Complementarity based a posteriori error estimates and their properties

The paper is devoted to complementary approaches in a posteriori error estimation for a diffusion‐reaction model problem. These approaches provide sharp and guaranteed upper bounds for the energy norm of the error and they are independent from the way how the approximate solution is obtained. In particular, the estimator naturally includes all sources of errors of any conforming approximation, like the discretization error, the error in the solver of linear algebraic systems, the quadrature error, etc. The paper recapitulates three complementarity approaches, proves sufficient and necessary conditions for the efficiency and asymptotic exactness of the error estimators, constructs an approximation by the method of hypercircle such that its error can be computed exactly, and presents numerical tests showing robustness of these approaches.