A globalized inexact semismooth Newton method for nonsmooth fixed--point equations involving variational inequalities
Authors
- Alphonse, Amal
ORCID: 0000-0001-7616-3293 - Christof, Constantin
- Hintermüller, Michael
ORCID: 0000-0001-9471-2479 - Papadopoulos, Ioannis
2020 Mathematics Subject Classification
- 35J86 47J20 49J40 49J52 49M15
Keywords
- Semismooth Newton method, quasi-variational inequality, thermoforming, nonsmooth analysis, obstacle problem, Newton differentiability, semismoothness, superlinear convergence
DOI
Abstract
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.
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