Estimating and localizing the algebraic and total numerical errors using flux reconstructions

Abstract

This paper presents a methodology for computing upper and lower bounds for both the algebraic and total errors in the context of the conforming finite element discretization and an arbitrary iterative algebraic solver. The derived bounds are based on the flux reconstruction techniques, do not contain any unspecified constants, and allow estimating the local distribution of both errors over the computational domain. We also discuss bounds on the discretization error, their application for constructing mathematically justified stopping criteria for iterative algebraic solvers, global and local efficiency of the total error upper bound, and the relationship to the previously published estimates on the algebraic error. Theoretical results are illustrated on numerical experiments for higher-order finite element approximations and the preconditioned conjugate gradient method. They in particular witness that the proposed methodology yields a tight estimate on the local distribution of the algebraic and total errors over thecomputational domain and illustrate the associate cost.