# A Posteriori Error Estimation in Finite Element Analysis

@inproceedings{Ainsworth2000APE, title={A Posteriori Error Estimation in Finite Element Analysis}, author={Mark Ainsworth and J. Tinsley Oden}, year={2000} }

This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in… Expand

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#### References

SHOWING 1-10 OF 173 REFERENCES

A Posteriori Error Estimators for the Stokes and Oseen Equations

- Mathematics
- 1997

The problem of obtaining /posteriori/ estimates of the discretization error when one uses finite element methods to approximate problems with an incompressibility constraint is discussed. A general… Expand

An a posteriori error estimate for finite element approximations of the Navier-Stokes equations

- Mathematics
- 1994

Abstract In this paper, an a posteriori error estimate for the steady state Navier-Stokes equations, based on element residual methods, is developed. Here, a special norm is designed to measure the… Expand

A procedure for a posteriori error analysis for the finite element method which contains a bounding measure

- Mathematics
- 1989

Abstract Post-processing procedures are applied to estimate the error in energy, displacement and stress in finite element solutions for problems in linear elasticity. The error estimates are based… Expand

A posteriori error analysis and adaptive processes in the finite element method: Part I—error analysis

- Computer Science
- 1983

It is shown that an extremely high rate of convergence is reached in practical problems using p-convergent methods, and applications to realistic stress analysis and potential problems are presented. Expand

A posteriori error estimate for the mixed finite element method

- Computer Science, Mathematics
- Math. Comput.
- 1997

A computable error bound for mixed finite element methods is established in the model case of the Poisson-problem to control the error in the H(div,Ω) x L 2 (Ω)-norm. The reliable and efficient a… Expand

A POSTERIORI FINITE ELEMENT ERROR BOUNDS FOR NON-LINEAR OUTPUTS OF THE HELMHOLTZ EQUATION

- Mathematics
- 1999

A Neumann subproblem a posteriori finite element procedure for the efficient and accurate calculation of rigorous, constant-free upper and lower bounds for non-linear outputs of the Helmholtz… Expand

A Posteriori Error Analysis of Finite Element Solutions for One-Dimensional Problems

- Mathematics
- 1981

The paper develops a theory of a posteriors error estimates under the $L_p $-energy norm for $2 \leqq p \leqq \infty $. The theory is based on a general concept of error indicators and error… Expand

A unified approach to a posteriori error estimation using element residual methods

- Mathematics
- 1993

SummaryThis paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations. It is shown that, by solving… Expand

A procedure for a posteriori error estimation for h-p finite element methods

- Mathematics
- 1992

Abstract A new approach to a posteriori error estimation is outlined which is applicable to general h-p finite element approximations of general classes of boundary value problems. The approach makes… Expand

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

- Mathematics
- 1987

Abstract This paper is the first in a series of two in which we discuss some theoretical and practical aspects of a feedback finite element method for solving systems of linear second-order elliptic… Expand