A Globalized Inexact Semismooth Newton Method for Nonsmooth Fixed-point Equations involving Variational Inequalities
<|reference_start|>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 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.<|reference_end|>
@article{alphonse2024a,
    title={A Globalized Inexact Semismooth Newton Method for Nonsmooth Fixed-point
  Equations involving Variational Inequalities},
    author={Amal Alphonse, Constantin Christof, Michael Hinterm"uller, Ioannis P.
  A. Papadopoulos},
    journal={arXiv preprint arXiv:2409.19637},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.19637},
    primaryClass={math.NA cs.NA math.AP}
}