New robust nonconforming finite elements of higher order

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 systems.