On the multilevel preconditioning of Crouzeix–Raviart elliptic problems

We consider robust hierarchical splittings of finite element spaces related to non‐conforming discretizations using Crouzeix–Raviart type elements. As is well known, this is the key to the construction of efficient two‐ and multilevel preconditioners. The main contribution of this paper is a theoretical and an experimental comparison of three such splittings. Our starting point is the standard method based on differences and aggregates (DA) as introduced in Blaheta et al. ( Numer. Linear Algebra Appl. 2004; 11 :309–326). On this basis we propose a more general (GDA) splitting, which can be viewed as the solution of a constraint optimization problem (based on certain symmetry assumptions). We further consider the locally optimal (ODA) splitting, which is shown to be equivalent to the first reduce (FR) method from Blaheta et al. ( Numer. Linear Algebra Appl. 2004; 11 :309–326). This means that both, the ODA and the FR splitting, generate the same subspaces, and thus the local constant in the strengthened Cauchy–Bunyakowski–Schwarz inequality is minimal for the FR (respectively ODA) splitting. Moreover, since the DA splitting corresponds to a particular choice in the parameter space of the GDA splitting, which itself is an element in the set of all splittings for which the ODA (or equivalently FR) splitting yields the optimum, we conclude that the chain of inequalities γ ⩽γ ⩽γ ⩽3/4 holds independently of mesh and/or coefficient anisotropy. Apart from the theoretical considerations, the presented numerical results provide a basis for a comparison of these three approaches from a practical point of view.