The superconvergent patch recovery (SPR) and adaptive finite element refinement

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

doi:10.1016/0045-7825(92)90023-d

Cited by 369 publications

(205 citation statements)

References 13 publications

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“…A closed-form solution to the linear elasticity system exists in the case of an infinitely large, thin plate with a circular hole, and the stress normal to the vertical plane of symmetry at point P (see Fig. 10 ( [30], Eq. (36), with r = a and θ = π/2).…”

Section: Plate With a Hole Under Tractionmentioning

confidence: 99%

A posteriori dual-mixed adaptive finite element error control for Lamé and Stokes equations

2005

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A unified and robust mathematical model for compressible and incompressible linear elasticity can be obtained by rephrasing the Herrmann formulation within the Hellinger-Reissner principle. This quasi-optimally converging extension of PEERS (Plane Elasticity Element with Reduced Symmetry) is called Dual-Mixed Hybrid formulation (DMH). Explicit residual-based a posteriori error estimates for DMH are introduced and are mathematically shown to be locking-free, reliable, and efficient. The estimator serves as a refinement indicator in an adaptive algorithm for effective automatic mesh generation. Numerical evidence supports that the adaptive scheme leads to optimal convergence for Lamé and Stokes benchmark problems with singularities.

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Section: Plate With a Hole Under Tractionmentioning

confidence: 99%

A posteriori dual-mixed adaptive finite element error control for Lamé and Stokes equations

2005

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“…This so-called Zienkiewicz-Zhu (ZZ) estimator has been invented in [26] and later been improved in [27]; many more variants have been developed since. The basic idea consists in computing an improvement of the gradient of the numerical solution by some post-processing procedure.…”

Section: Introductionmentioning

confidence: 99%

Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes

Kunert

Nicaise

2003

ESAIM: M2AN

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Abstract. We consider a posteriori error estimators that can be applied to anisotropic tetrahedral finite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large. Two kinds of Zienkiewicz-Zhu (ZZ) type error estimators are derived which originate from different backgrounds. In the course of the analysis, the first estimator turns out to be a special case of the second one, and both estimators can be expressed using some recovered gradient. The advantage of keeping two different analyses of the estimators is that they allow different and partially novel investigations and results. Both rigorous analytical approaches yield the equivalence of each ZZ error estimator to a known residual error estimator. Thus reliability and efficiency of the ZZ error estimation is obtained. The anisotropic discretizations require analytical tools beyond the standard isotropic methods. Particular attention is paid to the requirements on the anisotropic mesh. The analysis is complemented and confirmed by extensive numerical examples. They show that good results can be obtained for a large class of problems, demonstrated exemplary for the Poisson problem and a singularly perturbed reaction diffusion problem.Mathematics Subject Classification. 65N15, 65N30, 65N50.

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“…Basically, a posteriori error estimators can be categorised in two main groups namely the recovery type (e.g., [20,21]) and the residual type (e.g., [5,7]). Here, the residual-type error estimators are employed.…”

Section: Introductionmentioning

confidence: 99%

p-Version error estimation for linear elasticity

2008

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An error estimator, formulated earlier for h-adaptive strategies, is extended for use in the p-version finite element analysis. The estimation of error is based on solving a series of local problems, based on patches consisting of elements surrounding each node, with prescribed homogeneous essential boundary conditions. Unlike the original approach in which a patch was constructed based on one element, each patch in the present scheme is automatically formed based on a number of elements surrounding a corresponding node. The present scheme, based on enhancing the degree of interpolation, provides a better estimate than the original h-scheme while still preserving the original lower bound property. The capability of the new scheme is investigated in some numerical examples in terms of its global and local performance.

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The superconvergent patch recovery (SPR) and adaptive finite element refinement

Cited by 369 publications

References 13 publications

A posteriori dual-mixed adaptive finite element error control for Lamé and Stokes equations

A posteriori dual-mixed adaptive finite element error control for Lamé and Stokes equations

Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes

p-Version error estimation for linear elasticity

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