This paper is the first in a series in which we discuss computational methodologies for checking the quality of a posteriori error estimators for finite element approximations of linear elliptic problems. In this first part we study the asymptotic properties of error estimators in the interior of patchwise uniform grids of triangles. A completely numerical methodology for the analysis of the quality of estimators is presented. Results from the application of the methodology to the study of the quality of several well-known error estimators are reported. In subsequent papers we shall discuss methods to study the properties of estimators for meshes of quadrilaterals, non-uniform grids, at boundaries, grid-interfaces and near-singular points. © 1994.