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Journal of Scientific Computing

, Volume 77, Issue 1, pp 443–466 | Cite as

Fully Computable a Posteriori Error Bounds for Hybridizable Discontinuous Galerkin Finite Element Approximations

  • Mark Ainsworth
  • Guosheng FuEmail author
Article

Abstract

We derive a posteriori error estimates for the hybridizable discontinuous Galerkin (HDG) methods, including both the primal and mixed formulations, for the approximation of a linear second-order elliptic problem on conforming simplicial meshes in two and three dimensions. We obtain fully computable, constant free, a posteriori error bounds on the broken energy seminorm and the HDG energy (semi)norm of the error. The estimators are also shown to provide local lower bounds for the HDG energy (semi)norm of the error up to a constant and a higher-order data oscillation term. For the primal HDG methods and mixed HDG methods with an appropriate choice of stabilization parameter, the estimators are also shown to provide a lower bound for the broken energy seminorm of the error up to a constant and a higher-order data oscillation term. Numerical examples are given illustrating the theoretical results.

Keywords

HDG A posteriori error analysis Computable error bounds 

Mathematics Subject Classification

65N30 65Y20 65D17 68U07 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

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