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**Numerical aspects of parabolic free boundary and hysteresis problems.**
*(English)*
Zbl 0819.35155

Visintin, Augusto (ed.), Phase transitions and hysteresis. Lectures given at the 3rd session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy, July 13-21, 1993. Berlin: Springer-Verlag. Lect. Notes Math. 1584, 213-284 (1994).

This research exposition developed on the basis of lectures given at a summer course in 1993 is concerned with a detailed numerical analysis for parabolic free boundary problems and equations with hysteresis.

In the first section the weak (enthalpy) formulation of the two-phase Stefan problem is given together with properties of the solution (existence, regularity, continuous dependence on the data). Due to the low regularity of such problems smoothing procedures in the constitutive relations are discussed. In this connection travelling waves are considered for one-dimensional problems to understand the smoothing effects. Finally, the weak formulation of parabolic problems with a continuous hysteresis relation is recalled.

In the second section discrete-time schemes are introduced for the above mentioned problems. On one hand, nonlinear methods using backward- differences are considered, on the other hand linearization algorithms by means of the nonlinear Chernoff formulas are suggested.

The third section deals with the study of fully discrete schemes based on finite elements for the space discretization. The corresponding schemes are analyzed in detail and in addition to that the approximation of the free boundary is investigated.

In order to improve the accuracy and efficiency adaptive finite element methods are discussed for the two-phase Stefan problem in a further section.

In the last section the motion of surfaces by mean curvature and its approximation is studied.

Finally, the reader can find an extensive list of references, which are carefully cited in the text.

For the entire collection see [Zbl 0801.00028].

In the first section the weak (enthalpy) formulation of the two-phase Stefan problem is given together with properties of the solution (existence, regularity, continuous dependence on the data). Due to the low regularity of such problems smoothing procedures in the constitutive relations are discussed. In this connection travelling waves are considered for one-dimensional problems to understand the smoothing effects. Finally, the weak formulation of parabolic problems with a continuous hysteresis relation is recalled.

In the second section discrete-time schemes are introduced for the above mentioned problems. On one hand, nonlinear methods using backward- differences are considered, on the other hand linearization algorithms by means of the nonlinear Chernoff formulas are suggested.

The third section deals with the study of fully discrete schemes based on finite elements for the space discretization. The corresponding schemes are analyzed in detail and in addition to that the approximation of the free boundary is investigated.

In order to improve the accuracy and efficiency adaptive finite element methods are discussed for the two-phase Stefan problem in a further section.

In the last section the motion of surfaces by mean curvature and its approximation is studied.

Finally, the reader can find an extensive list of references, which are carefully cited in the text.

For the entire collection see [Zbl 0801.00028].

Reviewer: J.Steinbach (München)

### MSC:

35R35 | Free boundary problems for PDEs |

35K65 | Degenerate parabolic equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |