Journal of Scientific Computing

, Volume 64, Issue 1, pp 1–34 | Cite as

Algebraic and Discretization Error Estimation by Equilibrated Fluxes for Discontinuous Galerkin Methods on Nonmatching Grids

  • Vít Dolejší
  • Ivana Šebestová
  • Martin Vohralík


We derive a posteriori error estimates for the discontinuous Galerkin method applied to the Poisson equation. We allow for a variable polynomial degree and simplicial meshes with hanging nodes and propose an approach allowing for simple (nonconforming) flux reconstructions in such a setting. We take into account the algebraic error stemming from the inexact solution of the associated linear systems and propose local stopping criteria for iterative algebraic solvers. An algebraic error flux reconstruction is introduced in this respect. Guaranteed reliability and local efficiency are proven. We next propose an adaptive strategy combining both adaptive mesh refinement and adaptive stopping criteria. At last, we detail a form of the estimates avoiding any practical reconstruction of a flux and only working with the approximate solution, which simplifies greatly their evaluation. Numerical experiments illustrate a tight control of the overall error, good prediction of the distribution of both the discretization and algebraic error components, and efficiency of the adaptive strategy.


Linear diffusion problems Discontinuous Galerkin method A posteriori error estimate Flux reconstruction  Stopping criteria Algebraic error 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Vít Dolejší
    • 1
  • Ivana Šebestová
    • 1
  • Martin Vohralík
    • 2
  1. 1.Faculty of Mathematics and PhysicsCharles University in PraguePraha 8Czech Republic
  2. 2.INRIA Paris-RocquencourtLe ChesnayFrance

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