# Homotopy Methods for Nonlinear Magnetostatics

## Stefan Mühlböck

**Jan. 19, 2010, 4:15 p.m. P 215**

We consider a nonlinear system of equations. One possibility to solve this problem is Newtons method, which is only locally convergent. In order to obtain an appropriate starting value for Newtons method we want to consider so-called homotopy methods. Thereby a crucial point is to ﬁnd a continuous transition from a easily solvable problem to the original nonlinear problem by introducing a so-called homotopy parameter. In other words a known (or easy computable) solution is connected with the exact solution by a continuous function which we want to follow numerically. Therefore we present several strategies for following such curves.

Another main part of this talk is how to control the step size of such a method. This is based on asymptotic expansion.

The derived method is applied to solve a one dimensional nonlinear magnetostatic problem. Using numerical results we compare the diﬀerent strategies for following such curves.