An efficient, reliable and robust error estimator for elliptic problems in R3

In this article, we develop and analyze a hierarchical-type error estimator for a general class of second-order linear elliptic boundary value problems in bounded three-dimensional domains. This type of indicator automatically satisfies a global lower bound inequality, thereby giving efficiency, without regularity assumptions beyond those giving well-posedness of the continuous and discrete problems. The main focus of the paper is then to establish the reverse reliability result: a global upper bound on the error in terms of the error estimate (plus an oscillation term), again without additional regularity assumptions. The proof of this inequality depends on a clever choice of the space in which the error indicator lies and a moment-capturing quasi-interpolation result. We finish the article with a series of numerical experiments to illustrate the behavior predicted by the theoretical results.