We present a non-standard local approach to Richardson extrapolation,when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in RN,N≥2.The main feature of the approach is that it does not rely on a traditional asymptotic error expansion,but rather depends on a more easily proved weaker a priori estimate,derived by Schatz,called an asymptotic error expansion inequality.In order to use this inequality to verify that the Richardson procedure works at a point,we require a local condition which links the different subspaces used for extrapolation.Roughly speaking this condition says that the subspaces are similar about a point,i.e.,any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point.Examples of finite element subspaces that occur in practice and satisfy this condition are given.