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

  • Yuping Zeng und Shangyou Zhang ORCID logo EMAIL logo
Veröffentlicht/Copyright: 24. Februar 2025
Journal of Numerical Mathematics
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: 

Acknowledgments

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

  1. Research Ethics: The work is original and is not published elsewhere in any form or language.

  2. Informed consent: This article does not contain any studies involving animals. This article does not contain any studies involving human participants.

  3. Author contributions: All authors made equal contribution in this work.

  4. Use of Large Language Models, AI and Machine Learning Tools: There is no use of Large Language Models, AI and Machine Learning Tools.

  5. Conflict of interest: There is no potential conflict of interest.

  6. 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).

  7. 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

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. A subspace framework for L ∞ model reduction
  3. Long time behavior of the field–road diffusion model: an entropy method and a finite volume scheme
  4. A medius error analysis for the conforming discontinuous Galerkin finite element methods
  5. A Robin–Robin strongly coupled partitioned method for fluid–poroelastic structure interaction
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

Artikel in diesem Heft

  1. Frontmatter
  2. A subspace framework for L ∞ model reduction
  3. Long time behavior of the field–road diffusion model: an entropy method and a finite volume scheme
  4. A medius error analysis for the conforming discontinuous Galerkin finite element methods
  5. A Robin–Robin strongly coupled partitioned method for fluid–poroelastic structure interaction
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