A new equilibrated residual method improving accuracy and efficiency of flux-free error estimates

== Abstract ==

This paper presents a new methodology to compute guaranteed upper bounds for the energy norm of the error in the context of linear finite element approximations of the reaction–diffusion equation. The new approach revisits the ideas in Parés et al. (2009) [6, 4], with the goal of substantially reducing the computational cost of the ''flux-free'' method while retaining the good quality of the bounds. The new methodology provides also a technique to compute equilibrated boundary tractions improving the quality of standard equilibration strategies. The ''zeroth-order'' equilibration conditions are imposed using an alternative less restrictive form of the ''first-order'' equilibration conditions, along with a new efficient minimization criterion. This new equilibration strategy provides much more accurate upper bounds for the energy and requires only doubling the dimension of the local linear systems of equations to be solved.

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