A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator

Babuška, Ivo; Miller, Alan S.

doi:10.1016/0045-7825(87)90114-9

Cited by 262 publications

(163 citation statements)

References 3 publications

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“…Since the startling papers [2,3] the understanding and practical realization of adaptive refinement schemes in a finite element context has been documented in numerous publications [3,4,5,13,36]. A key ingredient in most adaptive algorithms are a-posteriori error estimators or indicators derived from the current residual or the solution of local problems.…”

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confidence: 99%

Adaptive wavelet methods for elliptic operator equations: Convergence rates

2000

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Abstract. This paper is concerned with the construction and analysis of wavelet-based adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution u of the equation by a linear combination of N wavelets. Therefore, a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of N wavelets (so called N -term approximation), which would be obtained by keeping the N largest wavelet coefficients of the real solution (which of course is unknown). The main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error O(N −s ) in the energy norm, whenever such a rate is possible by N -term approximation. The range of s > 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator. Moreover, it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to N . The adaptive algorithm applies to a wide class of elliptic problems and wavelet bases. The analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization.

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confidence: 99%

Adaptive wavelet methods for elliptic operator equations: Convergence rates

2000

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“…This principle is used in all adaptive approaches. For an analysis of this and similar approaches, we refer to [4], [7].…”

Section: Adaptivitymentioning

confidence: 99%

Untitled

2000

Math. Comp.

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Abstract. In this paper we construct elliptic boundary value problems whose standard finite element approximations converge arbitrarily slowly in the energy norm, and show that adaptive procedures cannot improve this slow convergence. We also show that the L 2 -norm and the nodal point errors converge arbitrarily slowly. With the L 2 -norm two cases need to be distinguished, and the usual duality principle does not characterize the error completely. The constructed elliptic problems are one dimensional.

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“…Indeed methods based on the use of SA-field, like the error in constitutive equation (Ladevèze, 1975;Ladevèze and Leguillon, 1983) or like the equilibrated residuals (Babuška and Rheinboldt, 1978;Babuška and Miller, 1987), provide efficient error estimators, for global or local quantities (Ladevèze and Moës, 1999;Prudhomme and Oden, 1999;Ohnimus et al, 2001;Becker and Rannacher, 2001), even in some nonlinear contexts (Babuška and Rheinboldt, 1982;Ladevèze et al, 1986;Ladevèze, 2001;Louf et al, 2003) and domain decomposition methods (Parret-Fréaud et al, 2010).…”

Section: Introductionmentioning

confidence: 99%

Study of the strong prolongation equation for the construction of statically admissible stress fields: Implementation and optimization

Rey¹,

Gosselet²,

Rey³

2014

Computer Methods in Applied Mechanics and Engineering

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This paper focuses on the construction of statically admissible stress fields (SA-fields) for a posteriori error estimation. In the context of verification, the recovery of such fields enables to provide strict upper bounds of the energy norm of the discretization error between the known finite element solution and the unavailable exact solution. The reconstruction is a difficult and decisive step insofar as the effectiveness of the estimator strongly depends on the quality of the SA-fields. This paper examines the strong prolongation hypothesis, which is the starting point of the Element Equilibration Technique (EET). We manage to characterize the full space of SA-fields satisfying the prolongation equation so that optimizing inside this space is possible. The computation exploits topological properties of the mesh so that implementation is easy and costs remain controlled. In this paper, we describe the new technique in details and compare it to the classical EET and to the flux-free technique for different 2D mechanical problems. The method is explained on first degree triangular elements, but we show how extensions to different elements and to 3D are straightforward.

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A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator

Cited by 262 publications

References 3 publications

Adaptive wavelet methods for elliptic operator equations: Convergence rates

Adaptive wavelet methods for elliptic operator equations: Convergence rates

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Study of the strong prolongation equation for the construction of statically admissible stress fields: Implementation and optimization

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