Guaranteed energy error bounds for the Poisson equation using a flux-free approach: solving the local problems in subdomains

Abstract

A method to compute guaranteed upper bounds for the energy norm of the exact error in the finite element solution of the Poisson equation is presented. The bounds are guaranteed for any finite element mesh however coarse it may be, not just in the asymptotic regime. The bounds are constructed by employing a subdomain based a posteriori error estimate which yields self-equilibrated residual loads in stars (patches of elements). The proposed approach is an alternative to standard equilibrated residual methods providing sharper bounds. The use of a flux-free error estimator improves the effectivities of the upper bounds for the energy while retaining the certainty of the bounds.