A functional type a posteriori error analysis for the Ramberg‐Osgood model

Bildhauer, Michael; Fuchs, Martin; Repin, Sergey

doi:10.1002/zamm.200710350

Cited by 6 publications

(2 citation statements)

References 24 publications

(8 reference statements)

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“…One of the first publications presenting this method was [24] where the a posteriori estimates were derived for a deformation plasticity model with hardening. Recently, the method was applied to the Ramberg-Osgood model (sometimes also called Norton-Hoff) in the theory of nonlinear solid media, see [6]. Also, we note close publications [5] and [12] where such estimates were derived for nonlinear viscous flow problems.…”

Section: Introductionmentioning

confidence: 58%

Functional a posteriori error estimates for incremental models in elasto-plasticity

Repin

Valdman

2009

Open Mathematics

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We consider a convex variational problem related to a time-step problem in elasto-plastic models with isotropic hardening. Our goal it to derive a posteriori error estimate of the difference between the exact solution and any function in the admissible (energy) class of the problem considered. The estimates are obtained by a advanced version of the variational approach earlier used for linear boundary-value problems and nonlinear variational problems with convex functionals (see [20, 21] and the monography [18]). They do no contain mesh-dependent constants and are valid for any conforming approximations regardless of the method used for their derivation. It is shown that the structure of error majorant reflects properties of the exact solution so that the majorant vanishes only if an approximate solution coincides with the exact one. Moreover, it possesses necessary continuity properties, so that any sequence of approximations converging to the exact solution in the energy space generates a sequence of positive numbers (explicitly computable by the majorant functional) that tends to zero.

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Section: Introductionmentioning

confidence: 58%

Functional a posteriori error estimates for incremental models in elasto-plasticity

Repin

Valdman

2009

Open Mathematics

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“…Extensions of the method to various models of viscous fluids are presented in [4,13,17,20,23,27,28], to diffusion problems with reaction and convection terms in [11,14,17,24], and to mixed approximations of elliptic problems in [16,25,26]. The method was also applied to certain classes of nonlinear problems: variational inequalities (see [19,21]), nonlinear models in solid mechanics (e.g., see [5,30,31]). …”

Section: Introductionmentioning

confidence: 99%

Advanced forms of functional a posteriori error estimates for elliptic problems

Repin

2008

Russian Journal of Numerical Analysis and Mathematical Modelling

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Functional a posteriori estimates have been derived for elliptic and linear parabolic problems in [12][13][14][15] and some other publications. They provide computable upper bounds of the difference between the exact solution u and any approximation v lying in the admissible (energy) class. This paper is concerned with advanced forms of these estimates, which are discussed within the paradigm of the reaction-diffusion problem. In the first part of the paper, we derive guaranteed and computable upper bounds for problems which admit the decomposition of the domain into a set of simple (e.g., simplicial or polyhedral) subdomains. For this case an upper bound which involves constants in the Poincaré inequality for the corresponding subdomains is obtained. Estimates of this type can be helpful if computations are performed by the domain decomposition method. The second part of the paper is devoted to the derivation of two-sided error bounds in terms of weighted norms. Our analysis shows that the approach based on certain transformations of the basic integral identity earlier developed for energy error norms (see [13,14]) can be successfully applied to error estimation in weighted norms.

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A Posteriori Estimates for Variational Inequalities

Repin

2010

Numerical Mathematics and Advanced Applications 2009

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A functional type a posteriori error analysis for the Ramberg‐Osgood model

Cited by 6 publications

References 24 publications

Functional a posteriori error estimates for incremental models in elasto-plasticity

Functional a posteriori error estimates for incremental models in elasto-plasticity

Advanced forms of functional a posteriori error estimates for elliptic problems

A Posteriori Estimates for Variational Inequalities

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