This paper is the final in a series of three in which we have discussed a finite element post-processing technique. Here we shall deal with the questions of adaptive mesh selection and a posteriori error estimation. Some numerical examples computed by the FEARS program will be used to illustrate the approaches taken.
1 Introduction
This is the final in a series of three papers in which we have sought to show how a suitable post-processing of a finite element solution can yield accurate pointwise .values for quantities such as displacements, stresses, flow rates and stress intensity factors. In References 1 and 2 we derived a number of extraction expressions for such quantities in the setting of some simple model problems, and saw how these expressions could serve as the bases of effective postprocessing techniques. We also carried out an error analysis for such post-processing computations. This analysis showed that the accuracy of the post-processed value could be related to how well the space of finite element functions is able to approximate both the solution of the basic problem and the solution of a related auxiliary problem. This auxiliary problem is of the same form as the basic problem, although with different loading data. (See Subsections 2.5 and 3.4 of Reference 1, and Section 4 of Reference 2.)
Hitherto, except for a few qualitative remarks, we have said little concerning the issues of (a) choosing a finite element subspace for calculating the approximate solution which is to be subsequently post-processed, and (b) estimating, a posteriori, the error in a computed postprocessed value.
The significance of both (a) and (b) is quite clear. In practice, the goal of any post-processing computation is to obtain a post-processed value of a specified accuracy at a minimal total computational cost. An estimate as in (b) provides a means of determining when the specified accuracy has been attained, whereas the choice in (a) largely determines the efficiency of the overall numerical procedure.
In this paper we propose a post-processing algorithm. It is based on the extraction techniques of References 1 and 2, and includes features that enable (a) and (b) to be handled quite effectively. Our discussion will be in the context of the 'membrane' model probleps already introduced in Section 5 of Reference 1 and Section 6 of Reference 2. In these, the order of t This research was partially supported by ONR Contract N00014-77-C-0623. The computations were carried out with support from the Computer Science Center at the