In this talk we shall discuss the monolithic Newton multigrid FEM
approach for the simulation of general nonlinear incompressible flow problems.
The governing equations arise from model problems with non-isothermal behavior
and pressure and shear dependent viscosity, in viscoelastic fluids. The well-known
HWNP is overcome with LCR formulation wich guarantees the positive definiteness
of the discrete conformation tensor. The coupling between the velocity gradient and
the elastic stress, which leads to the restriction for the choice of FEM approximation
spaces, and the hyperbolic nature of the problem are handled with edge-oriented
stabilization. The resulting linearized system inside of an outer Newton solver is
a typical nonsymmetric saddle point problem solved using the geometrical multigrid
with a Vanka-like smoother. We validate quantitatively this approach for the
well-known benchmark of flow around cylinder for two particular models namely
Oldroyd-B and Giesekus.