Output bounds for reduced-order approximations of elliptic partial differential equations

We present an a posteriori finite element procedure that provides inexpensive, rigorous, accurate, and constant-free lower and upper bounds for the error in the outputs – engineering quantities of interest – predicted by (Lagrangian) reduced-order approximations to coercive elliptic partial differential equations. The bound calculation requires (i) the reduced-order approximation for the primal and dual field variables, (ii) a lower bound for the minimum eigenvalue of the symmetric part of the operator, and (iii) the solution of purely local symmetric Neumann subproblems defined on small decoupled nodal overlapping patches. There are two critical components to the Neumann subproblems: a partition-of-unity attenuated local residual which eliminates hybrid fluxes from both the construction and analysis of the resulting estimators while simultaneously preserving the global bound property; and a L 2-regularization term which provides stability despite the absence of local nodal equilibrium of the reduced-order primal and dual solutions. The estimator bounding property and optimal convergence rate (as the reduced-order basis is enriched) are proven, and corroborating numerical results are presented for two examples: a heat conduction fin (symmetric) problem; and a conjugate advection-diffusion/multi-material heat transfer (non-symmetric) problem.