Averaging technique for FE – a posteriori error control in elasticity. Part I: Conforming FEM

Carstensen, Carsten; Funken, Stefan A.

doi:10.1016/s0045-7825(00)00248-6

Cited by 75 publications

(43 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…but are independent of u, f or any mesh-size in T. The reliability proof goes back essentially to the dominance of the edge contributions in standard residual-based error control by [27] and can be found in [13,23] for the Poisson problem at hand and in [15,[17][18][19] for related problems.…”

Section: Q(t) ⊆ P K (T) ∩ H(div ω)mentioning

confidence: 97%

“…Their theoretical foundation is less obvious and, in many applications, the use of averaging schemes remains indeed doubtful; see [7,[14][15][16][17][18][19] for positive results. The simplest setting for an explanation of dual and primal variables and their averaging is the 2D Poisson problem with given right-hand side f ∈ L (Ω) in a polygonal Lipschitz domain Ω and a unique weak solution u ∈ H (Ω) to…”

Section: Averaging Of the Dual Variablementioning

confidence: 99%

See 1 more Smart Citation

Reliable Averaging for the Primal Variable in the Courant FEM and Hierarchical Error Estimators on Red-Refined Meshes

Carstensen

Eigel

2016

Computational Methods in Applied Mathematics

Self Cite

View full text Add to dashboard Cite

A hierarchical a posteriori error estimator for the first-order finite element method (FEM) on a redrefined triangular mesh is presented for the 2D Poisson model problem. Reliability and efficiency with some explicit constant is proved for triangulations with inner angles smaller than or equal to π . The error estimator does not rely on any saturation assumption and is valid even in the pre-asymptotic regime on arbitrarily coarse meshes. The evaluation of the estimator is a simple post-processing of the piecewise linear FEM without any extra solve plus a higher-order approximation term. The results also allow the striking observation that arbitrary local averaging of the primal variable leads to a reliable and efficient error estimation. Several numerical experiments illustrate the performance of the proposed a posteriori error estimator for computational benchmarks.

show abstract

Section: Q(t) ⊆ P K (T) ∩ H(div ω)mentioning

confidence: 97%

Section: Averaging Of the Dual Variablementioning

confidence: 99%

Reliable Averaging for the Primal Variable in the Courant FEM and Hierarchical Error Estimators on Red-Refined Meshes

Carstensen

Eigel

2016

Computational Methods in Applied Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Carstensen and Funken [31,32] presented an error estimator, based on recoverytype error estimators, providing upper bounds of the error in energy norm, under certain assumptions of smoothness of the solution which permit to neglect the lack of internal equilibrium of the recovered solution. Díez et al [33] presented a methodology to obtain computable upper bounds of the error in the energy norm considering the stress recovered field provided by the SPR-C technique and taking into account the lack of internal equilibrium of the recovered stresses.…”

Section: Introductionmentioning

confidence: 99%

A recovery-explicit error estimator in energy norm for linear elasticity

Nadal

Dı́ez

Ródenas

et al. 2015

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

Significant research effort has been devoted to produce one-sided error estimates for Finite Element Analyses, in particular to provide upper bounds of the actual error. Typically, this has been achieved using residual-type estimates. One of the most popular and simpler (in terms of implementation) techniques used in commercial codes is the recovery-based error estimator. This technique produces accurate estimations of the exact error but is not designed to naturally produce upper bounds of the error in energy norm. Some attempts to remedy this situation provide bounds depending on unknown constants. Here, a new step towards obtaining error bounds from the recoverybased estimates is proposed. The idea is 1) to use a locally equilibrated recovery technique to obtain an accurate estimation of the exact error, 2) to add an explicit-type error bound of the lack of equilibrium of the recovered stresses in order to guarantee a bound of the actual error and 3) to efficiently and accurately evaluate the constants appearing in the bounding expressions, thus providing asymptotic bounds. The the numerical tests with h-adaptive refinement process show that the bounding property holds even for coarse meshes, providing upper bounds in practical applications.Error bounding, recovery techniques, explicit residual error estimator

show abstract

“…On the other hand, it is well known that averaging or gradient recovery techniques are a popular tool for improved convergence or superconvergence of finite element methods in elliptic partial differential equations with applications to a posteriori error control (see for example [21,22,23,24]). For nonconvex minimization problems such techniques have not been recommended as the energy minimization process enforces finer and finer oscillations.…”

Section: Introductionmentioning

confidence: 99%

An averaging scheme for macroscopic numerical simulation of nonconvex minimization problems

Carstensen

2007

Bit Numer Math

Self Cite

View full text Add to dashboard Cite

Abstract. Averaging or gradient recovery techniques, which are a popular tool for improved convergence or superconvergence of finite element methods in elliptic partial differential equations, have not been recommended for nonconvex minimization problems as the energy minimization process enforces finer and finer oscillations and hence at the first glance, a smoothing step appears even counterproductive. For macroscopic quantities such as the stress field, however, this counterargument is no longer true. In fact, this paper advertises an averaging technique for a surprisingly improved convergence behavior for nonconvex minimization problems. Similar to a finite volume scheme, numerical experiments on a double-well benchmark example provide empirical evidence of superconvergence phenomena in macroscopic numerical simulations of oscillating microstructures.

show abstract

Averaging technique for FE – a posteriori error control in elasticity. Part I: Conforming FEM

Cited by 75 publications

References 9 publications

Reliable Averaging for the Primal Variable in the Courant FEM and Hierarchical Error Estimators on Red-Refined Meshes

Reliable Averaging for the Primal Variable in the Courant FEM and Hierarchical Error Estimators on Red-Refined Meshes

A recovery-explicit error estimator in energy norm for linear elasticity

An averaging scheme for macroscopic numerical simulation of nonconvex minimization problems

Contact Info

Product

Resources

About