On the Finite Element Approximation of an Elliptic Variational Inequality Arising from an Implicit Time Discretization of the Stefan Problem

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Abstract
A semi-discretization in time of the weak formulation of the two phase Stefan moving boundary problem results in an elliptic boundary value problem with a non-linear jump discontinuity which can be set as an elliptic variational inequality. The purpose of this paper is to consider a finite element approximation to the inequality. Assuming that the solution is in H2 and that the length of the free boundary is finite an error estimate is proved. The resulting algebraic problem is one of solving a system of nonlinear equations associated with a diagonal multivalued monotone mapping. An S.O.R. method is given and shown to be globally convergent.

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