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 (Algebraic multilevel preconditioning methods, I. Numer. Math., 56 (1989)). The derivation of optimal convergence rate estimates in this context is mainly governed by the constant $\gamma\in (0,1)$ 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 $\gamma$ 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 $\gamma$ in the strengthened CBS inequality for anisotropic problems using the aforementioned finite element type was given by Blaheta, Margenov and Neytcheva (Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non-conforming FEM systems. NLAA 11 (2004)). In this study we show that the estimates of the constant $\gamma$ 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.