TY - JOUR
T1 - A unified study of continuous and discontinuous Galerkin methods
AU - Hong, Qingguo
AU - Wang, Fei
AU - Wu, Shuonan
AU - Xu, Jinchao
N1 - Funding Information:
Acknowledgements The last author was supported by US Department of Energy (Grant No. DE-SC-0009249), as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials, US Department of Energy (Grant No. DE-SC0014400) and National Science Foundation of USA (Grant No. DMS-1522615). The second author was partially supported by National Natural Science Foundation of China (Grant No. 11771350).
Funding Information:
The last author was supported by US Department of Energy (Grant No. DE-SC-0009249), as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials, US Department of Energy (Grant No. DE-SC0014400) and National Science Foundation of USA (Grant No. DMS-1522615). The second author was partially supported by National Natural Science Foundation of China (Grant No. 11771350).
Publisher Copyright:
© 2018, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.
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.
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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 -