A medius error analysis for the conforming discontinuous Galerkin finite element methods

Yuping Zeng; Shangyou Zhang

doi:10.1515/jnma-2024-0005

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A medius error analysis for the conforming discontinuous Galerkin finite element methods

Yuping Zeng und Shangyou Zhang

Veröffentlicht/Copyright: 24. Februar 2025

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Aus der Zeitschrift Journal of Numerical Mathematics Band 33 Heft 3

Abstract

In this paper, we derive an improved error estimate of a conforming discontinuous Galerkin (CDG) method for both second and fourth order elliptic problems, assuming only minimal regularity on the exact solutions. The result we established is called a medius error estimate since it relies on both a priori and a posteriori analysis. Compared with the standard interior penalty discontinuous Galerkin (IPDG) method, when choosing the standard DG norm, an additional term ‖∇u − ∇_w v _h‖₀ is incorporated in the CDG formulation for second order elliptic equation, while for the case of fourth order equation, this term becomes ‖ Δ u − Δ w v h ‖ 0 . These terms disappear if we directly analyze the nonstandard error formulations ‖∇u − ∇_w u _h‖₀ and ‖ Δ u − Δ w u h ‖ 0 . Extensive numerical results are also carried out to validate our theoretical findings.

Keywords: conforming discontinuous Galerkin method; elliptic problems; second and fourth order; minimal regularity; a medius error estimate

MSC 2010 Classification: 65N15; 65N30

Corresponding author: Shangyou Zhang, Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA, E-mail: szhang@udel.edu

Acknowledgments

The authors thank three anonymous referees for their valuable comments and suggestions which helped to improve the quality of this article.

Research Ethics: The work is original and is not published elsewhere in any form or language.
Informed consent: This article does not contain any studies involving animals. This article does not contain any studies involving human participants.
Author contributions: All authors made equal contribution in this work.
Use of Large Language Models, AI and Machine Learning Tools: There is no use of Large Language Models, AI and Machine Learning Tools.
Conflict of interest: There is no potential conflict of interest.
Research funding: This work is supported by Guangdong Basic and Applied Basic Research Foundation (No. 2020A1515011032) and National Natural Science Foundation of China (No. 11526097).
Data availability: This research does not use any external or author-collected data.

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Received: 2024-01-10

Accepted: 2024-11-04

Published Online: 2025-02-24

Published in Print: 2025-09-25

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Artikel in diesem Heft

https://doi.org/10.1515/jnma-2024-0005

Schlagwörter für diesen Artikel

conforming discontinuous Galerkin method; elliptic problems; second and fourth order; minimal regularity; a medius error estimate