Numerische Mathematik

, Volume 138, Issue 3, pp 681–721 | Cite as

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

  • J. Papež
  • Z. Strakoš
  • M. Vohralík


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 \({\mathbf {H}}(\mathrm{div},\varOmega )\), whereas the lower algebraic and total error bounds rely on locally constructed \(H^1_0(\varOmega )\)-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.


Numerical solution of partial differential equations Finite element method A posteriori error estimation Algebraic error Discretization error Stopping criteria Spatial distribution of the error 

Mathematics Subject Classification

65N15 65N30 76M10 65N22 65F10 



The authors wish to thank Ivana Pultarová, in particular for pointing out to us the inequality (5.9) including its proof. The authors are also grateful to anonymous referees for their numerous helpful comments.


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Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  2. 2.Institute of Computer ScienceCzech Academy of SciencesPragueCzech Republic
  3. 3.Inria Paris, 2 rue Simone IffParisFrance
  4. 4.Université Paris-Est, CERMICS (ENPC)Marne-la-Vallée 2France

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