A Globalized Inexact Semismooth Newton Method for Nonsmooth Fixed-point Equations involving Variational Inequalities
We develop a semismooth Newton framework for the numerical solution of fixed-point equations that are posed in Banach spaces. The framework is motivated by applications in the field of obstacle-type quasi-variational inequalities and implicit obstacle problems. It is discussed in a general functiona...
Gespeichert in:
Hauptverfasser: | , , , |
---|---|
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext bestellen |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Zusammenfassung: | We develop a semismooth Newton framework for the numerical solution of
fixed-point equations that are posed in Banach spaces. The framework is
motivated by applications in the field of obstacle-type quasi-variational
inequalities and implicit obstacle problems. It is discussed in a general
functional analytic setting and allows for inexact function evaluations and
Newton steps. Moreover, if a certain contraction assumption holds, we show that
it is possible to globalize the algorithm by means of the Banach fixed-point
theorem and to ensure $q$-superlinear convergence to the problem solution for
arbitrary starting values. By means of a localization technique, our Newton
method can also be used to determine solutions of fixed-point equations that
are only locally contractive and not uniquely solvable. We apply our algorithm
to a quasi-variational inequality which arises in thermoforming and which not
only involves the obstacle problem as a source of nonsmoothness but also a
semilinear PDE containing a nondifferentiable Nemytskii operator. Our analysis
is accompanied by numerical experiments that illustrate the mesh-independence
and $q$-superlinear convergence of the developed solution algorithm. |
---|---|
DOI: | 10.48550/arxiv.2409.19637 |