TABLE OF CONTENTS The a posteriori error estimators in the preceding section provide a bound of the error up to a generic constant; cf. Theorem 8.1. The theorem of Prager and Synge (Theorem 5.1) admits the computation of an error bound without such a generic constant. The essential idea is that a comparison of an approximate solution of the primal variational problem with a feasible function of the dual mixed problem (cf. (5.5) ?) yields an estimate. An elaborated theory was provided by Neittaanm aki and Repin [2004]. Since Theorem 5.1 looks like Pythagoras rule in an infinite dimensional space, the term hypercircle method is frequently found.
We consider the Poisson equation (Example II.2.10) as the simplest case. Given a finite element solution u h of the primal problem, for applying Theorem 5.1 an auxiliary function ? ? H(div) with
is required. Following Braess and Sch berl [2006] we demonstrate that such a function ? can be constructed by the solution of cheap local problems based on the knowledge of u h.
Let 
be a triangulation of a polygonal domain ? ? 
2. A crucial step is the evaluation of the error with respect to the solution for the differential equation with a right-hand side f h that is piecewise constant on the triangulation. We know from (8.8) that there is only an extra term ch f ? f h if we approximate...
