Local residual error estimator and adaptive finite element analysis of Poisson's problems

A local residual error estimator based on the least-square residuals of the balance equation, constitutive relation and irrotationality condition is presented to enable adaptive finite element analyses of Poisson's problems. The efficiency of the residual error indicator and the quality of the L-2 error indicator, based on the difference between recovered and direct approaches, are investigated in problems with sharp and relatively sharp concentration of gradients. Both error estimators are highly dependent on the accuracy of the post-processed derivatives. Two types of post-processing are adopted: the L-2 projection and a local macroelement technique. In the proposed macroelement post-processing technique, the derivatives are recovered by solving local variational problems involving the residuals of the balance equation, irrotationality condition and constitutive relation at special superconvergence points.