A posteriori error estimation techniques in practical finite element analysis

Abstract

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.

A dual weighted residual method applied to complex periodic gratings

“…However, the areas of high stress gradients cannot be guaranteed to coincide with the areas of interest and this may lead to pollution error. Hence, the goal-oriented error estimation method could be improved by using a balance between the local error and the global error so that the mesh is refined to assure a high level of accuracy of the quantity of interest and at the same time to not underestimate the effect of the global error [14].…”

Measuring the convergence behavior of shell analysis schemes

“…Of course, we considered in this paper only a priori error estimates and corresponding convergence behaviors of shell elements, the a posteriori error estimation in shell analyses is another large and very important field, where much progress is still needed [27].…”

Convergence of goal‐oriented adaptive finite element methods for nonsymmetric problems

“…Our analysis approach is signficantly different from that of Mommer and Stevenson [18], combining the recent contraction frameworks of Cascon, Kreuzer, Nochetto and Siebert [7], of Nochetto, Siebert and Veeser [19], and of Holst, Tsogtgerel and Zhu [16]. We also give some numerical results comparing our goal-oriented method both to the one presented in [18] and the dual weighted residual (DWR) method as in [2,4,9,13,14,10], among others. Unlike the existing literature on the DWR method, we prove strong convergence of our goal-oriented method.…”

“…We also show a number of the adaptive meshes below, demonstrating that in many cases, the compared methods produce similar performance from qualitatively different adaptive refinements. A discussion of the DWR method may be found in [2,4,9,13,14,10] for example. In our DWR implementation, the finite element space for the primal problem V T k employs linear Lagrange elements as do HP and MS for both the primal and dual spaces.…”