Finite element approximation of a Stefan problem with degenerate Joule heating
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 4, p. 633-652

We consider a fully practical finite element approximation of the following degenerate system tρ(u)-.(α(u)u)σ(u)|φ| 2 ,.(σ(u)φ)=0 subject to an initial condition on the temperature, u, and boundary conditions on both u and the electric potential, φ. In the above ρ(u) is the enthalpy incorporating the latent heat of melting, α(u)>0 is the temperature dependent heat conductivity, and σ(u)0 is the electrical conductivity. The latter is zero in the frozen zone, u0, which gives rise to the degeneracy in this Stefan system. In addition to showing stability bounds, we prove (subsequence) convergence of our finite element approximation in two and three space dimensions. The latter is non-trivial due to the degeneracy in σ(u) and the quadratic nature of the Joule heating term forcing the Stefan problem. Finally, some numerical experiments are presented in two space dimensions.

DOI : https://doi.org/10.1051/m2an:2004030
Classification:  35K55,  35K65,  35R35,  65M12,  65M60,  80A22
Keywords: Stefan problem, Joule heating, degenerate system, finite elements, convergence
@article{M2AN_2004__38_4_633_0,
     author = {Barrett, John W. and N\"urnberg, Robert},
     title = {Finite element approximation of a Stefan problem with degenerate Joule heating},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {4},
     year = {2004},
     pages = {633-652},
     doi = {10.1051/m2an:2004030},
     zbl = {1072.80010},
     mrnumber = {2087727},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2004__38_4_633_0}
}
Barrett, John W.; Nürnberg, Robert. Finite element approximation of a Stefan problem with degenerate Joule heating. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 4, pp. 633-652. doi : 10.1051/m2an:2004030. http://www.numdam.org/item/M2AN_2004__38_4_633_0/

[1] J.W. Barrett and C.M. Elliott, A finite element method on a fixed mesh for the Stefan problem with convection in a saturated porous medium, in Numerical Methods for Fluid Dynamics, K.W. Morton and M.J. Baines Eds., Academic Press (London) (1982) 389-409. | Zbl 0527.76096

[2] J.W. Barrett and R. Nürnberg, Convergence of a finite element approximation of surfactant spreading on a thin film in the presence of van der Waals forces. IMA J. Numer. Anal. 24 (2004) 323-363. | Zbl 1143.76473

[3] C.M. Elliott, On the finite element approximation of an elliptic variational inequality arising from an implicit time discretization of the Stefan problem. IMA J. Numer. Anal. 1 (1981) 115-125. | Zbl 0469.65042

[4] C.M. Elliott, Error analysis of the enthalpy method for the Stefan problem. IMA J. Numer. Anal. 7 (1987) 61-71. | Zbl 0638.65088

[5] C.M. Elliott and S. Larsson, A finite element model for the time-dependent Joule heating problem. Math. Comp. 64 (1995) 1433-1453. | Zbl 0846.65047

[6] R.F. Gariepy, M. Shillor and X. Xu, Existence of generalized weak solutions to a model for in situ vitrification. European J. Appl. Math. 9 (1998) 543-559. | Zbl 0939.35197

[7] S.S. Koegler and C.H. Kindle, Modeling of the in situ vitrification process. Amer. Ceram. Soc. Bull. 70 (1991) 832-835.

[8] J. Simon, Compact sets in the space L p (0,T;B). Ann. Math. Pura. Appl. 146 (1987) 65-96. | Zbl 0629.46031

[9] X. Xu, A compactness theorem and its application to a system of partial differential equations. Differential Integral Equations 9 (1996) 119-136. | Zbl 0843.35049

[10] X. Xu, Existence for a model arising from the in situ vitrification process. J. Math. Anal. Appl. 271 (2002) 333-342. | Zbl 1011.35135