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Superconvergence of Direct Discontinuous Galerkin Methods: Eigen-structure Analysis Based on Fourier Approach

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Abstract

This paper investigates superconvergence properties of the direct discontinuous Galerkin (DDG) method with interface corrections and the symmetric DDG method for diffusion equations. We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes. Based on the eigen-structure analysis, we carry out error estimates of the DDG solutions, which can be decomposed into three parts: (i) dissipation errors of the physically relevant eigenvalue, which grow linearly with the time and are of order 2k for \(P^k\ (k=2,3)\) approximations; (ii) projection error from a special projection of the exact solution, which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue; (iii) dissipative errors of non-physically relevant eigenvalues, which decay exponentially with respect to the spatial mesh size \(\Delta {x}\). We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree \(P^2\) approximations, but are not for odd degree \(P^3\) approximations. Numerical experiments are provided to verify the theoretical results.

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Acknowledgements

Research work of H. Wang is partially supported by the National Natural Science Foundation of China (Grant Nos. 11871428 and 12071214), and the Natural Science Foundation for Colleges and Universities of Jiangsu Province of China (Grant No. 20KJB110011). Research work of J. Yan is supported by the National Science Foundation (Grant No. DMS-1620335) and the Simons Foundation (Grant No. 637716). Research work of X. Zhong is partially supported by the National Natural Science Foundation of China (Grant Nos. 11871428 and 12272347).

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Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, Zhejiang, China

    Xuechun Liu & **nghui Zhong

  2. School of Science, Nan**g University of Posts and Telecommunications, Nan**g, 210023, Jiangsu, China

    Hai** Wang

  3. Department of Mathematics, Iowa State University, Ames, 50011, IA, USA

    Jue Yan

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  1. Xuechun Liu
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Correspondence to **nghui Zhong.

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This paper is dedicated to the memory of Professor Ching-Shan Chou.

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Liu, X., Wang, H., Yan, J. et al. Superconvergence of Direct Discontinuous Galerkin Methods: Eigen-structure Analysis Based on Fourier Approach. Commun. Appl. Math. Comput. 6, 257–278 (2024). https://doi.org/10.1007/s42967-022-00246-6

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  • DOI: https://doi.org/10.1007/s42967-022-00246-6

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