Superconvergent recovery of gradients on subdomains from piecewise linear finite‐element approximations

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Abstract
Engineers have been aware for some time of the phenomenon of superconvergence, whereby there exist (stress) points at which the accuracy of a finite‐element solution is superior to that of the approximation generally. This phenomenon has been treated in recent years by mathematicians who have proved, for certain two‐dimensional secondorder elliptic problems, superconvergent error estimates for retrieved finite‐element derivatives. These results have demanded high global regularity of the solutions of the bondary value problems. In this present article cut‐off functions are used to prove similar superconvergence results over interior subdomains. This allows superconvergence estimates to be derived for problems with solutions of low global regularity, particularly those involving singularities.