A HIERARCHICAL DUALITY APPROACH TO BOUNDS FOR THE OUTPUTS OF PARTIAL-DIFFERENTIAL EQUATIONS

We present a technique for generating lower and upper bounds to output s which are linear functionals of the solutions to (finite-element dis cretizations of) symmetric or nonsymmetric coercive linear partial dif ferential equations. The method is based upon the construction of an a ugmented Lagrangian which integrates (i) a quadratic 'energy' reformul ation of the desired output as the objective to be minimized, with (ii ) the finite-element equilibrium equations and (conforming) 'hybridize d' intersubdomain continuity conditions as the constraints to be satis fied. The bounds are then derived by appealing to the associated dual unconstrained max min problem evaluated for optimally chosen candidate Lagrange multipliers generated by a less expensive approximation, suc h as a low-dimensional finite-element discretization. As in many a pos teriori error estimation techniques, the bound calculation requires on ly the solution of subdomain-local symmetric (Neumann) problems on the refined 'truth' mesh. The technique is presented and illustrated for the case of the one-dimensional convection-diffusion equation. (C) 199 8 Elsevier Science S.A.