A posteriori goal-oriented bounds for the Poisson problem using potential and equilibrated flux reconstructions: application to the hybridizable discontinuous Galerkin method

TL;DR

This paper introduces a general framework for computing guaranteed bounds on linear functionals of the Poisson problem's solution, utilizing potential and flux reconstructions, applicable to various numerical methods including HDG, with demonstrated superconvergence.

Contribution

It develops a novel, general approach for deriving a posteriori bounds for the Poisson problem using potential and flux reconstructions, applicable to any numerical method, and demonstrates its effectiveness with HDG.

Findings

Achieves superconvergence on the bound gap.

Provides accurate bounds even on coarse meshes.

Demonstrates effectiveness through numerical experiments.

Abstract

We present a general framework to compute upper and lower bounds for linear-functional outputs of the exact solutions of the Poisson equation based on reconstructions of the field variable and flux for both the primal and adjoint problems. The method is devised from a generalization of the complementary energy principle and the duality theory. Using duality theory, the computation of bounds is reduced to finding independent potential and equilibrated flux reconstructions. A generalization of this result is also introduced, allowing to derive alternative guaranteed bounds from nearly-arbitrary H(div;{\Omega}) flux reconstructions (only zero-order equilibration is required). This approach is applicable to any numerical method used to compute the solution. In this work, the proposed approach is applied to derive bounds for the hybridizable discontinuous Galerkin (HDG) method. An attractive feature of the proposed approach is that superconvergence on the bound gap is achieved, yielding accurate bounds even for very coarse meshes. Numerical experiments are presented to illustrate the performance and convergence of the bounds for the HDG method in both uniform and adaptive mesh refinements.