Generalized aggregation-based multilevel preconditioning of Crouzeix-Raviart FEM elliptic problems

Preconditioners based on various multilevel extensions of two-level finite element methods (FEM) are well-known to yield iterative methods of optimal order complexity with respect to the size of the system, as was first shown by Axelsson and Vassilevski [4]. The derivation of optimal convergence rate estimates in this context is mainly governed by the constant γ ∈ (0, 1) in the so-called Cauchy-Bunyakowski-Schwarz (CBS) inequality, which is associated with the angle between the two subspaces obtained from a (recursive) two-level splitting of the finite element space. Accurate quantitative bounds, especially for the upper bound of γ are required for the construction of proper multilevel extensions of the related two-level methods. In this paper, an improved algebraic preconditioning algorithm for second-order elliptic boundary value problems is presented, where the discretization and formulation is based on Crouzeix-Raviart linear finite elements and the two-level splitting is based on differentiation and aggregation (DA). A uniform estimate of γ in the strengthened CBS inequality for anisotropic problems using the aforementioned finite element type was given by Blaheta, Margenov and Neytcheva [5]. In this study we show that the estimates of the constant γ can significantly be improved for the DA-algorithm by utilizing a minimum angle condition, where the latter is naturally used in mesh generators for practical problems. The results obtained herein can be used to set up a self-adaptive aggregation based mulitlevel preconditioner for elliptic problems based on Crouzeix-Raviart finite elements