# zbMATH — the first resource for mathematics

Error estimates for multidimensional Stefan problems with general boundary conditions. (English) Zbl 0593.35094
Free boundary problems: applications and theory, Vol. III, Proc. Int. Colloq., Maubuisson/France 1984, Res. Notes Math. 120, 50-60 (1985).
Summary: [For the entire collection see Zbl 0578.35003.]
In this paper we deal with a variational procedure based on the use of an integral test function which allows us to get $$L^ 2$$-type error estimates for the enthalpy formulation of two-phase Stefan problems in several space variables. We can improve some known results for linear boundary conditions and extend the analysis to nonlinear flux conditions.
The numerical approach consists in a regularization procedure combined with a $$C^ 0$$ piecewise-linear finite element scheme in space and the implicit Euler scheme in time. The difficulty is to obtain error estimates for the global approximation.
The set $$A_{\epsilon}(u)=\{0<u<\epsilon \}$$ (u temperature) plays an important role in this analysis. We present a class of multidimensional two-phase Stefan problems for which $$| A_{\epsilon}(u)| <c_{\epsilon}$$. These problems have further regularity properties: the free boundary is Lipschitz-continuous and mushy regions cannot appear spontaneously.

##### MSC:
 35R35 Free boundary problems for PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations