The lower bound property of the Morley element eigenvalues

Yang, Yidu; Li, Hao; Bi, Hai

doi:10.1016/j.camwa.2016.06.004

Cited by 5 publications

(5 citation statements)

References 34 publications

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“…This section presents a conforming companion in 3D to apply the Babuška-Osborn convergence analysis [BO91] for the discrete eigenvalue problem (1.3) and the standard Morley eigenvalue problem (2.11). For the latter the paper [YLB16] for n ě 2 follows [Ran79] for n " 2 and utilizes the trace inequality for second order derivatives B α u{Bx α for |α| " 2 under the regularity assumption u P W 3,p pΩq for 4{3 ă p ď 2. Those terms arise in an integration by parts in the classical a priori error analysis of the Morley FEM.…”

Section: Convergence Rates In 3dmentioning

confidence: 99%

Direct guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian

Carstensen¹,

Puttkammer²

2021

Preprint

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An extra-stabilised Morley finite element method (FEM) directly computes guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplace Dirichlet eigenvalues. The smallness assumption mintλ h , λuh 4 max ď 15.0864 on the maximal mesh-size h max makes the computed k-th discrete eigenvalue λ h ď λ a lower eigenvalue bound for the k-th Dirichlet eigenvalue λ. This holds for multiple and clusters of eigenvalues and serves for the localisation of the bi-Laplacian Dirichlet eigenvalues in particular for coarse meshes. The analysis requires interpolation error estimates for the Morley FEM with explicit constants in any space dimension n ě 2, which are of independent interest. The convergence analysis in 3D follows the Babuška-Osborn theory and relies on a companion operator for the Morley finite element method. This is based on the Worsey-Farin 3D version of the Hsieh-Clough-Tocher macro element with a careful selection of center points in a further decomposition of each tetrahedron into 12 sub-tetrahedra. Numerical experiments in 2D support the optimal convergence rates of the extrastabilised Morley FEM and suggest an adaptive algorithm with optimal empirical convergence rates.

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Section: Convergence Rates In 3dmentioning

confidence: 99%

Direct guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian

Carstensen¹,

Puttkammer²

2021

Preprint

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“…When the test domain is the L‐shaped domain, the C 0 IPG adaptive method can give an upper bound of the exact first eigenvalue (see ) and when the degree of freedom equals to 1,588,387 the eigenvalue equals 6703.991 (see table 2 in ); the adaptive Morley element can give a lower bound of the exact first eigenvalue (see ) and when the degree of freedom is 324,886 the eigenvalue is equal to 6702.3 (see table 5 in ); using Scheme 4.1 in this paper we get the eigenvalue 6703.621 with the degree of freedom 194,564. By comparison we see that we can get satisfactory results by relative small degree of freedom by using our schemes.…”

Section: Numerical Experimentsmentioning

confidence: 81%

Local and parallel finite element method for solving the biharmonic eigenvalue problem of plate vibration

Zhao

Yang

2018

Numerical Methods Partial

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In this paper, we establish a new local and parallel finite element discrete scheme based on the shifted-inverse power method for solving the biharmonic eigenvalue problem of plate vibration. We prove the local error estimation of finite element solution for the biharmonic equation/eigenvalue problem and prove the error estimation of approximate solution obtained by the local and parallel scheme. When the diameters of three grids satisfy H 4 = (w 2 ) = (h), the approximate solutions obtained by our schemes can achieve the asymptotically optimal accuracy. The numerical experiments show that the computational schemes proposed in this paper are effective to solve the biharmonic eigenvalue problem of plate vibration. KEYWORDS biharmonic eigenvalue, finite element, local error estimate, local and parallel algorithms, shifted-inverse power method INTRODUCTIONAs one of the essential aspects of science and engineering computing today, the parallel computing has become a focal point of many scholars. In 2000, Xu and Zhou [1] propose some local and parallel finite element algorithms by combining the two-grid finite element discretization scheme with the local defect correction technique for elliptic boundary value problems. Later, this parallel-computing technique has been developed (see, e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). In 2013, Bi et al. [22] propose a local finite element discretization based on the shifted-inverse power method for the second-order elliptic eigenvalue problem. Based on the above studies, this paper discusses the local and parallel finite element discrete Numer Methods Partial Differential Eq. 2019;35:851-869. wileyonlinelibrary.com/journal/num

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“…Using the trace inequality [ 5 ] proves the above estimate under the case . Using the arguments in [ 5 ], we can obtain the above estimate under the case (also see [ 11 ]).…”

Section: Preliminarymentioning

confidence: 99%

“…In the following algorithms, we have to provide an initial shape regular triangulation and a parameter . Also, from [ 10 , 11 ] we know that replacing with and replacing f with in ( 5.1 ), we can obtain the error estimator of Algorithms 1 and 1M . By Lemma 4.1 we can deduce that replacing with and replacing f with in ( 5.1 ), we can obtain the error estimator of Algorithms 2 – 3 and Algorithms 2M – 3M .…”

Section: Adaptive Algorithmsmentioning

confidence: 99%

“…The Morley element has been employed to solve the biharmonic eigenvalue problem, including the vibration of a plate; and [ 9 ] studied its a priori error estimate. [ 10 , 11 ] studied a posteriori error estimate and the adaptive method, [ 12 ] adopted a new method dispensing with any additional regularity assumption to study the error estimates and adaptive algorithms. This paper further studies the adaptive Morley element method and has the following features: The adaptive finite element methods, which were first proposed by Babuska and Rheinboldt [ 13 ], have gained an extensive attention in academia.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive Morley element algorithms for the biharmonic eigenvalue problem

Yang

2018

J Inequal Appl

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This paper is devoted to the adaptive Morley element algorithms for a biharmonic eigenvalue problem in R n (n ≥ 2). We combine the Morley element method with the shifted-inverse iteration including Rayleigh quotient iteration and the inverse iteration with fixed shift to propose multigrid discretization schemes in an adaptive fashion. We establish an inequality on Rayleigh quotient and use it to prove the efficiency of the adaptive algorithms. Numerical experiments show that these algorithms are efficient and can get the optimal convergence rate. MSC: 65N25; 65N30; 65N15

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The lower bound property of the Morley element eigenvalues

Cited by 5 publications

References 34 publications

Direct guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian

Direct guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian

Local and parallel finite element method for solving the biharmonic eigenvalue problem of plate vibration

Adaptive Morley element algorithms for the biharmonic eigenvalue problem

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