The contraction number of a class of two-level methods; an exact evaluation for some finite element subspaces and model problems

Maitre, Jean‐François; Musy, François

doi:10.1007/bfb0069942

Cited by 66 publications

(61 citation statements)

References 1 publication

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“…constants γ for linear finite elements are presented e.g. in [2], [5]. For two levels of hierarchy, the constant γ is 1 2 in the case of the isotropic Laplacian and rectangular finite elements, …”

Section: Comparison To Linear Elementsmentioning

confidence: 99%

The strengthened C.B.S. inequality constant for second order elliptic partial differential operator and for hierarchical bilinear finite element functions

Pultarová

2005

Appl Math

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Section: Comparison To Linear Elementsmentioning

confidence: 99%

The strengthened C.B.S. inequality constant for second order elliptic partial differential operator and for hierarchical bilinear finite element functions

Pultarová

2005

Appl Math

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“…We see from this definition that this new preconditioner generalizes the method from Vassilevski [15] (v -X,pv = X -t) and that from Axelsson and Vassilevski [3] (ko = X), and can be considered as an alternative algebraic generalization of the classical two-level methods studied in Bank and Dupont [5], Braess [7,8], Axelsson [1], Axelsson and Gustafsson [2], and Maitre and Musy [13]. A special variant of the present method for ko = 2 was considered in Vassilevski [16], which generalizes an earlier result of Kuznetsov [12].…”

Section: Introductionmentioning

confidence: 94%

Hybrid $V$-cycle algebraic multilevel preconditioners

Vassilevski¹

1992

Math. Comp.

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Abstract. We consider an algebraic derivation of multilevel preconditioners which are based on a sequence of finite element stiffness matrices. They correspond to a sequence of triangulations obtained by successive refinement and the associated finite element discretizations of second-order selfadjoint elliptic boundary value problems. The stiffness matrix at a given discretization level is partitioned into a natural hierarchical two-level two-by-two block form. Then it is factored into block triangular factors. The resulting Schur complement is then replaced (approximated) by the stiffness matrix on the preceding (coarser) level. This process is repeated successively for a fixed number ko > 1 of steps. After each /cq steps, the preconditioner so derived is corrected by a certain polynomial approximation, a properly scaled and shifted Chebyshev matrix polynomial which involves the preconditioner and the stiffness matrix at the considered level. The hybrid F-cycle preconditioner thus derived is shown to be of optimal order of complexity for 2-D and 3-D problem domains. The relative condition number of the preconditioner is bounded uniformly with respect to the number of levels and with respect to possible jumps of the coefficients of the considered elliptic bilinear form as long as they occur only across edges (faces in 3-D) of elements from the coarsest triangulation. In addition, an adaptive implementation of our hybrid K-cycle preconditioners is proposed, and its practical behavior is demonstrated on a number of test problems.

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“…At any rate, one cannot disentangle this dependence upon u from ∇ε in (5.23). Maitre and Musy [25] have estimated β 2 solely in terms of the triangulation. In fact, they show that…”

Section: Theorem 59 For V ∈ Hmentioning

confidence: 99%

On estimators for eigenvalue/eigenvector approximations

Grubišić

Ovall

2008

Math. Comp.

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Abstract. We consider a large class of residuum based a posteriori eigenvalue/eigenvector estimates and present an abstract framework for proving their asymptotic exactness. Equivalence of the estimator and the error is also established. To demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for Laplace eigenvalue problems in planar polygonal regions. To this end we develop new error analysis for the Galerkin approximation which avoids the use of the strengthened CauchySchwarz inequality and the saturation assumption, and gives reasonable and explicitly computable upper bounds on the discretization error. A brief discussion is also given concerning the design of estimators which are in the same spirit, but are based on different a posteriori techniques-notably, those of gradient recovery type.

show abstract

The contraction number of a class of two-level methods; an exact evaluation for some finite element subspaces and model problems

Cited by 66 publications

References 1 publication

The strengthened C.B.S. inequality constant for second order elliptic partial differential operator and for hierarchical bilinear finite element functions

The strengthened C.B.S. inequality constant for second order elliptic partial differential operator and for hierarchical bilinear finite element functions

Hybrid $V$-cycle algebraic multilevel preconditioners

On estimators for eigenvalue/eigenvector approximations

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