## Abstract

The change of variable for the temperature Θ in the one‐phase Stefan problem leads to the evolution inequality, (u_{t} – Δu – f)(v – u) ⩾ 0 for all regular v ⩾ 0, where u ⩾ 0 is required. This inequality is to hold over a space‐time domain D = Ω × (0, T) with a Dirichlet boundary condition imposed on ∂ Ω × (0, T) and a zero initial condition. The free boundary phase interface is given in one space dimension by The fully implicit divided difference scheme leads to a sequence of elliptic variational inequalities for {u_{m}}. The sequence {u_{m}} may be interpolated linearly in t to obtain an approximation U_{Δt} of u. The following results are obtained in this paper: (i) a two‐sided weak maximum principle for u_{m} – u_{m‐1} in N space dimensions, hence the free boundary approximation for N = 1, is a monotone increasing step function; (ii) the uniform convergence of U_{Δt} and ∇U_{Δt}, to u and ∇u, respectively, on D; (iii) the uniform convergence to the Hölder continuous, monotone increasing free boundary x on [0, T] of the piecewise linear sequence x_{Δt}, where x_{Δt} interpolates x_{Δt}, in one space dimension; (iv) a constructive existence proof for u and x in prescribed regularity classes.

Original language | English (US) |
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Pages (from-to) | 149-167 |

Number of pages | 19 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 2 |

Issue number | 2 |

DOIs | |

State | Published - 1980 |

## ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)