In this paper the basic concepts to obtain a posteriori error estimates for the finite element method are reviewed. Explicit residual-based, implicit (namely subdomain and element) residual-based, hierarchical-based, recovery-base and functional-based error estimators as well as goal oriented error estimators are presented for a test elliptic boundary value problem. These notes are an introductory presentation, reviewing in a not-too-technical way the fundamental concepts involved in the subject and do not aim at being exhaustive or complete but rather simple and easy to follow. For more detailed explanations, we refer the interested reader to [3] and eventually to [4],[9],[15],[27],[30], chapter 4 of [37],[38],[39] – and the references therein – where most of the material contained in this report can be found.

finite element method, a-posteriori error estimation, goal-oriented error estimation"
Finite element methods (msc 74S05), Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (msc 65N30)
Energy (theme 4)
Modelling, Analysis and Simulation [MAS]
Multiscale Dynamics

Ferracina, L. (2007). Everything you always wanted to know about a-posteriori error estimation in finite element methods, but were afraid to ask. Modelling, Analysis and Simulation [MAS]. CWI.