A simple error estimator and adaptive procedure for practical engineerng analysis

Zienkiewicz, O. C.; Zhu, J.Z.

doi:10.1002/nme.1620240206

Cited by 2,149 publications

(1,184 citation statements)

References 9 publications

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“…It is obtained by a comparison of the piecewise constant approximation of the primal variable with a higher order finite element solution arising from a modified nonconforming approach. Finally, we show that the difference between the piecewise constant and the nonconforming approximation is equivalent to a formulation that can be obtained by using some local averaging techniques (cf., e.g., [11,24,32,34]). …”

Section: Error Estimators For Raviart-thomas Elements 1349mentioning

confidence: 99%

“…In contrast to the hierarchical basis error estimator η H and the error estimator η L based on the solution of local subproblems, we do not have to solve additional defect problems. In the standard conforming case, error estimators obtained by some postprocessing of the approximation have been introduced by Zienkiewicz and Zhu in [32,34] and have been further analyzed by Rodriguez [24]. In the mixed setting there is some work of Brandts [11].…”

Section: Theorem 24mentioning

confidence: 99%

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A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

Wohlmuth

Hoppe

1999

Math. Comp.

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Abstract. We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimators relying on the solution of local subproblems and on a superconvergence result, respectively. Finally, we examine the relationship between the presented error estimators and compare their local components.

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Section: Error Estimators For Raviart-thomas Elements 1349mentioning

confidence: 99%

Section: Theorem 24mentioning

confidence: 99%

A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

Wohlmuth

Hoppe

1999

Math. Comp.

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“…The adaptive mesh refinement is done by estimating the error through a recoverybased projection method of Zienkiewicz & Zhu [8]. The true (stress) error is defined pointwise as e σ := σ − σ h , where σ the real (accurate) and σ h the calculated stress specify.…”

Section: Scale Adaptivity 21 Error Estimation and Mesh Refinementmentioning

confidence: 99%

“…First a h-adaptive mesh refinement method is used to refine an initial coarse mesh merely in regions where the error exceeds a preset tolerance related to a recovery-based error estimation [8,9]. For a better analysis of the composite substructure in nonlinear regimes and near highly stressed regions, the mesh refinement needs to be complemented by a model adaptivity method.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Multi‐scale Modeling Error Estimation for Composites

Hajibeik

Temizer

Löhnert

et al. 2011

Proc Appl Math and Mech

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Analysing composite materials through efficient higher-order homogenization techniques requires accurate simulations of highly heterogeneous domains. However the validity of the homogenization response in regions with highly localized variations is still questionable. A scale adaptation strategy is developed for a more accurate analysis of generally inelastic macrostructures, where a transition from a homogenized description to an explicit micro-structural resolution is pursued. The designated zones of interest are indicated through two different methods of error estimation. First for mesh refinement approaches, the discretization error is estimated using an indicator developed by Zienkiewicz-Zhu and studying L2 norm. Furthermore a second adaptation zone is identified based on a post-processing step on the homogenized solution and corresponds to regions with high strain-gradients, the gradient of the deformation gradient. In the designated sub-domains a micro-structure representation will replace the homogenized area. The introduced scale-adaptivity method returns an inexpensive and more accurate analysis of composite materials.

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“…(4) averaging methods: Use some local averaging technique for error estimation (cf., e.g., [6,21,29,30]). …”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations

Verfürth¹

1994

Math. Comp.

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Abstract. We give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems. In a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual. In a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite-dimensional approximation of the residual. This quantity can easily be evaluated, and for many practical applications sharp explicit upper and lower bounds are readily obtained. The general results are then applied to finite element discretizations of scalar quasi-linear elliptic partial differential equations of 2nd order, the eigenvalue problem for scalar linear elliptic operators of 2nd order, and the stationary incompressible Navier-Stokes equations. They immediately yield a posteriori error estimates, which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution.

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A simple error estimator and adaptive procedure for practical engineerng analysis

Cited by 2,149 publications

References 9 publications

A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

Adaptive Multi‐scale Modeling Error Estimation for Composites

A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations

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