Finite element approximation of a Stefan problem with degenerate Joule heating

Barrett, John W.; Nürnberg, Robert

ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 4, pp. 633-652.

We consider a fully practical finite element approximation of the following degenerate system

$$\frac{\partial }{\partial t}\rho \left(u\right)-\nabla .\left(\alpha \left(u\right)\nabla u\right)\ni \sigma \left(u\right){\left|\nabla \phi \right|}^2,\nabla .\left(\sigma \left(u\right)\nabla \phi \right)=0$$

subject to an initial condition on the temperature, $u$, and boundary conditions on both $u$ and the electric potential, $\phi$. In the above $\rho \left(u\right)$ is the enthalpy incorporating the latent heat of melting, $\alpha \left(u\right)>0$ is the temperature dependent heat conductivity, and $\sigma \left(u\right)\ge 0$ is the electrical conductivity. The latter is zero in the frozen zone, $u\le 0$, 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 $\sigma \left(u\right)$ and the quadratic nature of the Joule heating term forcing the Stefan problem. Finally, some numerical experiments are presented in two space dimensions.

[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

[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

[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

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

[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

[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

[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

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

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