Equilibrated Patch Recovery error estimates: simple and accurate upper bounds of the error

Author (s): Diez, P., Rodenas, J.J. and Zienkiewicz, O.C.
Journal: International Journal for Numerical Methods in Engineering

Volume: 69, Issue 10
Pages: 2075 – 2098
Date: 2007

This paper introduces a new recovery-type error estimator ensuring local equilibrium and yielding a guaranteed upper bound of the error. The upper bound property requires the recovered solution to be both statically equilibrated and continuous. The equilibrium is obtained locally (patch by patch) and the continuity is enforced by a postprocessing based on the partition of the unity concept. This postprocess is expected to preserve the features of the locally equilibrated stress field.
Nevertheless, the postprocess phase modifies the equilibrium, which is no longer exactly fulfilled. A new methodology is introduced that yields upper bound estimates by taking into account this lack of equilibrium. This requires computing the L2
norm of the error or relating it with the energy norm.
The guaranteed upper bounds are obtained by using a pessimistic bound of the error L2 norm, derived from an eigenvalue problem. Nevertheless, these bounds are not sharp. An additional strategy based on a more accurate assessment of the error L2 norm is introduced, providing sharp estimates which are practical upper bounds as it is demonstrated in the numerical tests.



@article {NME:NME1837,
author = {Díez, P., Ródenas, J.J. and Zienkiewicz, O.C.},
title = {Equilibrated patch recovery error estimates: simple and accurate upper bounds of the error},
journal = {International Journal for Numerical Methods in Engineering},
volume = {69},
number = {10},
publisher = {John Wiley & Sons, Ltd.},
issn = {1097-0207},
url = {http://dx.doi.org/10.1002/nme.1837},
doi = {10.1002/nme.1837},
pages = {2075--2098},
keywords = {error estimates, stress recovery, equilibrated stresses, upper bounds},
year = {2007},