Guaranteed and locally computable a posteriori error estimate

A new approach, based on the combination of the equilibrated residual method and the method of hypercircle, is proposed for a posteriori error estimation. Computer implementation of the equilibrated residual method is fast, but it does not produce guaranteed estimates. On the other hand, the method of hypercircle delivers guaranteed estimates, but it is not fast because it involves solving a global linear algebraic system. The combination of these two methods leads to guaranteed and locally computable a posteriori error estimator. This combined method is applied to linear elliptic problem in two dimensions with mixed boundary conditions and non-negative absolute terms.