• linear-functional outputs;
  • exact/guaranteed/strict bounds;
  • stabilization methods;
  • error estimation;
  • goal-oriented adaptivity;
  • advection–diffusion–reaction equation


The paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection–diffusion–reaction equation. The bounds are computed using implicit a posteriori error estimators from stabilized finite element approximations of the exact solution. The new methodology extends the a posteriori error estimates yielding bounds for the standard Galerkin formulation to be able to obtain bounds for stabilized formulations. This methodology is combined with both hybrid-flux and flux-free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the flux-free technique. Copyright © 2012 John Wiley & Sons, Ltd.