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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.

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
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