A posteriori estimation and adaptive control of the pollution error in the h-version of the finite element method

In References 1 and 2 we showed that the error in the finite-element solution has two parts, the local error and the pollution error, and we studied the effect of the pollution error on the quality of the local error-indicators and the quality of the derivatives recovered by local post-processing. Here we show that it is possible to construct a posteriori estimates of the pollution error in any patch of elements by employing the local error-indicators over the mesh outside the patch. We also give an algorithm for the adaptive control of the pollution error in any patch of elements of interest.     

A posteriori error estimates and adaptivity for finite element solutions in finite elasticity

Methods for a posteriori error estimation for finite element solutions are well established and widely used in engineering practice for linear boundary value problems. In contrast here we are concerned with finite elasticity and error estimation and adaptivity in this context. In the paper a brief outline of continuum theory of finite elasticity is first given. Using the residuals in the equilibrium conditions the discretization error of the finite element solution is estimated both locally and globally. The proposed error estimator is physically interpreted in the energy sense. We then present and discuss the convergence behaviour of the discretization error in uniformly and adaptively refined finite element sequences.     

Application of Variational a-Posteriori Multiscale Error Estimation to Higher-Order Elements

An explicit a-posteriori error estimator based on the variational multiscale method is extended to higher-order elements. The technique is based on a recently derived explicit formula of the fine-scale Green’s function for higher-order elements. For the class of element-edge exact methods, the technique is able to predict the error exactly in any desired norm. It is shown that for elements of order k, the exact error depends on the k−1 derivative of the residual. The technique is applied to one-dimensional examples of fluid transport computed with stabilized methods.     

A constitutive relation error estimator based on traction‐free recovery of the equilibrated stress

A new methodology for recovering equilibrated stress fields is presented, which is based on traction‐free subdomains' computations. It allows a rather simple implementation in a standard finite element code compared with the standard technique for recovering equilibrated tractions. These equilibrated stresses are used to compute a constitutive relation error estimator for a finite element model in 2D linear elasticity. A lower bound and an upper bound for the discretization error are derived from the error in the constitutive relation. These bounds in the discretization error are used to build lower and upper bounds for local quantities of interest. Copyright © 2008 John Wiley & Sons, Ltd.     

基于对偶理论的椭圆变分不等式的后验误差分析（英）

本文基于对偶理论对椭圆变分不等式的正则化方法提供一个相对全面的后验误差分析．我们分别考虑了摩擦接触问题和障碍问题,通过选取一种不同形式的有界算子和泛函,推导了其对偶形式并给出了正则化方法的 $H^1$ 范后验误差估计．最后,利用凸分析中的对偶理论建立了障碍问题的残量型后验误差估计的一般框架．同时我们选取一种特殊的对偶变量和泛函形式得到该问题的残量型误差估计及其有效性．数值解的后验误差估计是发展有效自适应算法的基础而模型误差的后验误差估计在分析问题中数据的不确定影响时是非常有用的．     

Goal-Oriented Error Estimates Based on Different FE-Spaces for the Primal and the Dual Problem with Applications to Fracture Mechanics

The objective of this paper is to derive goal-oriented a posteriori error estimators for the error obtained while approximately evaluating the nonlinear J-integral as a fracture criterion in linear elastic fracture mechanics (LEFM) using the finite element method. Such error estimators are based on the well-established technique of solving an auxiliary dual problem. In a straightforward fashion, the solution to the discretized dual problem is sought in the same FE-space as the solution to the original (primal) problem, i.e. on the same mesh, although it merely acts as a weight of the discretization error only. In this paper, we follow the strategy recently proposed by Korotov et al. [J Numer Math 11:33–59, 2003; Comp Lett (in press)] and derive goal-oriented error estimators of the averaging type, where the discrete dual solution is computed on a different mesh than the primal solution. On doing so, the FE-solutions to the primal and the dual problems need to be transferred from one mesh to the other. The necessary algorithms are briefly explained and finally some illustrative numerical examples are presented.