Gradient schemes for the Stefan problem

Robert Eymard 1 Pierre Féron 1 Thierry Gallouët 2 Raphaèle Herbin 2 Cindy Guichard 3, 4
3 COFFEE - COmplex Flows For Energy and Environment
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR7351
4 LJAD
JAD - Laboratoire Jean Alexandre Dieudonné
Abstract : We show in this paper that the gradient schemes (which encompass a large family of discrete schemes) may be used for the approximation of the Stefan problem $\partial_t \bar u - \Delta \zeta (\bar u) = f$. The convergence of the gradient schemes to the continuous solution of the problem is proved thanks to the following steps. First, estimates show (up to a subsequence) the weak convergence to some function $u$ of the discrete function approximating $\bar u$. Then Alt-Luckhaus' method, relying on the study of the translations with respect to time of the discrete solutions, is used to prove that the discrete function approximating $\zeta(\bar u)$ is strongly convergent (up to a subsequence) to some continuous function $\chi$. Thanks to Minty's trick, we show that $\chi = \zeta(u)$. A convergence study then shows that $u$ is then a weak solution of the problem, and a uniqueness result, given here for fitting with the precise hypothesis on the geometric domain, enables to conclude that $u = \bar u$. This convergence result is illustrated by some numerical examples using the Vertex Approximate Gradient scheme.
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Robert Eymard, Pierre Féron, Thierry Gallouët, Raphaèle Herbin, Cindy Guichard. Gradient schemes for the Stefan problem. International Journal on Finite Volumes, Institut de Mathématiques de Marseille, AMU, 2013, Volume 10 special. ⟨hal-00751555⟩

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