Abstract
The present work is devoted to the a posteriori error estimation for mixed approximations of linear self-adjoint elliptic problems. New guaranteed upper and lower bounds for the error measured in the natural product norm are derived, and individual sharp upper bounds are obtained for approximation errors in each of the physical variables. All estimates are reliable and valid for any approximate solution from the class of admissible functions. The estimates contain only global constants depending solely on the domain geometry and the given operators. Moreover, it is shown that, after an appropriate scaling of the coordinates and the equation, the ratio of the upper and lower bounds for the error in the product norm never exceeds 3. The possible methods of finding the approximate mixed solution in the class of admissible functions are discussed. The estimates are computationally very cheap and can also be used for the indication of the local error distribution. As applications, the diffusion problem as well as the problem of linear elasticity are considered.
Repin, S