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**The use of linear approximation scheme for solving the Stefan problem.**
*(English)*
Zbl 0902.65072

The paper deals with a linear approximation scheme based on Rothe’s method for certain parabolic moving boundary value problems in weak formulation. In particular, the two-phase Stefan problem on a domain consisting of two components with imperfect contact is studied. As an application, a model of the heat transfer in the course of solidification of steel in the process of continuous casting with imperfect contact between the mold and the slab is discussed.

Convergence of the Rothe functions to the weak solution of the problem under consideration is proved, an iterative procedure for their construction is designed and its properties are studied. The full discretization scheme is briefly discussed as well. A numerical example documents the applicability of the method.

The paper employs and modifies the approach of J. Kačur, A. Handlovičová and M. Kačurová [SIAM J. Numer. Anal. 30, No. 6, 1703-1722 (1993; Zbl 0792.65070)] and compares the effectivity of the proposed numerical method with other similar procedures based on Rothe’s method.

Convergence of the Rothe functions to the weak solution of the problem under consideration is proved, an iterative procedure for their construction is designed and its properties are studied. The full discretization scheme is briefly discussed as well. A numerical example documents the applicability of the method.

The paper employs and modifies the approach of J. Kačur, A. Handlovičová and M. Kačurová [SIAM J. Numer. Anal. 30, No. 6, 1703-1722 (1993; Zbl 0792.65070)] and compares the effectivity of the proposed numerical method with other similar procedures based on Rothe’s method.

Reviewer: P.Přikryl (Praha)

### MSC:

65Z05 | Applications to the sciences |

80A22 | Stefan problems, phase changes, etc. |

35R35 | Free boundary problems for PDEs |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

35K05 | Heat equation |

### Keywords:

moving boundary value problem; Stefan problem; heat equation; imperfect contact; Rothe’s method; solidification of steel; numerical example### Citations:

Zbl 0792.65070
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\textit{P. Dzurenda}, Appl. Math., Praha 42, No. 1, 35--56 (1997; Zbl 0902.65072)

### References:

[1] | J. L. Desbiolles, J. J. Droux, J. Rapaz, M. Rapaz: Simulation of solidification of alloys by the finite element method. An advanced course on the Finite Element Methods in Physics, European Association of Physics, Lausanne (Switzerland), 1987. |

[2] | A. Handlovičová: Errors estimates of a linear approximation schemes for nonlinear difussion problems. Acta Math. Univ. Comenianae LXI, 1 (1992), 27-39. · Zbl 0820.65055 |

[3] | W. Jäger, J. Kačur: Solution of porous medium type systems by linear approximation schemes. Preprint 540, Universität Heidelberg SFB 1233, 1989. · Zbl 0744.65060 |

[4] | J. Kačur, A. Handlovičová, M. Kačurová: Solution of nonlinear diffusion problems by linear approximation schemes. SIAM Num. Anal. 30 (1993), 1703-1722. · Zbl 0792.65070 |

[5] | J. Kačur, S. Luckhaus: Approximation of degenerate parabolic systems by nondegenerate elliptic parabolic systems. Preprint M2-91, Comenius University, Bratislava. · Zbl 0894.65043 |

[6] | R. H. Nochetto: Error estimates for multidimensional Stefan problems with general boundary conditions. Research Notes in Math. 120, Pitman, Boston, 1985, pp. 50-60. · Zbl 0593.35094 |

[7] | R. H. Nochetto, C. Verdi: An efficent linear scheme to approximate parabolic free boundary problems: error estimates and implementation. American Math. Society, 1988, pp. 27-53. · Zbl 0657.65131 |

[8] | R. H. Nochetto, C. Verdi: Approximation of degenerate parabolic problems using numerical integration. SIAM J. numer. anal. 35 (1988), no. 4, 784-814. · Zbl 0655.65131 |

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