Local error estimation in FEM for an optimal recovery problem

In this talk we consider optimal recovery of the solution to an elliptic boundary value problem.  It is assumed that boundary data are unknown.  Compensating information is provided in the form of a finite number of measurements of the solution, but even with this additional information solution of this problem is not unique.  We will discuss an optimal recovery framework for determining a "best possible" version of the solution and define its finite element approximation. We then present error estimates which explain how placement of the measurement points in the interior of the domain versus on the boundary can lead to better approximation of the solution.