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Gradient recovery based a posteriori error estimator for the adaptive direct discontinuous Galerkin method

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Abstract

In this paper, we propose a gradient recovery method for the direct discontinuous Galerkin (DDG) method. A quadratic polynomial is obtain by using the local discrete least-squares fitting to the gradient of numerical solution at certain sampling points. The recovered gradient is defined on a piecewise continuous space, and it may be discontinuous on the whole domain. Based on the recovered gradient, we introduce a posteriori error estimator which takes the \(L^2\) norm of the difference between the direct and post-processed approximations. Some benchmark test problems with typical difficulties are carried out to illustrate the superconvergence of the recovered gradient and validate the asymptotic exactness of the recovery-based a posteriori error estimator. Most of the test problems are from the US National Institute for Standards and Technology (NIST).

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Acknowledgements

Cao’s research was supported by Hunan Provincial Innovation Foundation for Postgraduate(CX20200619). Huang’s reserach was partially supported by NSFC Project (11971410) and China’s National Key R &D Programs (2020YFA0713500). Yi’s reserach was partially supported by NSFC Project (12071400), Hunan Provincial NSF Project (2021JJ40189).

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, People’s Republic of China

    Huihui Cao

  2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, People’s Republic of China

    Yunqing Huang & Nianyu Yi

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  1. Huihui Cao
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  2. Yunqing Huang
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Correspondence to Nianyu Yi.

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Cao, H., Huang, Y. & Yi, N. Gradient recovery based a posteriori error estimator for the adaptive direct discontinuous Galerkin method. Calcolo 60, 18 (2023). https://doi.org/10.1007/s10092-023-00513-9

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