 40.2 (2006): 353-366. <http://eudml.org/doc/249716>.

@article{Fernández2006,

abstract = {
In this work, the quasistatic thermoviscoelastic thermistor problem is
considered. The thermistor model describes the combination of the effects due to
the heat, electrical current conduction and Joule's heat generation. The variational
formulation leads to a coupled system of nonlinear variational equations for which
the existence of a weak solution is recalled.
Then, a fully discrete algorithm is introduced based on the finite element
method to approximate the spatial variable and an Euler scheme to discretize
the time derivatives. Error estimates are derived and, under suitable
regularity assumptions, the linear convergence of the scheme is deduced.
Finally, some numerical simulations are performed in order to show the behaviour
of the algorithm.
},

author = {Fernández, José R.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Thermoviscoelastic thermistor; error estimates; finite elements; numerical
simulations.; variational formulation; existence; weak solution; finite element method},

language = {eng},

month = {6},

number = {2},

pages = {353-366},

publisher = {EDP Sciences},

title = {Numerical analysis of the quasistatic thermoviscoelastic thermistor problem},

url = {http://eudml.org/doc/249716},

volume = {40},

year = {2006},

}

TY - JOUR

AU - Fernández, José R.

TI - Numerical analysis of the quasistatic thermoviscoelastic thermistor problem

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2006/6//

PB - EDP Sciences

VL - 40

IS - 2

SP - 353

EP - 366

AB -
In this work, the quasistatic thermoviscoelastic thermistor problem is
considered. The thermistor model describes the combination of the effects due to
the heat, electrical current conduction and Joule's heat generation. The variational
formulation leads to a coupled system of nonlinear variational equations for which
the existence of a weak solution is recalled.
Then, a fully discrete algorithm is introduced based on the finite element
method to approximate the spatial variable and an Euler scheme to discretize
the time derivatives. Error estimates are derived and, under suitable
regularity assumptions, the linear convergence of the scheme is deduced.
Finally, some numerical simulations are performed in order to show the behaviour
of the algorithm.

LA - eng

KW - Thermoviscoelastic thermistor; error estimates; finite elements; numerical
simulations.; variational formulation; existence; weak solution; finite element method

UR - http://eudml.org/doc/249716

ER -

## References

top- W. Allegretto and H. Xie, A non-local thermistor problem. Eur. J. Appl. Math.6 (1995) 83–94.
- W. Allegreto, Y. Lin and A. Zhou, A box scheme for coupled systems resulting from microsensor thermistor problems. Dynam. Contin. Discret. S.5 (1999) 209–223.
- W. Allegreto, Y. Lin and S. Ma, Existence and long time behaviour of solutions to obstacle thermistor equations. Discrete Contin. Dyn. S.8 (2002) 757–780.
- S.N. Antontsev and M. Chipot, The thermistor problem: existence, smoothness, uniqueness, blowup. SIAM J. Math. Anal.25 (1994) 1128–1156.
- A.R. Bahadir, Application of cubic B-spline finite element technique to the thermistor problem. Appl. Math. Comput.149 (2004) 379–387.
- A. Bermúdez, M.C. Muñiz and P. Quintela, Numerical solution of a three-dimensional thermoelectric problem taking place in an aluminum electrolytic cell. Comput. Method Appl. M.106 (1993) 129–142.
- O. Chau, J.R. Fernández, W. Han and M. Sofonea, A frictionless contact problem for elastic-viscoplastic materials with normal compliance and damage. Comput. Method Appl. M.191 (2002) 5007–5026.
- X. Chen, Existence and regularity of solutions of a nonlinear degenerate elliptic system arising from a thermistor problem. J. Partial Differential Equations7 (1994) 19–34.
- P.G. Ciarlet, The finite element method for elliptic problems, in Handbook of Numerical Analysis, Vol. II, Part 1, P.G. Ciarlet and J.L. Lions Eds., North Holland (1991) 17–352.
- G. Cimatti, Remark on the existence and uniqueness for the thermistor problem under mixed boundary conditions. Quart. J. Mech. Appl. Math.47 (1989) 117–121.
- G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics, Springer, New-York (1976).
- J.R. Fernández, K.L. Kuttler, M.C. Muñiz and M. Shillor, A model and simulations of the thermoviscoelastic thermistor. Eur. J. Appl. Math. (submitted).
- W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, Americal Mathematical Society–International Press (2002).
- S.D. Howison, A note on the thermistor problem in two space dimension. Quart. J. Mech. Appl. Math.47 (1989) 509–512.
- S.D. Howison, J. Rodrigues and M. Shillor, Stationary solutions to the thermistor problem. J. Math. Anal. Appl.174 (1993) 573–588.
- S. Kutluay, A.R. Bahadir and A. Ozdeć, A variety of finite difference methods to the thermistor with a new modified electrical conductivity. Appl. Math. Comput.106 (1999) 205–213.
- S. Kutluay, A.R. Bahadir and A. Ozdeć, Various methods to the thermistor problem with a bulk electrical conductivity. Int. J. Numer. Method. Engrg.45 (1999) 1–12.
- S. Kutluay and E. Esen, A B-spline finite element method for the thermistor problem with the modified electrical conductivity. Appl. Math. Comput.156 (2004) 621–632.
- S. Kutluay and A.S. Wood, Numerical solutions of the thermistor problem with a ramp electrical conductivity. Appl. Math. Comput.148 (2004) 145–162.
- K.L. Kuttler, M. Shillor and J.R. Fernández, Existence for the thermoviscoelastic thermistor problem. Differential Equations Dynam. Systems (to appear).
- H. Xie and W. Allegretto, ${C}^{\alpha}\left(\overline{\Omega}\right)$ solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem. SIAM J. Math. Anal.22 (1991) 1491–1499.
- X. Xu, The thermistor problem with conductivity vanishing for large temperature. P. Roy. Soc. Edinb. A124 (1994) 1–21.
- X. Xu, On the existence of bounded temperature in the thermistor problem with degeneracy. Nonlinear Anal.42 (2000) 199–213.
- X. Xu, On the effects of thermal degeneracy in the thermistor problem. SIAM J. Math. Anal.35 (4) (2003) 1081–1098.
- X. Xu, Local regularity theorems for the stationary thermistor problem. P. Roy. Soc. Edinb. A134 (2004) 773–782.
- S. Zhou and D.R. Westbrook, Numerical solutions of the thermistor equations. J. Comput. Appl. Math.79 (1997) 101–118.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.