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

Abstract

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 $L_{2}$ 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 $L_{2}$ norm, derived from an eigenvalue problem. Nevertheless, these bounds are not sharp. An additional strategy based on a more accurate assessment of the error $L_{2}$ norm is introduced, providing sharp estimates, which are practical upper bounds as it is demonstrated in the numerical tests.