Recovery-based error estimator for stabilized finite element methods for the Stokes equation

Song, Lina; Hou, Yanren; Cai, Zhiqiang

doi:10.1016/j.cma.2014.01.004

Cited by 15 publications

(16 citation statements)

References 23 publications

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“…We can repeat this refinement process until the mesh meets a termination criterion, for example, a given maximum number of refinements, or the tolerance of global estimator such that < . For more details on the explanation and numerical property of this error estimator for Euclidean domains, see the works of Ainsworth et al 38 and Song et al 42 The above refinement process is based on comparing the continuous and discontinuous gradients as well as the local and global information. Therefore, the developed method works well for problems with large variations of the gradient of the solution.…”

Section: A Gradient Recovery-based Error Estimatormentioning

confidence: 99%

“…Compared with the residual-based estimators, the gradient recovery-based estimator has advantage in capturing large variations in the gradient of the solution [38][39][40][41] and has been successfully applied to solving incompressible flow problems. 42,43 In the work of Wei et al, 44 the authors show the effectiveness of the super-convergence patch gradient recovery technique for surface diffusion PDEs, which inspires us to use the gradient recovery-based estimator in the AMR process. Moreover, the refinement technique employed is based on a local Delaunay algorithm on the tangent plane of the surface.…”

Section: Introductionmentioning

confidence: 98%

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A gradient recovery–based adaptive finite element method for convection‐diffusion‐reaction equations on surfaces

Xiao

Feng

2019

Numerical Meth Engineering

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In this paper, we present an adaptive mesh refinement method for solving convection-diffusion-reaction equations on surfaces, which is a fundamental subproblem in many models for simulating the transport of substances on biological films and solid surfaces. The method considered is a combination of well-known techniques: the surface finite element method, streamline diffusion stabilization, and the gradient recovery-based Zienkiewicz-Zhu error estimator. The streamline diffusion method overcomes the instability issue of the finite element method for the dominance of the convection.The gradient recovery-based adaptive mesh refinement strategy enables the method to provide high-resolution numerical solutions by relatively fewer degrees of freedom. Moreover, the implementation detail of a surface mesh refinement technique is presented. Various numerical examples, including the convection-dominated diffusion problems with large variations of solutions, nearly singular solutions, discontinuous sources, and internal layers on surfaces, are presented to demonstrate the efficacy and accuracy of the proposed method. KEYWORDSadaptive strategy, recovery-based error estimator, streamline diffusion method, surface convection-diffusion-reaction equations, surface finite element method MSC CLASSIFICATION 65N30, 65N12, 65N50, 58J32, 76R05

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Section: A Gradient Recovery-based Error Estimatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

A gradient recovery–based adaptive finite element method for convection‐diffusion‐reaction equations on surfaces

Xiao

Feng

2019

Numerical Meth Engineering

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“…For stabilized

P_{1} - P_{0}

finite element method, residual‐based a posteriori error estimators have been studied by Kay and Silvester , Wang et al for the penalizing jump method and by Zheng et al for the projection method. Recovery‐based a posteriori error estimate for the Stokes equations is investigated in and for both stabilized methods. Moreover, a posteriori error estimates for low‐order nonconforming finite element methods have also been developed in .…”

Section: Introductionmentioning

confidence: 99%

“…Recovery‐based a posteriori error estimate for the Stokes equations is investigated in and for both stabilized methods. Moreover, a posteriori error estimates for low‐order nonconforming finite element methods have also been developed in .…”

Section: Introductionmentioning

confidence: 99%

Projection stabilized nonconforming finite element methods for the stokes problem

Achchab

Agouzal

Bouihat

et al. 2016

Numerical Methods Partial

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In this article we study a projection‐stabilized nonconforming finite element discretization of the Stokes problem. We present a priori error analysis and give a recovery‐based a posteriori error estimator for the considered problem. Numerical results illustrate the theoretical performance of the error estimator. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 218–240, 2017

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“…Let us finally note that previous works on the a posteriori estimates for stabilized methods have mostly been confined to low order methods or to methods with stabilizing pressure jump terms, cf. [8,11,12,14,19,18]. As mentioned in the introduction, our estimates are the same as in [18], but our analysis is more straightforward.…”

mentioning

confidence: 98%

On the Error Analysis of Stabilized Finite Element Methods for the Stokes Problem

Stenberg¹,

Videman²

2015

SIAM J. Numer. Anal.

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Abstract. For a family of stabilized mixed finite element methods for the Stokes equations a complete a priori and a posteriori error analysis is given.Key words. Stabilized finite element methods, Galerkin least squares methods, Stokes problem, incompressible elasticity, a priori error estimates, a posteriori error estimates AMS subject classifications. 65N301. Introduction. Stabilization of mixed finite element methods for saddle point problems [10,9,4,3] is by now a well-established technique to design stable methods with finite element spaces which do not have to satisfy the so-called Babuška-Brezzi condition. The idea is to add a properly weighted residual of the momentum balance equation to the variational bilinear form. This resembles the least squares method and hence the formulation has often been called the Galerkin least squares method. This is, however, a somewhat misleading name since the formulation does not lead to a minimization problem.In the paper by Franca and Stenberg [6] a unified stability and error analysis for this class of methods was given. The error estimates were obtained under the assumption that the solution is regular enough, and so far a general analysis has been missing.The purpose of this paper is to address this question. We will show that using a technique proposed recently by Gudi [7] it is possible to derive quasi-optimal a priori estimates. This technique uses estimates known from a posteriori error analysis.In addition to the a priori analysis, we discuss a posteriori estimates. Since the added stabilization term is exactly a weighted residual, the a posteriori analysis is very straightforward. Similar a posteriori estimates were given in [18]. Our analysis seems, however, more natural.The plan of the paper is as follows. In the next sections we recall the continuous Stokes problem and its discretization by stabilized finite element methods. Then we present the new a priori analysis. We end by deriving the a posteriori error estimates. We will use well established notation. In addition, we will use the shorthand notation A B for: there exist a positive constant C, independent of the mesh parameter h, such that A ≤ CB.

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Recovery-based error estimator for stabilized finite element methods for the Stokes equation

Cited by 15 publications

References 23 publications

A gradient recovery–based adaptive finite element method for convection‐diffusion‐reaction equations on surfaces

A gradient recovery–based adaptive finite element method for convection‐diffusion‐reaction equations on surfaces

Projection stabilized nonconforming finite element methods for the stokes problem

On the Error Analysis of Stabilized Finite Element Methods for the Stokes Problem

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