A unified study of continuous and discontinuous Galerkin methods

Qingguo Hong, Fei Wang, Shuonan Wu, Jinchao Xu

Research output: Contribution to journalReview article

2 Citations (Scopus)

Abstract

A unified study is presented in this paper for the design and analysis of different finite element methods (FEMs), including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs, discontinuous Galerkin (DG) methods, hybrid discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG and WG are shown to admit inf-sup conditions that hold uniformly with respect to both mesh and penalization parameters. In addition, by taking the limit of the stabilization parameters, a WG method is shown to converge to a mixed method whereas an HDG method is shown to converge to a primal method. Furthermore, a special class of DG methods, known as the mixed DG methods, is presented to fill a gap revealed in the unified framework.

Original languageEnglish (US)
JournalScience China Mathematics
Volume62
Issue number1
DOIs
StatePublished - Jan 1 2019

Fingerprint

Discontinuous Galerkin Method
Galerkin Method
Finite Element Method
Nonconforming Finite Element Method
Converge
Inf-sup Condition
Discontinuous Galerkin
Mixed Methods
Penalization
Mixed Finite Element Method
Hybrid Method
Galerkin
Stabilization
Mesh

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Hong, Qingguo ; Wang, Fei ; Wu, Shuonan ; Xu, Jinchao. / A unified study of continuous and discontinuous Galerkin methods. In: Science China Mathematics. 2019 ; Vol. 62, No. 1.
@article{f1bd1a7c18644d838881151c25c455b1,
title = "A unified study of continuous and discontinuous Galerkin methods",
abstract = "A unified study is presented in this paper for the design and analysis of different finite element methods (FEMs), including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs, discontinuous Galerkin (DG) methods, hybrid discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG and WG are shown to admit inf-sup conditions that hold uniformly with respect to both mesh and penalization parameters. In addition, by taking the limit of the stabilization parameters, a WG method is shown to converge to a mixed method whereas an HDG method is shown to converge to a primal method. Furthermore, a special class of DG methods, known as the mixed DG methods, is presented to fill a gap revealed in the unified framework.",
author = "Qingguo Hong and Fei Wang and Shuonan Wu and Jinchao Xu",
year = "2019",
month = "1",
day = "1",
doi = "10.1007/s11425-017-9341-1",
language = "English (US)",
volume = "62",
journal = "Science China Mathematics",
issn = "1674-7283",
publisher = "Science in China Press",
number = "1",

}

A unified study of continuous and discontinuous Galerkin methods. / Hong, Qingguo; Wang, Fei; Wu, Shuonan; Xu, Jinchao.

In: Science China Mathematics, Vol. 62, No. 1, 01.01.2019.

Research output: Contribution to journalReview article

TY - JOUR

T1 - A unified study of continuous and discontinuous Galerkin methods

AU - Hong, Qingguo

AU - Wang, Fei

AU - Wu, Shuonan

AU - Xu, Jinchao

PY - 2019/1/1

Y1 - 2019/1/1

N2 - A unified study is presented in this paper for the design and analysis of different finite element methods (FEMs), including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs, discontinuous Galerkin (DG) methods, hybrid discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG and WG are shown to admit inf-sup conditions that hold uniformly with respect to both mesh and penalization parameters. In addition, by taking the limit of the stabilization parameters, a WG method is shown to converge to a mixed method whereas an HDG method is shown to converge to a primal method. Furthermore, a special class of DG methods, known as the mixed DG methods, is presented to fill a gap revealed in the unified framework.

AB - A unified study is presented in this paper for the design and analysis of different finite element methods (FEMs), including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs, discontinuous Galerkin (DG) methods, hybrid discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG and WG are shown to admit inf-sup conditions that hold uniformly with respect to both mesh and penalization parameters. In addition, by taking the limit of the stabilization parameters, a WG method is shown to converge to a mixed method whereas an HDG method is shown to converge to a primal method. Furthermore, a special class of DG methods, known as the mixed DG methods, is presented to fill a gap revealed in the unified framework.

UR - http://www.scopus.com/inward/record.url?scp=85056357468&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85056357468&partnerID=8YFLogxK

U2 - 10.1007/s11425-017-9341-1

DO - 10.1007/s11425-017-9341-1

M3 - Review article

AN - SCOPUS:85056357468

VL - 62

JO - Science China Mathematics

JF - Science China Mathematics

SN - 1674-7283

IS - 1

ER -