In this talk we present some recent numerical results
concerning flows with pressure and shear dependent viscosity.
From the numerical point of view several problems arise, first from the difficulty of
approximating incompressible velocity fields and, second, from poor conditioning
and possible lack of differentiability of the involved nonlinear functions due
to the material laws.
The lack of differentiability can be treated by regularisation. Then the Newton methods
as linearization technique can be applied; moreover the presence of the pressure
in the viscosity function leads to an additional term introducing a new non
classical linear algebraic saddle point problem.
The difficulty related to the approximation of incompressible velocity fields was
treated by applying the Rannacher-Turek Stokes element, here also we were
facing another problem related to the nonconforming approximation for problems involving
the symmetric part of gradient. In fact, the classical discrete Korn`s Inequality is
out of being satisfied. A new and more general approach which involves the jump across the
interelement boundaries should be used, which means a small modification of the discrete
bilinear form by adding an interface term, penalizing the jump of the velocity.
This is a modified procedure in the derivation of Discontinuous Galerkin
formulation. A Vanka smoother as defect correction inside of a direct multigrid
approach is presented. The results of some computational experiments for realistic flow
configurations are provided, which contain a pressure dependent viscosity,
too. In particular we present results for modelling granular flow via the
``Schaeffer model``.