Gradient schemes for the Stefan problem

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