Finite element approximation of a Stefan problem with degenerate Joule heating

Barrett, John W., and Nürnberg, Robert. "Finite element approximation of a Stefan problem with degenerate Joule heating." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.4 (2004): 633-652. <http://eudml.org/doc/244791>.

@article{Barrett2004,
abstract = {We consider a fully practical finite element approximation of the following degenerate system\[ \hspace*\{-56.9055pt\} \{\frac\{\partial \}\{\partial t\}\} \rho (u) - \nabla . ( \,\alpha (u) \,\nabla u ) \ni \sigma (u)\,|\nabla \phi |^2 , \quad \nabla . (\, \sigma (u) \,\nabla \phi ) = 0 \]subject to an initial condition on the temperature, $u$, and boundary conditions on both $u$ and the electric potential, $\phi $. In the above $\rho (u)$ is the enthalpy incorporating the latent heat of melting, $\alpha (u) &gt;0 $ is the temperature dependent heat conductivity, and $\sigma (u)\ge 0$ 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 $\sigma (u)$ and the quadratic nature of the Joule heating term forcing the Stefan problem. Finally, some numerical experiments are presented in two space dimensions.},
author = {Barrett, John W., Nürnberg, Robert},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Stefan problem; Joule heating; degenerate system; finite elements; convergence},
language = {eng},
number = {4},
pages = {633-652},
publisher = {EDP-Sciences},
title = {Finite element approximation of a Stefan problem with degenerate Joule heating},
url = {http://eudml.org/doc/244791},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Barrett, John W.
AU - Nürnberg, Robert
TI - Finite element approximation of a Stefan problem with degenerate Joule heating
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 4
SP - 633
EP - 652
AB - We consider a fully practical finite element approximation of the following degenerate system\[ \hspace*{-56.9055pt} {\frac{\partial }{\partial t}} \rho (u) - \nabla . ( \,\alpha (u) \,\nabla u ) \ni \sigma (u)\,|\nabla \phi |^2 , \quad \nabla . (\, \sigma (u) \,\nabla \phi ) = 0 \]subject to an initial condition on the temperature, $u$, and boundary conditions on both $u$ and the electric potential, $\phi $. In the above $\rho (u)$ is the enthalpy incorporating the latent heat of melting, $\alpha (u) &gt;0 $ is the temperature dependent heat conductivity, and $\sigma (u)\ge 0$ 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 $\sigma (u)$ and the quadratic nature of the Joule heating term forcing the Stefan problem. Finally, some numerical experiments are presented in two space dimensions.
LA - eng
KW - Stefan problem; Joule heating; degenerate system; finite elements; convergence
UR - http://eudml.org/doc/244791
ER -