# Semismooth* Newton Methods

## DI Michael Mandlmayr

**June 22, 2021, 3:30 p.m. ZOOM**

We present the application of the general semismooth* Newton method, introduced by Gfrerer and Outrata, to three challenging problems:

•Quasi-Variaitonal Inequalities

•Tresca Friction

•Coloumb Friction

The novelty of this method is, that these problems are interpreted as generalized equations. This means that for a set valued function we are interested in finding a point, such that the image contains zero.

As the method is called a “Newton”-method, to fulfill the expectations some linearization has to happen at some point of the graph. The linearization of a set valued mapping is done via a so called normalcones to the graph, loosely speaking this normalcone can be seen as a generalization of normals. Furthermore, we need to have suitable point for the linearization. In contrast to the well known newton method for functions, where for an iteration point $x$ we just take $(x,F(x))$, for set valued mappings the choice is not trivial and is done in a so called Approximation step.

So this method consists of two parts:

•An approximation step that chooses a point of the graph

•A newton step, where we solve a linearized problem

We will show that under suitable assumptions this method converges locally superlinear to the solution.

Moreover, we will illustrate the construction of such a method on the three given examples.

Also we will present numerical evidence for this convergence speed.