We study second order nonconforming finite elements as members of a new family
of higher order approaches which behave optimally not only on multilevel refined grids,
but also on perturbed grids which are still shape regular but which consist no longer
of asymptotically affine equivalent mesh cells. We present two approaches to prevent
this order reduction: The first one is based on the use of nonparametric basis functions
which are defined as polynomials on the original mesh cell. In the second approach,
we define all basis functions on the reference element but add one or more nonconforming
cell bubble functions which can be removed at the end by static condensation. For
the last approach, we prove optimal estimates for the approximation and consistency
error and derive optimal estimates for the discretization error in the case of a Poisson
problem. Furthermore, we construct and analyze numerically corresponding geometrical
multigrid solvers which are based on the canonical full order grid transfer operators.
Based on several benchmark configurations, for scalar Poisson problems as well as for the
incompressible Navier-Stokes equations (representing the desired application field of these
nonconforming finite elements), we demonstrate the high numerical accuracy, flexibility
and efficiency of the discussed new approaches which have been successfully implemented
in the FeatFlow software (www.featflow.de). The presented results show that the proposed
FEM-multigrid combination (together with discontinuous pressure approximations)
appear to be very advantageous candidates for realistic flow simulation tools, particularly
on (parallel) high performance computing system