A posteriori error analysis for nonconforming approximation of multiple eigenvalues

Boffi, Daniele; Durán, Ricardo G.; Gardini, Francesca; Gastaldi, Lucia

doi:10.1002/mma.3452

Cited by 14 publications

(11 citation statements)

References 17 publications

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“…is the solution of Problem (2.4) by the Morley element and λ M is the i-th eigenvalue, it follows from the theory of nonconforming eigenvalue approximations, see for instance, [2,3,5,16,40] and the references therein, that there exists u ∈ M(λ) with λ = λ i such that…”

Section: The Corresponding Canonical Interpolation Operator πmentioning

confidence: 99%

New Fourth Order Postprocessing Techniques for Plate Bending Eigenvalues by Morley Element

Ma¹,

Tian²

2021

Preprint

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In this paper, we propose and analyze the extrapolation methods and asymptotically exact a posterior error estimates for eigenvalues of the Morley element. We establish an asymptotic expansion of eigenvalues, and prove an optimal result for this expansion and the corresponding extrapolation method. We also design an asymptotically exact a posterior error estimate and propose new approximate eigenvalues with higher accuracy by utilizing this a posteriori error estimate. Finally, several numerical experiments are considered to confirm the theoretical results and compare the performance of the proposed methods.

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Section: The Corresponding Canonical Interpolation Operator πmentioning

confidence: 99%

New Fourth Order Postprocessing Techniques for Plate Bending Eigenvalues by Morley Element

Ma¹,

Tian²

2021

Preprint

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“…To capture the singularities, a popular way is to use adaptive scheme. Several studies have shown that an effective adaptive strategy for multiple eigenvalues should consider all involved discrete eigenfunctions [3,9,17], otherwise the singularities of the target eigenfunctions may be resolved in a wrong way. However, the main issue is that in general it is not known a priori the multiplicity of an eigenvalue of the continuous problem, which in turn brings difficulties to the design of efficient adaptive methods.…”

Section: Square Ringmentioning

confidence: 99%

Solving Biharmonic Eigenvalue Problem With Navier Boundary Condition Via Poisson Solvers On Non-Convex Domains

Zhang¹,

Li²,

Zhang³

2021

Preprint

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It is well known that the usual mixed method for solving the biharmonic eigenvalue problem by decomposing the operator into two Laplacians may generate spurious eigenvalues on non-convex domains. To overcome this difficulty, we adopt a recently developed mixed method, which decomposes the biharmonic equation into three Poisson equations and still recovers the original solution. Using this idea, we design an efficient biharmonic eigenvalue algorithm, which contains only Poisson solvers. With this approach, eigenfunctions can be confined in the correct space and thereby spurious modes in non-convex domains are avoided. A priori error estimates for both eigenvalues and eigenfunctions on quasi-uniform meshes are obtained; in particular, a convergence rate of O(h 2α ) (0 < α < π/ω, ω > π is the angle of the reentrant corner) is proved for the linear finite element. Surprisingly, numerical evidence demonstrates a O(h 2 ) convergent rate for the quasi-uniform mesh with the regular refinement strategy even on non-convex polygonal domains.

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“…This result, however, does not directly apply to nonconforming finite element methods. A generalisation for the Crouzeix-Raviart FEM for the eigenvalues of the Laplacian is given in (Boffi et al, 2014) where it is used that the nonconforming finite element space has an H 1 -conforming subspace. The Morley finite element does not satisfy a corresponding condition; this paper develops a new technique which allows the proof of eigenvalue error estimates of the form |λ j − λ ℓ,j | max{λ j , λ ℓ,j } ≤ C sin 2 a,NC (W, W ℓ ).…”

Section: Introductionmentioning

confidence: 99%

Morley finite element method for the eigenvalues of the biharmonic operator

Gallistl

2014

IMA J Numer Anal

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This paper studies the nonconforming Morley finite element approximation of the eigenvalues of the biharmonic operator. A new C 1 conforming companion operator leads to an L 2 error estimate for the Morley finite element method which directly compares the L 2 error with the error in the energy norm and, hence, can dispense with any additional regularity assumptions. Furthermore, the paper presents new eigenvalue error estimates for nonconforming finite elements that bound the error of (possibly multiple or clustered) eigenvalues by the approximation error of the computed invariant subspace. An application is the proof of optimal convergence rates for the adaptive Morley finite element method for eigenvalue clusters.

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A posteriori error analysis for nonconforming approximation of multiple eigenvalues

Cited by 14 publications

References 17 publications

New Fourth Order Postprocessing Techniques for Plate Bending Eigenvalues by Morley Element

New Fourth Order Postprocessing Techniques for Plate Bending Eigenvalues by Morley Element

Solving Biharmonic Eigenvalue Problem With Navier Boundary Condition Via Poisson Solvers On Non-Convex Domains

Morley finite element method for the eigenvalues of the biharmonic operator

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