| Peer-Reviewed

Comparative Study of Weak Galerkin and Discontinuous Galerkin Finite Element Methods

Received: 24 December 2021     Accepted: 23 April 2022     Published: 31 May 2022
Views:       Downloads:
Abstract

FEM is a valuable approximation tool for the solution of Partial Differential Equations when the analytical solutions are difficult or impossible to obtain due to complicated geometry or boundary conditions. The Project work involved collecting facts related to WG and DG‑FEMs. WG‑FEM is a numerical method that was first proposed and analyzed by Wang and Ye (2013) for general second‑order elliptic BVPs on triangular and rectangular meshes. DG‑FEMs as developed by Cockburn et al. (1970) uses a discontinuous function space to approximate the exact solution of the equations. The comparison and numerical examples demonstrated that WG‑FEMs are viable and hold some advantages over DG‑FEMs, due to their properties. Numerical examples demonstrated that WGM generates a smaller linear system to solve than the DGMs. WG‑FEM have less unknowns, no need for choosing penalty factor and normal flux is continuous across element interfaces compared to DG‑FEMs and the implementation of WG‑FEMs is easier than that of DG‑FEMs based on error and convergence rate. The computations were done by hand and with the help of MATLAB 2021Rb.

Published in International Journal of Systems Science and Applied Mathematics (Volume 7, Issue 2)
DOI 10.11648/j.ijssam.20220702.11
Page(s) 23-38
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2022. Published by Science Publishing Group

Keywords

Finite Element Method (FEM), Galerkin Method, Weak Galerkin FEM, Discontinuous Galerkin FEM

References
[1] Arnold, D. N. 1982: An Interior Penalty Finite Element Method with discontinuous element, SIAM J. Numer. Anal., pp. 742–760.
[2] Babuska, I and Zlamal, M. 1973: Nonconforming Elements in the Finite Element Method with Penalty. SIAM J. Num. Anal., 10: 863–875.
[3] Bahriawati, C. and Carstensen, C. 2005. Three Matlab implementations of the lowest-order Raviart-Thomas MFEM with a posterior error control, Comput Methods Appl. Math, 5: 333–361.
[4] Baker, G. A, 1977: Finite element methods for elliptic equations using nonconforming.
[5] Bang, K. and Kwon, Y. W. 2000: The Finite Element Method Using MATLAB, 2nd Ed. CRC Press, USA.
[6] Barnhill, E. and Whiteman, R. (1973). Error analysis of Finite Element Methods on triangles for elliptic boundary value problems. The Mathematics of Finite Elements and Applications, Academic Press, London 4: 83-112.
[7] Bassi, F. Rebay, S. Savini, M. Mariotti, G. and Pedinotti, S. 1997: A high order accurate Discontinuous Finite Element Method for inviscid and viscous turbomachinery flow, in Proceedings of the Second European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, Antwerpeen, Belgium.
[8] Bastian, P. and Rivere, B. 2003. Super convergence and H (div) projection for Discontinuous Galerkin Methods, Int. J. Numer. Methods Fluids, 42: 1043–1057.
[9] Becker, E. Carey, B. F and Oden, J. T. 1981: Finite Elements an Introduction, Texas Institute for Computational Mechanics, UT Austin.
[10] Brenner, S. and Scott, L. 2008. The Mathematical Theory of Finite Element Methods, third edition, Springer-verlag.
[11] Brezzi, F., Manzini G, Marini D, et al. (2000): Discontinuous Galerkin approximations for elliptic problems, Numerical Methods for partial differential equations: 365-378.
[12] Castillo, P. (2002): Performance of discontinuous Galerkin methods for elliptic PDEs, Siam Journal on Scientific Computing: 524-547.
[13] Castillo, P., 2002: Performance of discontinuous Galerkin methods for elliptic PDE’s.
[14] Cavendish, J. C., Price, H. S. and Varga, R. S. 1969. Galerkin methods for the numerical solution of boundary value problems: Petroleum Eng. Soc. Jour., June, 204-22.
[15] Cocburn, B. Karniadakis, G. and Shu, C. W. 2000. Discontinuous Galerkin Methods: Theory, Computation and Applications Lecture Notes in Computational Science and Engineering, volume 11, Springer. Proceedings of the first DG Conference.
[16] Cockburn, B. and Shu, C. (2005). Foreword for the special issue on Discontinuous Galerkin methods. J. Sci. Comp., 22: 1–3.
[17] Cockburn, B. Gopalakrishnan, J. and Wang, H. 2007. Locally conservative fluxes for the continuous Galerkin Method. SIAM Journal of Numerical Analysis, 45: 1742–1776.
[18] Courant, R. 1943. Variational methods for the solution of problems of equilibrium and vibrations Bulletin of the American Mathematics Society.
[19] Dawson, C. 2006. Foreword for the special issue on Discontinuous Galerkin Methods. Comput. Methods Appl. Mech. Engrg., 195: 3.
[20] Douglas, J. and Dupont, T. 1970. Galerkin methods for parabolic equations: Indus, and Appl. Math. Soc. Jour. Numerical Analysis, v. 7, (4): 575-626. elements, Math. Comput., pp. 45–59.
Cite This Article
  • APA Style

