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

Summary

A parabolic problem of the following form is considered

$$\frac{\partial }{{\partial t}}\left[ {a\vartheta + w} \right] - \Delta \vartheta = f$$

((1))

$$w \varepsilon \Lambda (\vartheta ),$$

((2))

wherea 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 therelaxed equation

$$\varepsilon \frac{{\partial w}}{{\partial t}} + \Lambda ^{ - 1} (w) \mathrel\backepsilon \vartheta (\varepsilon : constant > 0).$$

((3))

The problem (1), (3) is then discretized in time by thesemi-explicit scheme

$$a\frac{{\vartheta ^n - \vartheta ^{n - 1} }}{\tau } + \frac{{w^n - w^{n - 1} }}{\tau } - \Delta \vartheta ^n = f^n$$

((4))

$$\varepsilon \frac{{w^n - w^{n - 1} }}{\tau } + \Lambda ^{ - 1} (w^n ) \mathrel\backepsilon \vartheta ^{n - 1} ;$$

((5))

a finite element space discretization and quadrature formulae are then introduced. Thus at each time-step (5) is replaced by a finite number ofindependent algebraic equations, which can be solved with respect to the barycentral values ofw ⁿ; then (4) is reduced to alinear system of algebraic equations having as unknowns the nodal values of ϑⁿ. 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.