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

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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.

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, 182 George St, Providence, RI, 02912, USA

    Mark Ainsworth & Guosheng Fu

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  1. Mark Ainsworth
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  2. Guosheng Fu
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Correspondence to Guosheng Fu.

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First author gratefully acknowledges the partial support of this work under AFOSR Contract FA9550-12-1-0399.

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Ainsworth, M., Fu, G. Fully Computable a Posteriori Error Bounds for Hybridizable Discontinuous Galerkin Finite Element Approximations. J Sci Comput 77, 443–466 (2018). https://doi.org/10.1007/s10915-018-0715-9

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  • DOI: https://doi.org/10.1007/s10915-018-0715-9

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