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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