Elastic deformation processes play an important role in solid mechanics.
In this talk, we consider nonlinear elastic behavior with a hyperelastic
material law. Combined with the equations of equilibrium this forms a
nonlinear first order system of partial differential equations for the
displacement 
and the first Piola-Kirchhoff stress tensor 
. In
order to solve this system, we consider a nonlinear least squares
functional, which has to be minimized. For the minimization we use the
iterative Gauss-Newton method, which results in a sequence of linear
least squares problems.
For the finite element approximation of the associated variational
problem we use quadratic Raviart-Thomas elements for the stress and
continuous quadratic finite elements for the displacement.
At the end of the talk we will give a numerical example and an outlook.