Error estimates for a semi-explicit numerical scheme for Stefan-type problems

A parabolic problem of the following form is considered {Mathematical expression} {Mathematical expression} where a is a positive constant, f is a datum and λ is a maximal monotone graph. This system contains the (weak formulation of the)Stefan problem as a particular case. Here the problem (1), (2) is approximated by coupling (1) with the relaxed equation {Mathematical expression} The problem (1), (3) is then discretized in time by the semi-explicit scheme {Mathematical expression} {Mathematical expression} a finite element space discretization and quadrature formulae are then introduced. Thus at each time-step (5) is replaced by a finite number of independent algebraic equations, which can be solved with respect to the barycentral values of wn; then (4) is reduced to a linear system of algebraic equations having as unknowns the nodal values of θ{symbol}n. Assuming the condition τ/ε≦a, the fully discrete scheme is stable and its solution converges to that of (1), (2). Error estimates are proved. The results of some numerical experiments are discussed; they show that the present method is faster than other classical procedures.