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**Smoothing methods and semismooth methods for nondifferentiable operator equations.**
*(English)*
Zbl 0979.65046

This paper deals with the numerical solution of nonlinear equations in Banach spaces with operators that are not differentiable. Using a concept of semismoothness in infinite-dimensional spaces, which is based upon the notion of slanting functions and slant differentiability and which coincides with the semismoothness in the case of a locally Lipschitz continuous function on \(\mathbb{R}^n\), analogues of Newton’s method and their superlinear convergence are studied. It is proved that a function is slantly differentiable at some point iff it is Lipschitz continuous.

Finally, the two-dimensional Poisson equation with homogeneous Dirichlet boundary and nonlinear, nonsmooth right-hand side depending on the solution serves as an example.

Finally, the two-dimensional Poisson equation with homogeneous Dirichlet boundary and nonlinear, nonsmooth right-hand side depending on the solution serves as an example.

Reviewer: Etienne Emmrich (Berlin)

### MSC:

65J15 | Numerical solutions to equations with nonlinear operators |

47J25 | Iterative procedures involving nonlinear operators |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

65N06 | Finite difference methods for boundary value problems involving PDEs |

65H10 | Numerical computation of solutions to systems of equations |