  This article is a review on basic concepts and tools devoted to a posteriori
error estimation for problems solved with the Finite Element Method. For the
sake of simplicity and clarity, we mostly focus on linear elliptic diffusion
problems, approximated by a conforming numerical discretization. The review
mainly aims at presenting in a unified manner a large set of powerful
verification methods, around the concept of equilibrium. Methods based on that
concept provide error bounds that are fully computable and mathematically
certified. We discuss recovery methods, residual methods, and duality-based
methods for the estimation of the whole solution error (i.e. the error in
energy norm), as well as goal-oriented error estimation (to assess the error on
specific quantities of interest). We briefly survey the possible extensions to
non-conforming numerical methods, as well as more complex (e.g. nonlinear or
time-dependent) problems. We also provide some illustrating numerical examples
on a linear elasticity problem in 3D.
