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 the energy norm, but techniques for other norms are also discussed.

A Posteriori Error Estimation in Finite Element Analysis

This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h4) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L2 projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L2 projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.

The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique

Basic error estimates for elliptic problems

In this paper the development and behaviour of a new finite element algorithm for viscous incompressible flow is presented. The stability and background theory are discussed and the numerical performance is considered for some benchmark problems. The Taylor–Galerkin approach naturally leads to a time-stepping algorithm which is shown to perform well for a wide range of Reynolds numbers (1 ⩽ Re ⩽ 400). Various modifications to the algorithm are investigated, particularly with respect to their effects on stability and accuracy.

A Taylor–Galerkin‐based algorithm for viscous incompressible flow

Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (II)

On justifie l'utilisation d'un schema simple pour determiner les gradients a partir de l'approximation a elements finis triangulaires lineaire par morceaux de la solution d'un probleme elliptique du second ordre. Le gradient obtenu est une estimation superconvergente du vrai gradient

/pdf/superconvergent-recovery-of-the-gradient-from-piecewise-2c0488tu0v.pdf

Superconvergent Recovery of the Gradient from Piecewise Linear Finite-element Approximations

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Superconvergent Recovery of the Gradient from Piecewise Linear Finite-element Approximations