Two‐Sided A Posteriori Error Estimates for Mixed Formulations of Elliptic Problems

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 the 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 the applications, the diffusion problem as well as the problem of linear elasticity are considered.

A Posteriori Error Estimates for Finite Volume Approximations

“…Similar result for mixed approximations of elliptic type boundary-value problems was established in [28]. Moreover, the relations (4.14) and (4.15) show that if the error is controlled in the norm |||[ (u − u h , p − q h ) ]||| then the efficiency index is always between 1 and 3.…”

“…To measure the quality of the obtained approximate solutions we apply the so-called functional a posteriori estimates (see [24,25,26,27,28]; a consequent exposition of the theory is presented in [17]). These estimates were derived by purely functional arguments without attracting Galerkin orthogonality or some other special properties of an approximate solution.…”

“…The case of diffusion-convection-reaction equations is postponed to a forthcoming paper and will be based on some ideas developed in [20]. The key idea is to obtain error bounds in a functional framework as presented in [17] (see also [24,25,26,27,28]). This point of view yields error majorants in a general framework.…”

Combineda posteriorimodeling-discretization error estimate for elliptic problems with complicated interfaces

“…Our method differs from these approaches and its derivation is based on our previous publications (see [19][20][21][22][23][24][25][26][27][28][29][30][31]), in which estimates of the difference between the exact solution of boundary value problems and arbitrary functions from the corresponding energy space has been derived by purely functional methods without requiring specific information on the approximating subspace and the numerical method used. As a result, the estimates contain no mesh dependent constants and are valid for any conforming approximation from the respective energy space.…”

“…As a result, the estimates contain no mesh dependent constants and are valid for any conforming approximation from the respective energy space. In the papers [30,31], these properties have been used for the analysis of modeling errors.…”