Nonsmooth exact penalization second-order methods for incompressible bi-viscous fluids

We consider the exact penalization of the incompressibility condition $$\text {div}(\mathbf {u})=0$$ for the velocity field of a bi-viscous 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.