Eigenvalue Approximation by Mixed and Hybrid Methods

Mercier, Bertrand; Osborn, John E.; Rappaz, Jacques; Raviart, Pierre Arnaud

doi:10.2307/2007651

Cited by 73 publications

(80 citation statements)

References 35 publications

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“…For H 1 = (ν^2(Ω)) 2 a similar statement had been made by Ladyzhenskaya and was proved for different classes of domains in a number of papers (see references in [6,18,23]). The considered space Η 1 contains the space Hj of functions u^ that vanish on all r f and R i9 and coincide with (\&\(Ω')), where Ω' may be assumed connected.…”

Section: Proofsupporting

confidence: 68%

“…The convergence of such projection methods for α = 0 is studied in a number of papers (see, for example, [2,19,20,23]), where it is shown, in particular, that each A,· from the problems (2.1) is the limit, as h-*Q, of m i eigenvalues 2^, r = l,...,m, of the problem (2.10), and for sufficiently small ft's the estimates 1 (2.11) hold, where = sup p(u;fi) §i is the direct sum of S(#°), Θ Η the 8 a P between S(Aj) and $ (recall that Θ Η = \\ P t -P t \\ , where P i9 Pi are orthoprojectors in Η onto S(A,·) and £ £ , respectively, and 0 H coincides with sup p(u; $) when dim S(A £ ) = dim ^ (see [23]). …”

Section: L~* =mentioning

confidence: 99%

“…Other possible choices of ft are also indicated there (see also [23] approximations of identity operators E l ,E 2 inH l ,H 2 . The condition (2.3'), as we show in [6], leads to the equivalence with respect to spectrum of L 2 and E 2 , and also of -L 2 , LiiL l2 and L 22 , where -L 2 =-aL 22 …”

Section: Proofmentioning

confidence: 99%

“…Therefore, more traditional work with quadrature formulae, curvilinear triangles in a triangulation of Ω, piecewise polynomial and singular basis functions, etc. can be considered by analogy to investigations carried out in [4,6,8,10,13,16,23], where one can find a more detailed bibliography.…”

Section: General Remarksmentioning

confidence: 99%

“…Other methods for reducing order were studied in a large number of papers, e.g. [4,16,23], but no essential progress was achieved in the minimization of W(s). Among related results on the investigation of projection methods and FEM one should distinguish [2,19,20,23].…”

mentioning

confidence: 99%

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Effective methods for solving eigenvalue problems with fourth-order elliptic operators

D'yakonov¹

1986

Russian Journal of Numerical Analysis and Mathematical Modelling

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A generalized eigenvalue problem with an elliptic fourth-order operator in a domain with a piecewise-smooth boundary is approximated by a finite element scheme. An efficient iterative method is suggested for evaluating the smallest eigenvalues of the constructed generalized algebraic eigenvalue problem, and an estimate is found for the complexity of the iterative method. The results are illustrated by using as examples three main types of problems arising in the theory of elastic plates. This paper is connected with the author's previous publications [6][7][8][9][10][11][12][13][14] and is devoted to effective numerical procedures of finding the smallest eigenvalue λ ί of eigenvalue problems with linear elliptic fourth-order operators in a bounded domain Ω on the plane with a piecewise-smooth boundary Γ. Such problems can be written in the operator form Lw = AMw, 0<λ ί <λ 2 <·-(0.1) with bounded symmetric linear operators L and M in Hubert space H, where L and M are self-adjoint and positive definite; L~1M is a compact operator. The Hubert space//can be of the form//! χ H 2 χ ··· χ // p ,where// r isasubspaceof the Sobolev space W\(£l) if 1 < r ^ k^ ^ p, and a subspace of W\(ty ifk 1

k( see [3,4,24]). The elements of H are denoted by w = {w { ,..., w p }. Let the eigenspace corresponding to λ ί consist of functions w such that Wr eW m(r) + y (Q), y>0 , l^r^p (0.2) where w r is the rth component of any eigenfunction w corresponding to m(r) = 2 if I k 1 . The main result of the paper is the construction, under proper conditions on the types of the boundary value problems, of some variants of the finite element method (FEM) and iterative methods (IM) for finding the smallest eigenvalues of the arising algebraic problems such that an approximation of λ± with 0(e 2 )-accuracy may be obtained at the computational cost of W(s) arithmetic operations, where W(s) = 0(s-2/ ?\lns\ l \ 1=12. (0.3)A generalization is possible if the condition Μ > 0 is not satisfied and λ 1 is either the smallest positive or the largest negative eigenvalue. For such problems, the estimates (0.3) are still valid. The admissible boundary conditions can be of a rather general nature. If, for example, either the equation Δ 2 νν> = λ\ν or Δ 2 νν = -ΑΔνν is considered, then the boundary conditions can be any of three main types known in the theory of elastic plates that correspond to the cases of clamped, simply supported, and free plates. Brought to you by | University of Arizona Authenticated Download Date | 7/5/15 3:09 PM Brought to you by | University of Arizona Authenticated Download Date | 7/5/15 3:09 PM

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Section: Proofsupporting

confidence: 68%

Section: L~* =mentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Section: General Remarksmentioning

confidence: 99%

mentioning

confidence: 99%

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Effective methods for solving eigenvalue problems with fourth-order elliptic operators

D'yakonov¹

1986

Russian Journal of Numerical Analysis and Mathematical Modelling

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Parallel iterative methods for Navier–Stokes equations and application to eigenvalue computation

Graham

Spence

Vainikko

2003

Concurrency and Computation

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SUMMARYWe describe the construction of parallel iterative solvers for finite-element approximations of the Navier-Stokes equations on unstructured grids using domain decomposition methods. The iterative method used is FGMRES, preconditioned by a parallel adaptation of a block preconditioner recently proposed by Kay et al. The parallelization is achieved by adapting the technology of our domain decomposition solver DOUG (previously used for scalar problems) to block-systems. The iterative solver is applied to shifted linear systems that arise in eigenvalue calculations. To illustrate the performance of the solver, we compare several strategies both theoretically and practically for the calculation of the eigenvalues of large sparse non-symmetric matrices arising in the assessment of the stability of flow past a cylinder.

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A multilevel correction method for Stokes eigenvalue problems and its applications

Lin

Luo

Xie

2013

Math Methods in App Sciences

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In this paper, a new multilevel correction scheme is proposed to solve Stokes eigenvalue problems by the finite element method. This new scheme contains a series of correction steps, and the accuracy of eigenpair approximation can be improved after each step. In each correction step, we only need to solve a Stokes problem on the corresponding fine finite element space and a Stokes eigenvalue problem on the coarsest finite element space. This correction scheme can improve the efficiency of solving Stokes eigenvalue problems by the finite element method. As applications of this multilevel correction method, a multigrid method and an adaptive finite element technique are introduced for Stokes eigenvalue problems. Some numerical results are given to validate our schemes.

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Eigenvalue Approximation by Mixed and Hybrid Methods

Cited by 73 publications

References 35 publications

Effective methods for solving eigenvalue problems with fourth-order elliptic operators

Effective methods for solving eigenvalue problems with fourth-order elliptic operators

Parallel iterative methods for Navier–Stokes equations and application to eigenvalue computation

A multilevel correction method for Stokes eigenvalue problems and its applications

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