Fully Reliable Localized Error Control in the FEM

Carstensen, Carsten; Funken, Stefan A.

doi:10.1137/s1064827597327486

Cited by 99 publications

(93 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(c) Estimates for the constants c 1 and c 2 may be found in [5]. For planar triangulations into right isoceles triangles (halved squares), we found c 1 , c 2 < 1 in [6]. (d) For details on red-green-blue refinement strategies cf., e.g., [23].…”

Section: Remark 51 (A) An Upper Bound Formentioning

confidence: 83%

See 1 more Smart Citation

Numerical Analysis of a Relaxed Variational Model of Hysteresis in Two-Phase Solids

Carstensen

Plecháč

2001

ESAIM: M2AN

Self Cite

View full text Add to dashboard Cite

Abstract. This paper presents the numerical analysis for a variational formulation of rate-independent phase transformations in elastic solids due to Mielke et al. The new model itself suggests an implicit time-discretization which is combined with the finite element method in space. A priori error estimates are established for the quasioptimal spatial approximation of the stress field within one time-step. A posteriori error estimates motivate an adaptive mesh-refining algorithm for efficient discretization. The proposed scheme enables numerical simulations which show that the model allows for hysteresis.Mathematics Subject Classification. 65N30, 73C05.

show abstract

Section: Remark 51 (A) An Upper Bound Formentioning

confidence: 83%

“…From the formulae of the previous section we obtain the stress field (6 are monotone decreasing and increasing, respectively. Define a := (E) and b := (F ) for two symmetric matrices E, F and notice that (E − F ), C(E 2 − E 1 ) = 2γ(b − a).…”

Section: Finite Element Approximation and Convergencementioning

confidence: 99%

Numerical Analysis of a Relaxed Variational Model of Hysteresis in Two-Phase Solids

Carstensen

Plecháč

2001

ESAIM: M2AN

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is important to emphasize that in mesh-free methods the reference error is not anymore u h − u H as in finite elements but only the solution of problem (11). This also implies that the fine solution u h may not be recovered from u H + e h .…”

Section: Error Equations and Reference Errormentioning

confidence: 99%

Bounds for quantities of interest and adaptivity in the element‐free Galerkin method

Vidal

Mariné

Dı́ez

et al. 2008

Numerical Meth Engineering

View full text Add to dashboard Cite

1 of estimators ensuring bounds for the error. Classical residual type estimators, which provide upper bounds of the error, require flux-equilibration procedures (hybrid-flux techniques) to properly set boundary conditions for local problems [10, 2]. Flux-equilibration requires a domain decomposition, which is natural in finite elements but not in mesh-free methods. Moreover, it is performed by a complex algorithm, strongly dependent on the finite element type and requiring a data structure that is not natural in a standard finite element code.The idea of using flux-free estimators, based on the partition-of-the-unity concept and using local subdomains different than elements, has been already proposed in [11,12,13,14] for finite elements. The main advantage of the flux-free approach is the simplicity in the implementation. Obviously, this is especially important in the 3D case.From the mesh-free point of view, another advantage is the fact that the local subdomains where the error equation is solved are the support of the functions characterizing the partition of unity. This is a concept that also exists in mesh-free methods and thus the extension is possible. Moreover, boundary conditions of the local problems are natural and the usual data structure of a code is directly employed.In the last few years, some research has been devoted to develop error estimation procedures for mesh-free methods. To the authors knowledge implicit residual based estimators have not been proposed for mesh-free methods. However, in [19] a strategy to assess the energy norm of the error for the Generalized Finite Element Method (GFEM) is presented and the possible generalizations to mesh-free methods are also devised. Implicit residual error estimators are now standard in finite elements because they are mathematically sound, more accurate and allow computing upper and lower bounds for energy norms as well as for functional outputs. In this paper the implicit residual-type flux-free error estimator proposed in [14], which is comparable in effectivity to the standard hybrid-flux estimators, is extended to the Element Free Galerkin Method. In the present work, the difficulties found when extending the implicit residual methods to mesh-free are related with the fact that the refined discretization do not induce nested functional spaces. This is specially important for the error representation. Moreover, the bounds for the specific quantity of interest must be in this case carefully computed. In reference [19] these difficulties are not encountered because the error estimator is performed in terms of the energy norm and in the GFEM the refined subspaces are nested.Mesh-free methods are especially well suited for adaptivity. Enriching the mesh-free discretization is straightforward because the new particles are added without any connectivity restriction. Here, a refinement scheme based on inserting particles following a quadtree structure in a background rectangular grid is proposed. The refinement criterion uses the information provided...

show abstract

“…Upper bounds on C rel for related estimators with a best value around 1 can be found in [CF1,CF2]. Remark 1.4.…”

Section: −1mentioning

confidence: 99%

All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable

Carstensen

2003

Math. Comp.

View full text Add to dashboard Cite

Abstract. All first-order averaging or gradient-recovery operators for lowestorder finite element methods are shown to allow for an efficient a posteriori error estimation in an isotropic, elliptic model problem in a bounded Lipschitz domain Ω in R d . Given a piecewise constant discrete flux p h ∈ P h (that is the gradient of a discrete displacement) as an approximation to the unknown exact flux p (that is the gradient of the exact displacement), recent results verify efficiency and reliability ofin the sense that η M is a lower and upper bound of the flux error p−p h L 2 (Ω) up to multiplicative constants and higher-order terms. The averaging space Q h consists of piecewise polynomial and globally continuous finite element functions in d components with carefully designed boundary conditions. The minimal value η M is frequently replaced by some averaging operator A : P h → Q h applied within a simple post-processing to p h . The result q h := Ap h ∈ Q h provides a reliable error bound withThis paper establishes η A ≤ C eff η M and so equivalence of η M and η A . This implies efficiency of η A for a large class of patchwise averaging techniques which includes the ZZ-gradient-recovery technique. The bound C eff ≤ 3.88 established for tetrahedral P 1 finite elements appears striking in that the shape of the elements does not enter: The equivalence η A ≈ η M is robust with respect to anisotropic meshes. The main arguments in the proof are Ascoli's lemma, a strengthened Cauchy inequality, and elementary calculations with mass matrices.

show abstract

Fully Reliable Localized Error Control in the FEM

Cited by 99 publications

References 13 publications

Numerical Analysis of a Relaxed Variational Model of Hysteresis in Two-Phase Solids

Numerical Analysis of a Relaxed Variational Model of Hysteresis in Two-Phase Solids

Bounds for quantities of interest and adaptivity in the element‐free Galerkin method

All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable

Contact Info

Product

Resources

About