A full step Newton method for the solution of a nonlinear magnetostatic optimization problem

We consider a nonlinear magnetostatic field problem, where in a certain subdomain a cost functional (i.e., the magnetic field has to fulfill a prescribed figure) is defined. The optimization problem is stated in such a way that the field problem is treated as a constraint. A Lagrange function is established depending on the magnetic vector potential A, the current density J in coil segments and the vector of Lagrange multipliers λ. The derivatives with respect to the variables are set to zero to obtain the optimality system. The optimality system is solved applying full Newton steps without line search. For the left hand side of the Newton system the second derivatives of the Lagrange function are required while the first derivatives of the Lagrange function are written into the right hand side. Employing a finite element scheme to discretize the problem domain, we end up with a linear system of equations that has to be solved for each Newton step