Least squares finite element methods are motivated, beside others, by the
fact that in contrast to standard mixed finite element methods, the choice of the finite
element spaces is not subject to the LBB stability condition and the corresponding
discrete linear system is symmetric and positive definite. We intend to benefit from
these two positive attractive features, in one hand, to use different types of elements
representing the physics as for instance the capillary forces and mass conservation
and, on the other hand, to show the flexibility of the geometric multigrid methods
to handle efficiently the resulting linear systems. We numerically solve the V-VP,
Vorticity-Velocity-Pressure, and S-V-P, Stress-Velocity-Pressure, formulations of
the incompressible Navier-Stokes equations based on the least squares principles
using different types of finite elements, conforming, nonconforming and discontinuous
of low as well as high order. For the discrete systems, we use a conjugate
gradient (CG) solver accelerated with a geometric multigrid preconditioner. In addition,
we employ a Krylov space smoother which allows a parameter-free smoothing.
Combining this linear solver with the Newton linearization results in a robust and
efficient solver.We analyze the application of this general approach, of using different
types of finite elements, and the efficient solver, geometric multigrid, for several
prototypical benchmark configurations (driven cavity, flow around obstacles), and
we investigate the effects of pressure jumps for the capillary force in multiphase
flow simulations (static bubble configuration).