A Short Theory of the Rayleigh–Ritz Method

Yserentant, Harry

doi:10.1515/cmam-2013-0013

Cited by 11 publications

(9 citation statements)

References 14 publications

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“…In that paper an error analysis of the Rayleigh-Ritz method without penalization or consistency errors is presented that shows how the error in an eigenvector (and eigenvalue) approximation can be bounded in terms of its best approximation in the ansatz space, in the same spirit as the more general results in [23]. We generalize the results of [43] in the sense that we allow penalization and consistency errors, i.e., we generalize the analysis of [43] to the abstract setting presented in section 4.1.…”

Section: Thus We Getmentioning

confidence: 76%

“…Remark 6.2. Even for the conforming case d λ (•, •) = 0, the bound in Theorem 6.3 differs from the one derived in [43]. In that paper stability (i.e., uniform boundedness) of the b h -orthogonal projection Q h in the energy norm is assumed.…”

Section: And Is Given Bymentioning

confidence: 86%

“…In this section we present an analysis of the errors in the eigenvector approximations resulting from the discrete problem (4.3). Our analysisis is based on the theory presented in [43]. In that paper an error analysis of the Rayleigh-Ritz method without penalization or consistency errors is presented that shows how the error in an eigenvector (and eigenvalue) approximation can be bounded in terms of its best approximation in the ansatz space, in the same spirit as the more general results in [23].…”

Section: Thus We Getmentioning

confidence: 99%

“…As shown in e.g. [23,11,43] the rate of convergence of the Rayleigh-Ritz method critically depends on whether the considered eigenvalue λ is well separated from the other eigenvalues or is part of a cluster of eigenvalues. To measure this, the quantity…”

Section: And Is Given Bymentioning

confidence: 99%

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Analysis of finite element methods for surface vector-Laplace eigenproblems

Reusken¹

2020

Preprint

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In this paper we study finite element discretizations of a surface vector-Laplace eigenproblem. We consider two known classes of finite element methods, namely one based on a vector analogon of the Dziuk-Elliott surface finite element method and one based on the so-called trace finite element technique. A key ingredient in both classes of methods is a penalization method that is used to enforce tangentiality of the vector field in a weak sense. This penalization and the perturbations that arise from numerical approximation of the surface lead to essential nonconformities in the discretization of the variational formulation of the vector-Laplace eigenproblem. We present a general abstract framework applicable to such nonconforming discretizations of eigenproblems. Error bounds both for eigenvalue and eigenvector approximations are derived that depend on certain consistency and approximability parameters. Sharpness of these bounds is discussed. Results of a numerical experiment illustrate certain convergence properties of such finite element discretizations of the surface vector-Laplace eigenproblem.

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Section: Thus We Getmentioning

confidence: 76%

Section: And Is Given Bymentioning

confidence: 86%

Section: Thus We Getmentioning

confidence: 99%

Section: And Is Given Bymentioning

confidence: 99%

See 2 more Smart Citations

Analysis of finite element methods for surface vector-Laplace eigenproblems

Reusken¹

2020

Preprint

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“…In every example, the Algorithm 1 is compared with the iterative Rayleigh–Ritz procedure . In the Rayleigh–Ritz procedure, m smallest eigenvectors are computed as well, and the size of the projected problem is the same as the size of the corresponding projected problem in the Algorithm 1, which is m + n .…”

Section: Numerical Examplesmentioning

confidence: 99%

Convergence theory of exact interpolation scheme for computing several eigenvectors

Pultarová

2015

Numerical Linear Algebra App

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An asymptotic convergence analysis of a new multilevel method for numerical solution of eigenvalues and eigenvectors of symmetric and positive definite matrices is performed. The analyzed method is a generalization of the original method that has recently been proposed by R. Kužel and P. Vaněk (DOI: 10.1002/nla.1975) and uses a standard multigrid prolongator matrix enriched by one full column vector, which approximates the first eigenvector. The new generalized eigensolver is designed to compute m > 1 eigenvectors. Their asymptotic convergence in terms of the generalized residuals is proved, and its convergence factor is estimated. The theoretical analysis is illustrated by numerical examples.

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Cut Finite Element Methods for Linear Elasticity Problems

Hansbo

Larson

Larsson

2017

Lecture Notes in Computational Science and Engineering

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We formulate a cut finite element method for linear elasticity based on higher order elements on a fixed background mesh. Key to the method is a stabilization term which provides control of the jumps in the derivatives of the finite element functions across faces in the vicinity of the boundary. We then develop the basic theoretical results including error estimates and estimates of the condition number of the mass and stiffness matrices. We apply the method to the standard displacement problem, the frequency response problem, and the eigenvalue problem. We present several numerical examples including studies of thin bending dominated structures relevant for engineering applications. Finally, we develop a cut finite element method for fibre reinforced materials where the fibres are modeled as a superposition of a truss and a Euler-Bernoulli beam. The beam model leads to a fourth order problem which we discretize using the restriction of the bulk finite element space to the fibre together with a continuous/discontinuous finite element formulation. Here the bulk material stabilizes the problem and it is not necessary to add additional stabilization terms. *

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A Short Theory of the Rayleigh–Ritz Method

Cited by 11 publications

References 14 publications

Analysis of finite element methods for surface vector-Laplace eigenproblems

Analysis of finite element methods for surface vector-Laplace eigenproblems

Convergence theory of exact interpolation scheme for computing several eigenvectors

Cut Finite Element Methods for Linear Elasticity Problems

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