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Volume 17, Issue 2
A Conforming Discontinuous Galerkin Finite Element Method: Part II

Xiu Ye & Shangyou Zhang

Int. J. Numer. Anal. Mod., 17 (2020), pp. 281-296.

Published online: 2020-02

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  • Abstract

A conforming discontinuous Galerkin (DG) finite element method has been introduced in [19] on simplicial meshes, which has the flexibility of using discontinuous approximation and the simplicity in formulation of the classic continuous finite element method. The goal of this paper is to extend the conforming DG finite element method in [19] so that it can work on general polytopal meshes by designing weak gradient ∇$w$ appropriately. Two different conforming DG formulations on polytopal meshes are introduced which handle boundary conditions differently. Error estimates of optimal order are established for the corresponding conforming DG approximation in both a discrete $H$1 norm and the $L$2 norm. Numerical results are presented to confirm the theory.

  • Keywords

Weak Galerkin, discontinuous Galerkin, stabilizer/penalty free, finite element methods, second order elliptic problem.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

xxye@ualr.edu (Xiu Ye)

szhang@udel.edu (Shangyou Zhang)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-17-281, author = {Xiu and Ye and xxye@ualr.edu and 13554 and Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA. and Xiu Ye and Shangyou and Zhang and szhang@udel.edu and 5543 and Department of Mathematics Science, University of Delaware, Newark 19716, USA and Shangyou Zhang}, title = {A Conforming Discontinuous Galerkin Finite Element Method: Part II}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2020}, volume = {17}, number = {2}, pages = {281--296}, abstract = {

A conforming discontinuous Galerkin (DG) finite element method has been introduced in [19] on simplicial meshes, which has the flexibility of using discontinuous approximation and the simplicity in formulation of the classic continuous finite element method. The goal of this paper is to extend the conforming DG finite element method in [19] so that it can work on general polytopal meshes by designing weak gradient ∇$w$ appropriately. Two different conforming DG formulations on polytopal meshes are introduced which handle boundary conditions differently. Error estimates of optimal order are established for the corresponding conforming DG approximation in both a discrete $H$1 norm and the $L$2 norm. Numerical results are presented to confirm the theory.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13651.html} }
TY - JOUR T1 - A Conforming Discontinuous Galerkin Finite Element Method: Part II AU - Ye , Xiu AU - Zhang , Shangyou JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 281 EP - 296 PY - 2020 DA - 2020/02 SN - 17 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/13651.html KW - Weak Galerkin, discontinuous Galerkin, stabilizer/penalty free, finite element methods, second order elliptic problem. AB -

A conforming discontinuous Galerkin (DG) finite element method has been introduced in [19] on simplicial meshes, which has the flexibility of using discontinuous approximation and the simplicity in formulation of the classic continuous finite element method. The goal of this paper is to extend the conforming DG finite element method in [19] so that it can work on general polytopal meshes by designing weak gradient ∇$w$ appropriately. Two different conforming DG formulations on polytopal meshes are introduced which handle boundary conditions differently. Error estimates of optimal order are established for the corresponding conforming DG approximation in both a discrete $H$1 norm and the $L$2 norm. Numerical results are presented to confirm the theory.

Xiu Ye & Shangyou Zhang. (2020). A Conforming Discontinuous Galerkin Finite Element Method: Part II. International Journal of Numerical Analysis and Modeling. 17 (2). 281-296. doi:
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