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We investigate in this paper improvements of the a posteriori error estimates in the finite element method discretization of the Poisson equation, introduced in (M. Vohral´ok, A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization, to appear in C. R. Math. Acad. Sci. Paris) and (M. Vohral´ok, Guaranteed and fully ro- bust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients, submitted). The estimates presented in these r are guaranteed in the sense that they feature no undetermined constants and fully computable but numerical experiments show that the effectivity index, i.e., the ratio of the estimated and actual error, does not approach the optimal value of one but rather a slightly bigger value. We identify in this paper the reason for this and introduce a possible remedy, which consists in performing a local minimization of the values of the estimators over patches of simplicial submesh elements sharing a given vertex. We then present a set of numerical experiments showing the improvements achieved and compare our estimators, both theoretically and numerically, with the classical residual ones. Resume. Estimations d'erreur a posteriori calculables pour la methode desements finis basees sur la conservativite locale de cette methode : amemployant une min- imisation locale. Nousdans cet article des ameliorations des estimations a posteriori pour la discretisation de l'´equation de Poisson par la methode deselements finis, introduites dans (M. Vohral´ok, A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization, ` paraˆotre dans C. R. Math. Acad. Sci. Paris) et (M. Vohral´ok, Guaran- teed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients, soumis). Les estimations presentees dans ces references sont garanties au sens ou elles ne comportent pas de constantes indeterminees et sont entierement calculables, mais des essais numeriques montrent que l'indice d'efficacite,donne par le rapport entre l'erreur estimee et l'erreur exacte, n'approche pas la valeur optimale de un mais une valeur legerement plusee. Dans cet article, nous en identifions la raison et nous introduisons une correction basee sur une minimisation locale des valeurs des estimateurs sur des ensembles de triangles ou de tetraedres d'un sous-maillage qui partagent un sommet donne. Nous presentons ensuite une serie d'essais numeriques et comparons, tant du point de vue theorique que numerique, nos estimations avec les estimations classiques par residu.

Computable a posteriori error estimates in the finite element method based on its local conservativity: improvements using local minimization