The effect of a minimum angle condition on the preconditioning of the pivot block arising from 2-level-splittings of Crouzeix-Raviart FE-spaces

The construction of efficient two- and multilevel preconditioners for linear systems arising from the finite element discretization of self-adjoint second order elliptic problems is known to be governed by robust hierarchical splittings of finite element spaces. In this study we consider such splittings of spaces related to nonconforming discretizations using Crouzeix-Raviart linear elements: We discuss the standard method based on differences and aggregates, a more general splitting and the first reduce method which is equivalent to a locally optimal splitting. All three splittings are shown to fit a general framework of differences and aggregates. Further, we show that the bounds for the spectral condition numbers related to the additive and multiplicative preconditioners of the coarse grid complement block of the hierarchical stiffness matrix for the three splittings can be significantly improved subject to a minimum angle condition.