A posteriori estimation of the error in the recovered derivatives of the finite element solution

This work addresses the accuracy of the solution derivatives which are recovered by local averaging of the finite element solution. The main results of the study are: (1) The error in the locally averaged derivatives (e.g. the derivatives which are recovered by the Zienkiewicz-Zhu Superconvergent Patch Recovery (ZZ-SPR) or other similar local recoveries) can be more than the error in the derivatives computed directly from the finite element solution, especially in the case of unsmooth solutions and/or coarse meshes. (2) In order to determine which solution derivatives should be relied upon, the locally averaged ones or the ones computed directly from the finite element solution, one must be able to estimate their errors. It is shown that one can obtain indicators of the error in the derivatives recovered by the ZZ-SPR by employing an additional local averaging of the recovered derivatives ( recycling of the ZZ-SPR) or by comparing the derivatives computed by the ZZ-SPR with the derivatives obtained using a different local averaging which takes into account the character of the exact solution ( harmonic averaging).