Abstract This contribution derives guaranteed upper bounds of the energy norm of the approximation error for linear elliptic partial differential systems. We generalize the complementarity error estimates known for scalar elliptic problems to general diffusion–convection–reaction linear elliptic systems. For systems we prove analogous properties of these error bounds as for the scalar case. A brief description how the presented general theory applies to linear elasticity is included as well as an application to chemical systems with reactions of at most first order. Numerical experiments showing the sharpness of the obtained upper bounds and their behavior in the adaptive procedure are presented, too.