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Description

In typical flow problems, engineers are rarely interested in the entire field solution; only some selected characteristic metrics---or outputs---of the system are relevant. Moreover, the ultimate interest of the analysis is not the forward problem but the design problem. This design problem is solved by repeated appeals to the forward problem; consequently, the numerical solution method for the partial differential equations must be sufficiently inexpensive to permit numerous evaluations, yet sufficiently fine to demonstrably represent the true performance of the system.

We propose a new finite element {em a posteriori} error control strategy which reconciles these two conflicting requirements. The technique provides lower and upper bounds for the output of interest that are inexpensive to compute, rigorous, quantitative, and sharp; furthermore, the bound gap permits a local (elemental) decomposition suitable for adaptive subsequent refinements. The method considerably generalizes earlier techniques in that we obtain quantitative constant-free bounds---contrary to earlier explicit techniques---for the output of interest---contrary to earlier implicit techniques.

The procedure may be viewed as an implicit Aubin-Nitsche construction. The computation is initiated by two global solves on a coarse mesh---one for the initial (primal) problem, and one for the adjoint (dual) problem; subsequent fine mesh {em local} projections eliminate the indefinite terms associated with the incompressibility constraint. Finally, a ``classical" hybridization technique permits to compute the estimators in terms of solutions of {em local} Neumann subproblems. Under a weak hypothesis, related to the relative magnitude of the $L^2$ and $H^1$ errors of the solution, we can prove the convergence of the lower and upper estimators to the true output from below and above.

To illustrate the capabilities of the technique, we consider the problem of free thermal convection in a complex enclosure; the outputs of interest are the mean temperature over parts of the domain boundary, and the kinetic energy of the flow. The methodology can be applied to a variety of other situations, including elasticity, the Helmholtz equation, eigenvalue problems, and time-dependent (parabolic) problems; an example of the latter is briefly described.