A hierarchical duality approach to bounds for the outputs of partial differential equations Original Research Article

We present a technique for generating lower and upper bounds to outputs which are linear functionals of the solutions to (finiteelement discretizations of) symmetric or nonsymmetric coercive linear partial differential equations. The method is based upon the construction of an augmented Lagrangian which integrates (i) a quadratic ‘energy’ reformulation of the desired output as the objective to be minimized, with (ii) the finite-element equilibrium equations and (conforming) ‘hybridized’ intersubdomain continuity conditions as the constraints to be satisfied. 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, such as a low-dimensional finite-element discretization. As in many a posteriori error estimation techniques, the bound calculation requires only 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.