Computing bounds for linear functionals of exact weak solutions to Poisson's equation

Abstract

We present a method for Poisson’s equation that computes guaranteed upper and lower bounds for the values of piecewise-polynomial linear functional outputs of the exact weak solution of the infinite-dimensional continuum problem with piecewise-polynomial forcing. The method results from exploiting the Lagrangian saddle point property engendered by recasting the output problem as a constrained minimization problem. Localization is achieved by Lagrangian relaxation and the bounds are computed by appeal to a local dual problem. The proposed method computes approximate Lagrange multipliers using traditional finite element approximations to calculate a primal and an adjoint solution along with well known hybridization techniques to calculate interelement continuity multipliers. The computed bounds hold uniformly for any level of refinement, and in the asymptotic convergence regime of the finite element method, the bound gap decreases at twice the rate of the energy norm measure of the error in the finite element solution. Given a finite element solution and its output adjoint solution, the method can be used to provide a certificate of precision for the output with an asymptotic complexity that is linear in the number of elements in the finite element discretization. The elemental contributions to the bound gap are always positive and hence lend themselves to be used as adaptive indicators, as we demonstrate with a numerical example.