Unified edge-oriented stabilization of nonconforming finite element methods for incompressible flow problems

Summary. This paper deals with various aspects of edge-oriented stabilization techniques for nonconforming finite element methods for the numerical solution of incompressible flow problems. We discuss two separate classes of problems which require appropriate stabilization techniques: First, the lack of coercivity for nonconforming low order approximations for treating problems with the symmetric deformation tensor instead of the gradient formulation in the momentum equation (‘Korn’s inequality’) which particularly leads to convergence problems of the iterative solvers for small Reynolds (Re) numbers. Second, numerical instabilities for high Re numbers or whenever convective operators are dominant such that the standard Galerkin formulation fails and leads to spurious oscillations. We show that the right choice of edge-oriented stabilization is able to provide simultanously excellent results regarding robustness and accuracy for both seemingly different cases of problems, and we discuss the sensitivity of the involved parameters w.r.t. mesh distortions and variations of the Re number. Moreover, we explain how efficient multigrid solvers can be constructed to circumvent the problems with the arising ‘non-standard ’ FEM data structures, and we provide several examples for the numerical efficiency for realistic flow configurations with benchmarking character.