A Posteriori Error Estimates of Approximate Solutions to Elliptic Boundary Value Problems in Terms of Local Norms and Cost Functionals

Functional relations are obtained for estimating the accuracy of approximate solutions in terms of measures differing significantly from energy norms usually used for this purpose. In particular, they are applicable to local norms and measures based on specially constructed linear functionals. The need for such error control tools arises if there is special interest in the solution behavior within a certain subdomain or in special properties of the solution. It is shown that a posteriori estimates of functional type used previously for global estimates can be adapted for the solution of this problem. Functional identities and estimates are obtained for estimating the error of any conformal approximation in terms of a wide class of measures, including local norms and goal-oriented functionals. The theoretical results are verified using a series of examples confirming the efficiency of the proposed method.