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A two-part paper evaluates the application of local a posteriori error estimation to steady-state flow problems (Part I) and elasticity problems (Part II). A posteriori error estimation for the finite element (FE) method is a functional tool for assessing the quality of the finite element results. This paper introduces two error estimators, the Zienkiewicz-Zhu (ZZ) and the Element Residual Method (ERM), within a general mathematical framework. These error estimators are used as drivers for a mesh adaptation process. Four steady-state flow problems using Poisson equation are included to compare the two error estimators and to demonstrate the functional use of a posteriori error estimators for obtaining FE solutions with a pre-specified error tolerance. Of the two methods, the ERM is shown to be more reliable in comparison to the ZZ estimator.