Superconvergence and a posteriori error estimates for the Stokes eigenvalue problems

Liu, Huipo; Gong, Wei; Wang, Shuanghu; Yan, Ningning

doi:10.1007/s10543-013-0422-8

Cited by 13 publications

(13 citation statements)

References 34 publications

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“…For this experiment, we consider λ 2 = 48.9844 as the reference value to the fourth eigenvalue. It is known that the dual problem has H ι+1 regularity where 0 < ι < 1 [4]. The convergence results obtained for the two-field problem are shown in Figure 7, where the reference values are given in Table 9 and 10 using P 1 and P 2 elements, respectively.…”

Section: Test 2: L-shaped Domainmentioning

confidence: 99%

“…It is well known that the inf-sup condition holds for the continuous problem (6), and the corresponding solution operator is compact. From the spectral theory ( [8]) it follows that (6) has a sequence of real eigenvalues (see also [2,4,15]) which are assumed to satisfy…”

Section: The Two-field Stokes Eigenproblemmentioning

confidence: 99%

“…A sample discretization of the problem domain using N = 5 is illustrated in Figure 1. As we have already mentioned, the exact solution is unknown, and we take λ 1 = 52.3447 as a reference to the minimum eigenvalue (see [2,4,20]). Figures 2 and 3 present the convergence of the minimum eigenvalue approximations to the reference value λ 1 for the two-field and three-field problems, re- spectively.…”

Section: Test 1: Square Domainmentioning

confidence: 99%

“…Researchers might want to avoid the use of schemes satisfying this condition. This demand has led to many recent studies devoted to develop robust and efficient stabilized techniques for approximating the Stokes eigenvalue problem [2,3,4,5,6]. It is worth noting that there are also alternative approaches.…”

Section: Introductionmentioning

confidence: 99%

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A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems

Türk

Boffi

Codina

2016

Computer Methods in Applied Mechanics and Engineering

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In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement–pressure) and the three-field (stress–displacement–pressure) formulations. The method presented is based on a subgrid scale concept, and depends on the approximation of the unresolvable scales of the continuous solution. In general, subgrid scale techniques consist in the addition of a residual based term to the basic Galerkin formulation. The application of a standard residual based stabilization method to a linear eigenvalue problem leads to a quadratic eigenvalue problem in discrete form which is physically inconvenient. As a distinguished feature of the present study, we take the space of the unresolved subscales orthogonal to the finite element space, which promises a remedy to the above mentioned complication. In essence, we put forward that only if the orthogonal projection is used, the residual is simplified and the use of term by term stabilization is allowed. Thus, we do not need to put the whole residual in the formulation, and the linear eigenproblem form is recovered properly. We prove that the method applied is convergent, and present the error estimates for the eigenvalues and the eigenfunctions. We report several numerical tests in order to illustrate that the theoretical results are validated.Peer ReviewedPostprint (author's final draft

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Section: Test 2: L-shaped Domainmentioning

confidence: 99%

Section: The Two-field Stokes Eigenproblemmentioning

confidence: 99%

Section: Test 1: Square Domainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems

Türk

Boffi

Codina

2016

Computer Methods in Applied Mechanics and Engineering

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“…Using the estimates (16) and (19), and Theorem 4.1, we obtain for h small enough |λ − λ * h | h 2k+4 .…”

Section: Local Post-processingmentioning

confidence: 91%

Arnold--Winther Mixed Finite Elements for Stokes Eigenvalue Problems

Gedicke¹,

Khan²

2018

SIAM J. Sci. Comput.

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This paper is devoted to study the Arnold-Winther mixed finite element method for two dimensional Stokes eigenvalue problems using the stress-velocity formulation. A priori error estimates for the eigenvalue and eigenfunction errors are presented. To improve the approximation for both eigenvalues and eigenfunctions, we propose a local post-processing. With the help of the local post-processing, we derive a reliable a posteriori error estimator which is shown to be empirically efficient. We confirm numerically the proven higher order convergence of the post-processed eigenvalues for convex domains with smooth eigenfunctions. On adaptively refined meshes we obtain numerically optimal higher orders of convergence of the post-processed eigenvalues even on nonconvex domains.

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Divergence-conforming discontinuous Galerkin finite elements for Stokes eigenvalue problems

Gedicke

Khan

2019

Numer. Math.

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In this paper, we present a divergence-conforming discontinuous Galerkin finite element method for Stokes eigenvalue problems. We prove a priori error estimates for the eigenvalue and eigenfunction errors and present a robust residual based a posteriori error estimator. The a posteriori error estimator is proven to be reliable and (locally) efficient in a meshdependent velocity-pressure norm. We finally present some numerical examples that verify the a priori convergence rates and the reliability and efficiency of the residual based a posteriori error estimator.

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Superconvergence and a posteriori error estimates for the Stokes eigenvalue problems

Cited by 13 publications

References 34 publications

A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems

A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems

Arnold--Winther Mixed Finite Elements for Stokes Eigenvalue Problems

Divergence-conforming discontinuous Galerkin finite elements for Stokes eigenvalue problems

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