Explicit upper bounds for dual norms of residuals

We derive upper bounds for the dual norms of residuals that are explicit in terms of local Poincaré constants. Residuals are continuous linear functionals that are orthogonal to a finite element space and have a singular part supported on the skeleton of the underlying mesh. Functionals of this type play a key role in a posteriori error estimation. Our main tools are a discrete partition of unity and suitably weighted trace and Poincare' inequalities. The technique is illustrated for negative first order Sobolev norms and a dual norm arising in convection-reaction-diffusion problems.