The Superconvergent Cluster Recovery Method

Huang, Yunqing; Yi, Nianyu

doi:10.1007/s10915-010-9379-9

Cited by 41 publications

(25 citation statements)

References 34 publications

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“…,Ω , where (λ h , u h ) is the corresponding approximation to an eigenpair (λ, u). The first term on the right side of the above identity can be approximated with high accuracy by gradient recovery techniques, such as polynomial preserving recovery techniques(PPR for short hereinafter) in [40], Zienkiewicz-Zhu superconvergence patch recovery techniques(SPR for short hereinafter) in [41] and the superconvergent cluster recovery method in [20]. Since the second term is of higher order, new approximate eigenvalues with higher accuracy can be obtained by the gradient recovery techniques.…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods

2014

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In this paper, three high-accuracy methods for eigenvalues of second order elliptic operators are proposed by using the nonconforming Crouzeix-Raviart(CR for short hereinafter) element and the nonconforming enriched Crouzeix-Raviart(ECR for short hereinafter) element. They are based on a crucial full one order superconvergence of the first order mixed Raviart-Thomas(RT for short hereinafter) element. The main ingredient of such a superconvergence analysis is to employ a discrete Helmholtz decomposition of the difference between the canonical interpolation and the finite element solution of the RT element. In particular, it allows for some vital cancellation between terms in one key sum of boundary terms. Consequently, a full one order superconvergence follows from a special relation between the CR element and the RT element, and the equivalence between the ECR element and the RT element for these two nonconforming elements. These superconvergence results improve those in literature from a half order to a full one order for the RT element, the CR element and the ECR element. Based on the aforementioned superconvergence of the RT element, asymptotic expansions of eigenvalues are established and employed to achieve high accuracy extrapolation methods for these two nonconforming elements. In contrast to a classic analysis in literature, the novelty herein is to use not only the canonical interpolations of these nonconforming elements but also that of the RT element to analyze such asymptotic expansions. Based on the superconvergence of these nonconforming elements, asymptotically exact a posteriori error estimators of eigenvalues are constructed and analyzed for them. Finally, two post-processing methods are proposed to improve accuracy of approximate eigenvalues by employing these a posteriori error estimators. Numerical tests are provided to justify and compare the performance of the aforementioned methods.

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Section: Introductionmentioning

confidence: 99%

Lower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods

2014

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“…Introduction. Gradient recovery [7,10,12,17,[20][21][22][23][24][25] is an effective and widely used post-processing technique in scientific and engineering computation. The main purpose of this techniques is to reconstruct a better numerical gradient from a finite element solution.…”

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confidence: 99%

Polynomial preserving recovery on boundary

Guo

Zhang

Zhao

et al. 2016

Journal of Computational and Applied Mathematics

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“…The iterative post-processing procedures are carried out until this estimator reaches a predefined tolerance. Some adaptive procedures based on finite element recovery techniques were utilized with different types of equations [14][15][16]. The adaptive technique begins by solving the considered problem on an initial mesh and a chosen gradient recovery technique is applied.…”

Section: Adaptive Finite Element Methodsmentioning

confidence: 99%

Weighted Technique for Finite Element Gradient Recovery at Boundary

Kashwaa¹,

Elsaid²,

El-Agamy³

2020

Matrix sci. math.

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In this paper, an improved technique is presented to recover the finite element gradient at boundaries. The proposed technique begins by evaluating the recovered gradient at the interior nodes using polynomial preserving recovery technique. Then we propose formula for weights to the recovered gradient at the interior nodes attached to boundary nodes. The sum of these weighted recovered gradients is utilized as an approximation for the gradient at the attached boundary node. The validity of the proposed technique is illustrated by some two-dimensional numerical examples.

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The Superconvergent Cluster Recovery Method

Cited by 41 publications

References 34 publications

Lower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods

Lower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods

Polynomial preserving recovery on boundary

Weighted Technique for Finite Element Gradient Recovery at Boundary

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