Generalized Aggregation-Based Multilevel Preconditioning of Crouzeix-Raviart FEM Elliptic Problems

Abstract

It is well-known that iterative methods of optimal order complexity with respect to the size of the system can be set up by utilizing preconditioners based on various multilevel extensions of two-level finite element methods (FEM), as was first shown in [5]. Thereby, the constant γ 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, plays a key role in the derivation of optimal convergence rate estimates. In this paper a generalization of an algebraic preconditioning algorithm for second-order elliptic boundary value problems is presented, where the domain is discretized using linear Crouzeix-Raviart finite elements and the two-level splitting is defined by differentiation and aggregation (DA). It is shown that the uniform estimate on the constant γ (as presented in [6]) can be improved if a minimum angle condition, which is an integral part in any mesh generator, is assumed to hold in the triangulation. The improved values of γ can then be exploited in the set up of more problem-adapted multilevel preconditioners with faster convergence rates.