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Preconditioners based on various multilevel extensions of two-level finite element methods (FEM) lead to iterative methods which have an optimal order computational complexity with respect to the size of the system. Such methods were first presented in Axelsson and Padiy (SIAM. J. Sci. Stat. Comp. 1990; 20:1807) and Axelsson and Vassilevski (Numer. Math. 1989; 56:157), and are based on (recursive) two-level splittings of the finite element space. The key role in the derivation of optimal convergence rate estimates is played by the constant γ in the so-called Cauchy–Bunyakowski–Schwarz (CBS) inequality, associated with the angle between the two subspaces of the splitting. It turns out that only existence of uniform estimates for this constant is not enough but accurate quantitative bounds for γ have to be found as well. More precisely, the value of the upper bound for γ∈(0,1) is part of the construction of various multilevel extensions of the related two-level methods. In this paper, an algebraic two-level preconditioning algorithm for second-order elliptic boundary value problems is constructed, where the discretization is done using Crouzeix–Raviart non-conforming linear finite elements on triangles. An important point to make is that in this case the finite element spaces corresponding to two successive levels of mesh refinements are not nested. To handle this, a proper two-level basis is considered, which enables us to fit the general framework for the construction of two-level preconditioners for conforming finite elements and to generalize the method to the multilevel case. The major contribution of this paper is the derived estimates of the related constant γ in the strengthened CBS inequality. These estimates are uniform with respect to both coefficient and mesh anisotropy. To our knowledge, the results presented in the paper are the first such estimates for non-conforming FEM systems. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY The choice of the preconditioner is a key factor to accelerate the convergence of eigensolvers for large-size sparse eigenproblems. Although incomplete factorizations with partial fill-in prove generally effective in sequential computations, the efficient preconditioning of parallel eigensolvers is still an open issue. The present paper describes the use of block factorized sparse approximate inverse (BFSAI) preconditioning for the parallel solution of large-size symmetric positive definite eigenproblems with both a simultaneous Rayleigh quotient minimization and the Jacobi-Davidson algorithm. BFSAI coupled with a block diagonal incomplete decomposition proves a robust and efficient parallel preconditioner in a number of test cases arising from the finite element discretization of 3D fluid-dynamical and mechanical engineering applications, outperforming FSAI even by a factor of 8 and exhibiting a satisfactory scalability. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The standard multigrid algorithm is widely known to yield optimal convergence whenever all high-frequency error components correspond to large relative eigenvalues. This property guarantees that smoothers like Gauss–Seidel and Jacobi will significantly dampen all the high-frequency error components, and thus, produce a smooth error. This has been established for matrices generated from standard discretizations of most elliptic equations. In this paper, we address a system of equations that is generated from a perturbation of the non-elliptic operator I-grad div by a negative ℇ Δ. For ℇnear to one, this operator is elliptic, but as ℇapproaches zero, the operator becomes non-elliptic as it is dominated by its non-elliptic part. Previous research on the non-elliptic part has revealed that discretizing I-grad div with the proper finite element space allows one to define a robust geometric multigrid algorithm. The robustness of the multigrid algorithm depends on a relaxation operator that yields a smooth error. We use this research to assist in developing a robust discretization and solution method for the perturbed problem. To this end, we introduce a new finite element space for tensor product meshes that is used in the discretization, and a relaxation operator that succeeds in dampening all high-frequency error components. The success of the corresponding multigrid algorithm is first demonstrated by numerical results that quantitatively imply convergence for any ℇis bounded by the convergence for ℇequal to zero. Then we prove that convergence of this multigrid algorithm for the case of ℇ equal to zero is independent of mesh size. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]