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.

翻译：我们提出了一种半光滑牛顿框架，用于求解巴拿赫空间中提出的不动点方程的数值解。该框架的动机来源于障碍型拟变分不等式和隐式障碍问题领域的应用。我们在一般泛函分析框架下讨论该框架，并允许不精确的函数求值和牛顿步。此外，如果满足特定的收缩假设，我们证明可以通过巴拿赫不动点定理实现算法的全局化，并确保对于任意初始值都能以$q$-超线性收敛速度逼近问题解。通过局部化技术，我们的牛顿法也可用于求解仅局部收缩且非唯一可解的不动点方程。我们将算法应用于热成型中出现的拟变分不等式，该问题不仅包含障碍问题作为非光滑性来源，还涉及含有不可微涅梅茨基算子的半线性偏微分方程。我们的分析辅以数值实验，验证了所开发求解算法的网格无关性和$q$-超线性收敛性。