IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v220y2024icp445-461.html
   My bibliography  Save this article

The uniform convergence of a weak Galerkin finite element method in the balanced norm for reaction–diffusion equation

Author

Listed:
  • Tao, Xia
  • Hao, Jiaxiong
  • Zhang, Yu

Abstract

In this paper, a weak Galerkin finite element method is implemented to solve the one-dimensional singularly perturbed reaction–diffusion equation. This weak Galerkin finite element scheme uses piecewise polynomial with degree k≥1 in the interior part of each element and piecewise constant function at the nodes of each element. The existence and uniqueness of the weak Galerkin finite element solution are proved. Based on the interpolation operator and the corresponding approximation properties, an ɛ−uniform error bound of O(N−(k+1)+ɛ(N−1lnN)k) in the energy-like norm is investigated rigorously. Furthermore, an ɛ−uniform error bound of O(N−(k+1/2)+(N−1lnN)k) in the balanced norm is established by the weighted local L2 projection and its corresponding approximation properties. Finally, numerical experiments validate the theoretical results. Moreover, numerical results show that this weak Galerkin finite element solution achieves the convergence rate of O((N−1lnN)k+1) in the L2 norm and the discrete L∞ norm uniformly with respect to the singular perturbation parameter ɛ.

Suggested Citation

  • Tao, Xia & Hao, Jiaxiong & Zhang, Yu, 2024. "The uniform convergence of a weak Galerkin finite element method in the balanced norm for reaction–diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 445-461.
    • Handle: RePEc:eee:matcom:v:220:y:2024:i:c:p:445-461
    DOI: 10.1016/j.matcom.2024.02.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475424000405
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2024.02.001?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matcom:v:220:y:2024:i:c:p:445-461. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.