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Accuracy-Enhancement of Discontinuous Galerkin Methods for PDEs Containing High Order Spatial Derivatives

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Abstract

In this paper, we consider the accuracy-enhancement of discontinuous Galerkin (DG) methods for solving partial differential equations (PDEs) with high order spatial derivatives. It is well known that there are highly oscillatory errors for finite element approximations to PDEs that contain hidden superconvergence points. To exploit this information, a Smoothness-Increasing Accuracy-Conserving (SIAC) filter is used to create a superconvergence filtered solution. This is accomplished by convolving the DG approximation against a B-spline kernel. Previous theoretical results about this technique concentrated on first- and second-order equations. However, for linear higher order equations, Yan and Shu (J Sci Comput 17:27–47, 2002) numerically demonstrated that it is possible to improve the accuracy order to \(2k+1\) for local discontinuous Galerkin (LDG) solutions using the SIAC filter. In this work, we firstly provide the theoretical proof for this observation. Furthermore, we prove the accuracy order of the ultra-weak local discontinuous Galerkin (UWLDG) solutions could be improved to \(2k+2-m\) using the SIAC filter, where \(m=[\frac{n}{2}]\), n is the order of PDEs. Finally, we computationally demonstrate that for nonlinear higher order PDEs, we can also obtain a superconvergence approximation with the accuracy order of \(2k+1\) or \(2k+2-m\) by convolving the LDG solution and the UWLDG solution against the SIAC filter, respectively.

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Data Availability

EThe datasets generated during the current study are available from the corresponding author on reasonable request. They support our published claims and comply with field standards.

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Funding

Research of Qi Tao is supported by the fellowship of China Postdoctoral Science Foundation, No: 2020TQ0030, 2021M700357. Research of Jennifer K. Ryan partially supported by Air Force Office of Scientific Research (AFOSR), Computational Mathematics program (Program Manager: Dr. Fariba Fahroo), under Grant Number FA 9550-20-1-0166 and the U.S. National Science Foundation under Grant Number DMS-2110745. Research of Yan Xu is supported by NSFC Grant No. 12071455.

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Authors and Affiliations

  1. Faculty of Science, Beijing University of Technology, Beijing, 100124, China

    Qi Tao

  2. Maths Department, Milton Keynes College, Milton Keynes, UK

    Liangyue Ji

  3. Department of Mathematics, KTH Royal Institute of Technology in Stockholm, Stockholm, Sweden

    Jennifer K. Ryan

  4. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Yan Xu

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  1. Qi Tao
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Correspondence to Yan Xu.

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Tao, Q., Ji, L., Ryan, J.K. et al. Accuracy-Enhancement of Discontinuous Galerkin Methods for PDEs Containing High Order Spatial Derivatives. J Sci Comput 93, 13 (2022). https://doi.org/10.1007/s10915-022-01967-9

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  • DOI: https://doi.org/10.1007/s10915-022-01967-9

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