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Error estimates for two-phase Stefan problems in several space variables. II: Nonlinear flux conditions. (English) Zbl 0606.65085
[For part I see ibid. 22, 457-499 (1985; reviewed above)]
We derive \(L^ 2\)-error-estimates for multidimensional two-phase Stefan problems involving further nonlinearities such as nonlinear flux conditions. We approximate the enthalpy formulation by a regularization procedure combined with a \(C^ 0\)-piecewise linear finite element scheme in space, and the implicit Euler scheme in time. Under some restrictions on the initial datum and on the finite element mesh, we obtain an \(L^ 2\)-rate of convergence of order \(\epsilon^{1/2}\) for regularized problems, and an \(L^ 2\)-rate of convergence of order (h\(| \log h|^ k+\tau +\epsilon)^{1/2}\) for the discrete problems \((k=0,1)\). These estimates lead to the choice \(h\sim \epsilon \sim \tau\) and yield a global \(L^ 2\)-rate essentially of order \(h^{1/2}.\)
The a priori relationship between the approximation parameters allows the discrete problem to satisfy a maximum principle (without a lumping procedure), from which a priori bounds and monotonicity properties of the discrete solutions are shown.

MSC:
65Z05 Applications to the sciences
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35K05 Heat equation
35R35 Free boundary problems for PDEs
80A17 Thermodynamics of continua
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