Guaranteed and robust error bounds for nonconforming approximations of elliptic problems

We present guaranteed, robust and computable a posteriori error estimates for nonconforming approximations of elliptic problems. Our analysis is based on a Helmholtz-type decomposition of the error expressed in terms of fluxes. Such a decomposition results in a gradient term and a divergence-free term, that are the exact solutions of two auxiliary problems. We suggest a new approach to deriving computable two-sided bounds of the norms of these solutions. The a posteriori estimates obtained in this paper differ from those that are based on projections of nonconforming approximations to a conforming space. Numerical experiments confirm that these new estimates provide very accurate error bounds, and can be efficiently exploited in practical computations.