Enhancing eigenvalue approximation by gradient recovery on adaptive meshes

Wu, Haijun; Zhang, Zhimin

doi:10.1093/imanum/drn050

Cited by 22 publications

(16 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another one is a mesh density 54 H. Wu condition. The two conditions are verified numerically by real-life adaptive meshes (see, e.g., [25,26]). The results in [25] have been applied to enhance the eigenvalue approximations by the finite element method (see [26]).…”

Section: Introductionmentioning

confidence: 85%

“…In this paper, we extend the results in [25] and [26] on PPR to the case of the modified polynomial preserving recovery (MPPR). We first consider the application of recovered gradient by MPPR to adaptive finite element methods for elliptic problems.…”

Section: Introductionmentioning

confidence: 87%

“…Gradient recovery has been widely used for a posteriori error estimates (see, e.g., [1,4,9,12,20,21,25,26,32,[34][35][36][37]). Comparing with the a posteriori error estimates of residual type (see, e.g., [1,2,7,10,16,17,22,24]), the a posteriori error estimates based on gradient recovery have the advantages of problem-independence and asymptotic exactness.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Modified Polynomial Preserving Recovery and Its Applications to a Posteriori Error Estimates

Wu¹

2010

NMTMA

View full text Add to dashboard Cite

A modified polynomial preserving gradient recovery technique is proposed. Unlike the polynomial preserving gradient recovery technique, the gradient recovered with the modified polynomial preserving recovery (MPPR) is constructed element-wise, and it is discontinuous across the interior edges. One advantage of the MPPR technique is that the implementation is easier when adaptive meshes are involved. Superconvergence results of the gradient recovered with MPPR are proved for finite element methods for elliptic boundary problems and eigenvalue problems under adaptive meshes. The MPPR is applied to adaptive finite element methods to construct asymptotic exact a posteriori error estimates. Numerical tests are provided to examine the theoretical results and the effectiveness of the adaptive finite element algorithms.

show abstract

Section: Introductionmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Modified Polynomial Preserving Recovery and Its Applications to a Posteriori Error Estimates

Wu¹

2010

NMTMA

View full text Add to dashboard Cite

show abstract

“…In [34], Naga, Zhang, and Zhou used Polynomial Preserving Recovery to enhance eigenvalue approximation. In [40], Wu and Zhang further showed polynomial preserving recovery can even enhance eigenvalue approximation on adaptive meshes. The idea was further studied in [31,15].…”

mentioning

confidence: 99%

“…Adaptive finite element method(AFEM) is a fundamental tool to overcome such difficulties. In the context of adaptive finite element method for elliptic eigenvalue problems, residual type a posteriori error estimators are analyzed in [14,20,28,39] and recovery type a posteriori error estimators are investigated by [29,40,27]. For all adaptive methods mentioned above, an algebraic eigenvalue problem has to be solved during every iteration, which is very time consuming.…”

mentioning

confidence: 99%

Superconvergent Two-Grid Methods for Elliptic Eigenvalue Problems

2016

View full text Add to dashboard Cite

Abstract. Some numerical algorithms for elliptic eigenvalue problems are proposed, analyzed, and numerically tested. The methods combine advantages of the two-grid algorithm [J. Xu and A. Zhou, Math. Comp, 70(2001) To reduce the computational cost of eigenvalue problems, Xu and Zhou introduced a two-grid discretization scheme [42]. Later on, similar ideas were applied to non self-adjoint eigenvalue problems [22] and semilinear elliptic eigenvalue problems [11]. Furthermore, it also has been generalized to three-scale discretization [16] and multilevel discretization [24]. Recently, a new shifted-inverse power method based two-grid scheme was proposed in [21,43].To improve accuracy of eigenvalue approximation, many methods have been proposed. In [37], Shen and Zhou introduced a defect correction scheme based on averaging recovery, like a global L 2 projection and a Clément-type operator. In [34], Naga, Zhang, and Zhou used Polynomial Preserving Recovery to enhance eigenvalue approximation. In [40], Wu and Zhang further showed polynomial preserving recovery can even enhance eigenvalue approximation on adaptive meshes. The idea was further studied in [31,15]. Alternatively, Racheva and Andreev proposed a two-space method to achieve better eigenvalue approximation [36] and it was also applied to biharmonic eigenvalue problem [1].In this paper, we propose some fast and efficient solvers for elliptic eigenvalue problems. We combine ideas of the two-grid method, two-space method, shiftedinverse power method, and PPR recovery enhancement to design our new algorithms. The first purpose is to introduce two superconvergent two-grid methods for eigenvalue problems. Our first algorithm is a combination of the shifted-inverse power based two-

show abstract

A Story on Adaptive Finite Element Computations for Elliptic Eigenvalue Problems

Międlar

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

View full text Add to dashboard Cite

Enhancing eigenvalue approximation by gradient recovery on adaptive meshes

Abstract: We utilize the recovered gradient by the polynomial-preserving recovery to enhance the eigenvalue approximation of the Laplace operator under adaptive meshes. Superconvergence rate is established and numerical tests on benchmark problems support our theoretical findings.

Cited by 22 publications

References 30 publications

A Modified Polynomial Preserving Recovery and Its Applications to a Posteriori Error Estimates

A Modified Polynomial Preserving Recovery and Its Applications to a Posteriori Error Estimates

Superconvergent Two-Grid Methods for Elliptic Eigenvalue Problems

A Story on Adaptive Finite Element Computations for Elliptic Eigenvalue Problems

Contact Info

Product

Resources

About