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

Abstract

We present a cost e#ective method for computing quantitative upper and lower bounds on linear functional outputs of exact weak solutions to the advection-di#usion-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 -norm of the error in the finite element solution