In this note we present some of our recent 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, Newton-like methods as linearization technique can be applied; however the presence of the pressure in the viscosity function leads to an additional term introducing a new
non-classical linear saddle point problem.
The difficulty related to the approximation of incompressible velocity fields is treated by applying the nonconforming Rannacher-Turek Stokes element. However, then we are facing
another problem related to the nonconforming approximation for problems involving the symmetric part of gradient: the classical discrete `Korn`s Inequality` is not satisfied.
A new and more general approach which involves the jump across the inter-element boundaries should be used, which requires a small modification of the discrete bilinear form by adding an interface term, penalizing the jump of the velocity over edges. This is
achieved via a modified procedure in the derivation of a Discontinuous Galerkin
formulation. As a solver for the discrete nonlinear systems, a Newton variant is discussed while a `Vanka-like` 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.