Identity for deviations from the exact solution of the problem $\Lambda^*\mathcal{A}\Lambda u+l=0$ and its implications

For elliptic equations of the form $\Lambda^*\mathcal{A}\Lambda u+l=0$, we examine how to compute the distance between the function $u$ and its arbitrary approximation $v$ from the corresponding energy space. The analysis is based on an identity that holds for the norms of the deviations from the exact solution of this problem and the exact solution of the dual problem. This identity has a number of implications. Specifically, with the help of it, the maximum and minimum distances to the exact solution can be estimated using only the given approximate solution, the data of the problem, and the solution of a specially constructed finite-dimensional problem. Moreover, there is no need to use Clément’s interpolation or flux equilibration. It is shown that the estimates are equivalent to corresponding norms of the distance to the solution and are applicable to a large class of approximations, including Galerkin ones and rather rough approximations of the exact solution. These results are checked using a series of numerical experiments that compare the efficiency of various methods.