    Desta Sodano Sheiso. (2022). Comparative Study of Weak Galerkin and Discontinuous Galerkin Finite Element Methods. International Journal of Systems Science and Applied Mathematics, 7(2), 23-38. https://doi.org/10.11648/j.ijssam.20220702.11

    Copy | Download

    ACS Style

    Desta Sodano Sheiso. Comparative Study of Weak Galerkin and Discontinuous Galerkin Finite Element Methods. Int. J. Syst. Sci. Appl. Math. 2022, 7(2), 23-38. doi: 10.11648/j.ijssam.20220702.11

    Copy | Download

    AMA Style

    Desta Sodano Sheiso. Comparative Study of Weak Galerkin and Discontinuous Galerkin Finite Element Methods. Int J Syst Sci Appl Math. 2022;7(2):23-38. doi: 10.11648/j.ijssam.20220702.11

    Copy | Download

  • @article{10.11648/j.ijssam.20220702.11,
      author = {Desta Sodano Sheiso},
      title = {Comparative Study of Weak Galerkin and Discontinuous Galerkin Finite Element Methods},
      journal = {International Journal of Systems Science and Applied Mathematics},
      volume = {7},
      number = {2},
      pages = {23-38},
      doi = {10.11648/j.ijssam.20220702.11},
      url = {https://doi.org/10.11648/j.ijssam.20220702.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20220702.11},
      abstract = {FEM is a valuable approximation tool for the solution of Partial Differential Equations when the analytical solutions are difficult or impossible to obtain due to complicated geometry or boundary conditions. The Project work involved collecting facts related to WG and DG‑FEMs. WG‑FEM is a numerical method that was first proposed and analyzed by Wang and Ye (2013) for general second‑order elliptic BVPs on triangular and rectangular meshes. DG‑FEMs as developed by Cockburn et al. (1970) uses a discontinuous function space to approximate the exact solution of the equations. The comparison and numerical examples demonstrated that WG‑FEMs are viable and hold some advantages over DG‑FEMs, due to their properties. Numerical examples demonstrated that WGM generates a smaller linear system to solve than the DGMs. WG‑FEM have less unknowns, no need for choosing penalty factor and normal flux is continuous across element interfaces compared to DG‑FEMs and the implementation of WG‑FEMs is easier than that of DG‑FEMs based on error and convergence rate. The computations were done by hand and with the help of MATLAB 2021Rb.},
     year = {2022}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Comparative Study of Weak Galerkin and Discontinuous Galerkin Finite Element Methods
    AU  - Desta Sodano Sheiso
    Y1  - 2022/05/31
    PY  - 2022
    N1  - https://doi.org/10.11648/j.ijssam.20220702.11
    DO  - 10.11648/j.ijssam.20220702.11
    T2  - International Journal of Systems Science and Applied Mathematics
    JF  - International Journal of Systems Science and Applied Mathematics
    JO  - International Journal of Systems Science and Applied Mathematics
    SP  - 23
    EP  - 38
    PB  - Science Publishing Group
    SN  - 2575-5803
    UR  - https://doi.org/10.11648/j.ijssam.20220702.11
    AB  - FEM is a valuable approximation tool for the solution of Partial Differential Equations when the analytical solutions are difficult or impossible to obtain due to complicated geometry or boundary conditions. The Project work involved collecting facts related to WG and DG‑FEMs. WG‑FEM is a numerical method that was first proposed and analyzed by Wang and Ye (2013) for general second‑order elliptic BVPs on triangular and rectangular meshes. DG‑FEMs as developed by Cockburn et al. (1970) uses a discontinuous function space to approximate the exact solution of the equations. The comparison and numerical examples demonstrated that WG‑FEMs are viable and hold some advantages over DG‑FEMs, due to their properties. Numerical examples demonstrated that WGM generates a smaller linear system to solve than the DGMs. WG‑FEM have less unknowns, no need for choosing penalty factor and normal flux is continuous across element interfaces compared to DG‑FEMs and the implementation of WG‑FEMs is easier than that of DG‑FEMs based on error and convergence rate. The computations were done by hand and with the help of MATLAB 2021Rb.
    VL  - 7
    IS  - 2
    ER  - 

    Copy | Download

Author Information
  • Department of Mathematics, Collage of Natural and Computational Science, Wolkite University, Wolkite, Ethiopia

  • Sections