  We consider the exact penalization of the incompressibility condition
$div(u)=0$ for the velocity field of a Bingham fluid in terms of the
$L^1$-norm. This penalization procedure results in a nonsmooth optimization
problem for which we propose an algorithm using generalized second-order
information. Our method solves the resulting nonsmooth problem by considering
the steepest descent direction and extra generalized second-order information
associated to the nonsmooth term. This method has the advantage that the
divergence-free property is enforced by the descent direction proposed by the
method without the need of build-in divergence-free approximation schemes. The
inexact penalization approach, given by the $L^2$-norm, is also considered in
our discussion and comparison.

Optimization and Control