A Posteriori Error Estimates for Non-Linear Problems

Verfürth, Rüdiger

doi:10.1007/978-3-663-14007-8_29

Cited by 19 publications

(33 citation statements)

References 14 publications

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“…We prove that the estimator is equivalent to the error up to higher order terms. We remark that in the above mentioned papers by Larson 7 and Verfürth 15 , the a posteriori error estimates are obtained assuming that the numerical solution is close enough to the exact one. We do not assume this and that is why we have to add explicit higher order terms in our reliability and efficiency estimates.…”

Section: Introductionmentioning

confidence: 98%

A Posteriori Error Estimates for the Finite Element Approximation of Eigenvalue Problems

Durán

Padra

Rodrı́guez

2003

Math. Models Methods Appl. Sci.

109

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This paper deals with a posteriori error estimators for the linear finite element approximation of second order elliptic eigenvalue problems in two or three dimensions. First, we give a simple proof of the equivalence, up to higher order terms, between the error and a residual type error estimator. Second, we prove that the volumetric part of the residual is dominated by a constant times the edge or face residuals, again up to higher order terms. This result was not known for eigenvalue problems.

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Section: Introductionmentioning

confidence: 98%

A Posteriori Error Estimates for the Finite Element Approximation of Eigenvalue Problems

Durán

Padra

Rodrı́guez

2003

Math. Models Methods Appl. Sci.

109

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“…So far, only the case of coinciding discrete and continuous boundary has been considered in the a posteriori analysis. From the vast literature on this subject we cite [BW,BR,Ve1,Ve2]. A robust algorithm which takes into account the error due to unavoidable data approximation is described and analyzed in [Dö1].…”

Section: Introductionmentioning

confidence: 99%

An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation

Dörfler

Rumpf

1998

Math. Comp.

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Abstract. We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No assumption is made on a geometrically fitting shape.A posteriori error estimates are given in the energy norm and the L 2 -norm, and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation. Furthermore, a strategy is presented to compute the effect of the non-discretized part of the domain on the error starting from a coarse mesh. This especially implies that parts of the domain, where the measured error is small, stay non-discretized. The presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary, the reliable computation of the a posteriori estimates and a mesh adaptation based on Delaunay techniques. Numerical examples illustrate that errors outside the initial discretization will be detected.

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“…Despite a number of significant advances in the field, much of the research to date has focused on source problems. In the context of the finite element approximation of second-order selfadjoint elliptic eigenvalue problems we mention the recent articles [20,21,33,37]; for related work, based on considering the eigenvalue problem as a parameter-dependent nonlinear equation, see Verfürth [43,44], for example. For earlier references devoted to the derivation of a posteriori error bounds for the finite element approximation of symmetric eigenvalue problems, we refer to [7,8], for example.…”

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confidence: 99%

Adaptivity and a Posteriori Error Control for Bifurcation Problems II: Incompressible Fluid Flow in Open Systems with Z 2 Symmetry

et al. 2011

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Access from the University of Nottingham repository:http://eprints.nottingham.ac.uk/1257/1/cliffe_et_al_z2_bifurcation.pdf Copyright and reuse:The Nottingham ePrints service makes this work by researchers of the University of Nottingham available open access under the following conditions. This article is made available under the University of Nottingham End User licence and may be reused according to the conditions of the licence. For more details see: http://eprints.nottingham.ac.uk/end_user_agreement.pdf A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. Abstract. In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork or Hopf bifurcation occurs when the underlying physical system possesses reflectional or Z 2 symmetry. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard DualWeighted-Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.Key words. Incompressible flows, bifurcation problems, a posteriori error estimation, adaptivity, discontinuous Galerkin methods, Z 2 symmetry 1. Introduction. Due to the highly nonlinear governing equations and complex geometries involved, understanding fluid flow remains one of the fundamental engineering challenges today. Often it is impossible to obtain analytical solutions to problems and numerical methods must instead be exploited. Broadly speaking, there are two numerical approaches to understanding the Navier-Stokes equations: the first involves direct 'simulation' of the time dependent problem, while the second, less commonly used approach focuses on applying nonlinear analysis to compute paths of steady solutions using numerical continuation methods and to determine their stability based on eigenvalue information. In this article we focus on the latter technique, specifically where we seek to understand how the solution structure changes as one parameter of interest is varied; in the case of the incompressible Navier-Stokes equations this parameter is typically the Reynolds number. We are particularly interested in the location of critical parameters at which a bifurcation first occurs; a review of techniques for bifurcation detection can be found in Cliffe et al. [16], for example.Over the past few decades, ...

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A Posteriori Error Estimates for Non-Linear Problems

Cited by 19 publications

References 14 publications

A Posteriori Error Estimates for the Finite Element Approximation of Eigenvalue Problems

A Posteriori Error Estimates for the Finite Element Approximation of Eigenvalue Problems

An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation

Adaptivity and a Posteriori Error Control for Bifurcation Problems II: Incompressible Fluid Flow in Open Systems with Z 2 Symmetry

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