Existence and approximation of weak solutions of the Stefan problem with nonmonotone nonlinearities

Abstract

Consider the equation, in the distribution sense, for the temperature in a two-phase multidimensional Stefan problem

$$\frac{{\partial u}}{{\partial t}} - \nabla .(k(u) \nabla u) + g(u) = f$$

((1))

on a space-time domain D=(O,T) × Ω with specified initial and boundary conditions and enthalpy discontinuity across the free boundary. Here the conductivity coefficient k is a positive function with compact range, defined and continuous on R except at O, and g is a Lipschitz body heating function, frequently encountered in welding problems, which is not assumed monotone. (We may take g such that g(u)u≥0).

Implicit two level time discretizations are employed in transformed versions of (1), giving a (finite) sequence of nonlinear elliptic boundary value problems (for each δt) which are solved by a Galerkin method. A subsequence of the step functions constructed on D is shown to converge weakly to a weak solution of the transformed equation. If, in addition, g is monotone, the entire sequence is strongly convergent to the unique solution.

Research supported by a grant from the Science Research Council, at Oxford University Computing Laboratory, 19 Parks Road, Oxford, OX1 3PL.