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

Abstract

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 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 of 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 ${\cal H}^1$-norm of the error in the finite element solution. The bounds are valid for any choice of model problem parameters for which an equilibrating approximate solution can befound, but they become increasingly pessimistic as the parameters tend toward the advection-dominated or reaction-dominated limits.