Computing Bounds for Linear Functionals of Exact Weak Solutions to the Advection-Diffusion-Reaction Equation

We present a cost effective method for computing quantitative upper and lower bounds on linear functional outputs of exact weak solutions to the advection-diffusion-reaction equation and we demonstrate a simple adaptive strategy by which such outputs can be computed to a prescribed precision. The bounds are computed from independent local subproblems resulting from a standard finite element approximation to the problem. At the heart of the method lies a local dual problem by which we transform an infinite dimensional minimization problem into a finite dimensional feasibility problem. The bounds hold for all levels of refinement on polygonal domains with piecewise polynomial forcing, and the bound gap converges at twice the rate of the H1-norm of the error in the finite element solution. AMS subject classifications. 65N15, 65N30, 49M29