In this paper we review the basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem. We give the basic ideas to establish global error estimates for the energy norm as well as goal-oriented error estimates. While we show how these error estimation techniques are employed for our simple model problem, the emphasis of the paper is on assessing whether these procedures are ready for use in practical linear finite element analysis. We conclude that the actually practical error estimation techniques do not provide mathematically proven bounds on the error and need to be used with care. The more accurate estimation procedures also do not provide proven bounds that, in general, can be computed efficiently. We also briefly comment upon the state of error estimations in nonlinear and transient analyses and when mixed methods are used.

We present some developments for the extension of goal-oriented error estimation procedures to the analysis of Navier-Stokes incompressible fluid flows with structural interactions. Particular focus is given on error assessment of specific quantities of interest defined on the structural part. The goal is to establish relatively coarse meshes to model the fluid flow but achieve acceptable accuracy in the quantities of interest. A nonlinear goal-oriented error estimation procedure is presented which is applicable to general nonlinear analyses. Some illustrative solutions using ADINA are given.

We show that local quantities of interest such as displacements or stresses of a FE-solution can be calculated with improved accuracy if fundamental solutions are employed. The approach is based on Betti's theorem and an integral representation of the local quantities via Green's function. The unknown Green's function is split into a regular part and a fundamental solution so that only the regular part must be approximated on the finite element ansatz space. Some numerical studies for linear elasticity will illustrate our approach. IntroductionIn finite element technology a posteriori recovery techniques for displacements or stresses are becoming increasingly popular. The best known technique in this regard is the method of Zienkiewicz and Zhu where discontinuous stresses are smoothed by an L 2 -projection, [16]. But often we are only interested in the behavior of the solution in certain regions of the problem domain. In such situations it would make no sense to recover the displacement or stress field from the raw FE-output in the whole domain but to minimize the error only in the region of interest. These so-called goal-oriented error estimates are based on duality techniques (Betti's theorem) where the improvement of local quantities of interest is achieved by an iterative local mesh refinement strategy, [3, 6-8, 11, 13, 14].A fundamental result of FE-technology is that any local quantity can be calculated by forming the L 2 -scalar product between the approximated Green's function corresponding to this quantity and the right-hand side of the governing differential equation. This insight is the starting point of our approach. It is based on an integral representation of the local quantity of interest via Green's function. The Green's function is split into an unknown regular part and a known fundamental solution of the adjoint differential operator. Since the exact singularity is used in the integral representation no regularization is necessary. Only the regular part of the Green's function must be approximated on the FE-ansatz space resulting in a higher accuracy of the local quantities. Also no mesh refinement is necessary to catch the singular part. Furthermore the new approach can easily be incorporated into existing FE-codes.The paper is organized as follows: In Sect. 2 we introduce a model problem, the plane stress state of 2-D linear elasticity. In Sect. 3 we briefly introduce the application of Green's function. In Sect. 4 and 5 we shall explain our strategy to improve the accuracy of finite element calculations by employing fundamental solutions. In Sect. 6 an upper error bound of the local quantities is presented and finally in Sect. 7 numerical results for 2-D problems are discussed. Model problemWe consider a problem of 2-D linear elasticity namely the plane stress state in a bounded plane body X & R 2 with a Lipschitz continuous boundary C ¼ C D þ C N . The boundary value problem consists of finding the solution u ¼ fu i g of the elliptic Lamé-equationwith body forces p i 2 L 2 ðXÞ and boundary ...

SUMMARYIn this paper, we first present a consistent procedure to establish influence functions for the finite element analysis of shell structures, where the influence function can be for any linear quantity of engineering interest. We then design some goal-oriented error measures that take into account the cancellation effect of errors over the domain to overcome the issue of over-estimation. These error measures include the error due to the approximation in the geometry of the shell structure. In the calculation of the influence functions we also consider the asymptotic behaviour of shells as the thickness approaches zero. Although our procedures are general and can be applied to any shell formulation, we focus on MITC finite element shell discretizations. In our numerical results, influence functions are shown for some shell test problems, and the proposed goal-oriented error estimation procedure shows good effectivity indices.

In this paper, we present a goal-oriented a posteriori error estimation technique for the pointwise error of finite element approximations using fundamental solutions. The approach is based on an integral representation of the pointwise quantity of interest using the corresponding Green's function, which is decomposed into an unknown regular part and a fundamental solution. Since only the regular part must be approximated with finite elements, very accurate results are obtained. The approach also allows the derivation of error bounds for the pointwise quantity, which are expressed in terms of the primal problem and the regular part problem. The presented technique is applied to linear elastic test problems in two-dimensions, but it can be applied to any linear problem for which fundamental solutions exist.