Classical Stefan problem
Eriksson and Johnson [ErikssonJohnson:91] developed the first a posteriori error estimators for finite element diskretizations of linear parabolic problems (like heat equation), where only computable terms like the discrete numerical solution enter. These methods were extended by both authors [ErikssonJohnson:95b] and by Verfürth [Verfuerth:95a] to classes of nonlinear parabolic equations. But degenerate parabolic problems like the classical Stefan problemu_t - \Delta \beta(u) = f mit \beta(s) = min(s,0) + max(s-1, 0)
are not covered by them. As \beta vanishes in the whole interval [0,1], the equation is not uniformly parabolic, so the above techniques can not be applied. In our work groupt, adaptive methods for degenerate parabolic problems were developed [NSV:97b]. In [NSV:97a], a-posteriori error estimators for the Stefan problem are developed, which give upper bounds for the error in temperature in $L^2(L^2)$ and for the error in enthalpy in $L^\infty(H^{-1})$. The methods are based on the use of a (linear, degenerate) dual problem. The adaptive finite element method was implemented in two and three space dimensions, with local mesh refinements based on the bisecion of mesh elements (triangles or tetrahedra), based on the finite element toolbox ALBERT. In [NSV:97a] we demonstrate the application to a two dimensional model problem, more simulations in two and three space dimensions are presented in [NSV:97g,NSV:97e].
Moving interface for a 3D Stefan problem
Continuous Casting
The continuous casting problem is a Stefan problem with given, dominant convection. It can be used to model the casting of steel. $L^2$-norm based methods lead here to an exponential dependence of the constants in the error estimator on the maximal convection. In [CNS:98a] we present a posteriori estimators for the $L^1(L^1)$-norm of the temperature error, which results in a much weeker dependence of constants from the convection. The next figure shows the temperature graph and the corresponding adaptive mesh (with a vertical zoom by factor 16) from a two dimensional simulation.
Temperature graph from a 2D continuous casting simulation
Corresponding finite element mesh (vertical zoom by factor 16)