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

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 of the Poisson model problem and an arbitrary iterative algebraic solver. The derived bounds do not contain any unspecified constants and allow estimating the local distribution of both errors over the computational domain. Combining these bounds, we also obtain guaranteed upper and lower bounds on the discretization error. This allows to propose novel mathematically justified stopping criteria for iterative algebraic solvers ensuring that the algebraic error will lie below the discretization one. Our upper algebraic and total error bounds are based on locally reconstructed fluxes in $${\mathbf {H}}(\mathrm{div},\varOmega )$$H(div,Ω), whereas the lower algebraic and total error bounds rely on locally constructed $$H^1_0(\varOmega )$$H01(Ω)-liftings of the algebraic and total residuals. We prove global and local efficiency of the upper bound on the total error and its robustness with respect to the approximation polynomial degree. Relationships to the previously published estimates on the algebraic error are discussed. 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 the computational domain and illustrate the associated cost.