##
**Numerical approximation of hysteresis problems.**
*(English)*
Zbl 0608.65082

The considered equations are of the form \(\partial (u+w)/\partial t+Au=f\); \(\partial u/\partial t+Au+w=f\) where A is an elliptic operator and f a datum, suitable initial and boundary conditions having to be provided. Hysteresis relations \(u\mapsto w\) are represented by means of continuous or (multivalued) discontinuous Volterra functionals. Existence results are given for the corresponding weak formulation. Implicit time discretization and finite elements are used for the numerical approximation.

The uniqueness is also proved. The numerical approximation of the above mentioned problems in a one-dimensional model situation is considered. The result of some numerical test is also reported. It is noted that from the numerical viewpoint no substantial difference arises between the cases of continuous and discontinuous hysteresis functionals. We must stress out the fact that the authors consider only the case in which hysteresis means a relation \(u\mapsto w\) such that w depends just on the range of u in [0,t] and on the order in which these values are assumed, not on the velocity v’(t); but in many real cases, encountered currently in electromagnetism for instance, the dependence on v’ is impossible to avoid.

The uniqueness is also proved. The numerical approximation of the above mentioned problems in a one-dimensional model situation is considered. The result of some numerical test is also reported. It is noted that from the numerical viewpoint no substantial difference arises between the cases of continuous and discontinuous hysteresis functionals. We must stress out the fact that the authors consider only the case in which hysteresis means a relation \(u\mapsto w\) such that w depends just on the range of u in [0,t] and on the order in which these values are assumed, not on the velocity v’(t); but in many real cases, encountered currently in electromagnetism for instance, the dependence on v’ is impossible to avoid.

Reviewer: E.V.Nicolau

### MSC:

65Z05 | Applications to the sciences |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N40 | Method of lines for boundary value problems involving PDEs |

35K20 | Initial-boundary value problems for second-order parabolic equations |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

78A25 | Electromagnetic theory (general) |