For many problems and many discretizations, a priori analysis allows to state that, if the number of degrees of freedom is large enough, the approximation of the solution will be good enough. The problem that has been the subject ofmany research and the subject of the current worshop is to guess and certify the right size of discrete spaces for a given

accuracy. Among the possible results that one want to certify lies some outputs, computed from the discrete solution of partial differential system. We define and explain how to get reliable and computable bounds on these outputs. The problem will consider include the Stokes and Navier Stokes equations in fluid dynamics and the Hartree Fock problem in quantum chemistry. The discretization we shall present will be either of classical finite element type but also of reduced basis type that, coupled with the error bounds can become reliable and serious alternative to more standard discretization methods. This work has been done through collaborations more particularly with A. T. Patera, E. M. Ronquist, D. Rovas and G. Turinici