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

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. Vohralík, A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization, C. R. Math. Acad. Sci. Paris 346 (2008), 687–690] and [M. Vohralík, Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients, submitted]. The estimates presented in these references 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. 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