Abstract : Over the last three decades , adaptive finite element methods have been very use ful for efficient numerical solutions . Their usefulness is especially apparent when the exact solution has strong , geometrically localized variations or present singularities . The key features are a posteriori error estimation and strategy of mesh refinement . Let's pointed out that the a posteriori error estimator are explicit quantities , de pending only on the computed numerical solution and the data of the problem . So that , the a posteriori error controls provides a practical , as well as mathematically sound , means of detecting singularities . In this thesis , we will focus on the study of indicators errors in three different situ ations . Firstly , it will be determined by pure residue and by pure residue and local problem by using the conforming FEM . Secondly , it deals with the a posteriori error estimator for non - conforming FEM , for a diffusion - reaction equation . Finally , we focus on the a posteriori error estimation for stabilization approximation of Stokes problem .