TY - JOUR
T1 - An extended Galerkin analysis for elliptic problems
AU - Hong, Qingguo
AU - Wu, Shuonan
AU - Xu, Jinchao
N1 - Funding Information:
This work was supported by Center for Computational Mathematics and Applications, The Pennsylvania State University. The second author was supported by National Natural Science Foundation of China (Grant No. 11901016) and the startup grant from Peking University. The third author was supported by National Science Foundation of USA (Grant No. DMS-1522615).
Publisher Copyright:
© 2020, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2020
Y1 - 2020
N2 - A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For the second-order elliptic equation −div(α∇u) = f, this framework employs four different discretization variables, uh, ph, ŭh and p⌣ h, where uh and ph are for approximation of u and p = −α∇u inside each element, and ŭh and p⌣ h are for approximation of residual of u and p · n on the boundary of each element. The resulting 4-field discretization is proved to satisfy two types of inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, many existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.
AB - A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For the second-order elliptic equation −div(α∇u) = f, this framework employs four different discretization variables, uh, ph, ŭh and p⌣ h, where uh and ph are for approximation of u and p = −α∇u inside each element, and ŭh and p⌣ h are for approximation of residual of u and p · n on the boundary of each element. The resulting 4-field discretization is proved to satisfy two types of inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, many existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.
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U2 - 10.1007/s11425-019-1809-7
DO - 10.1007/s11425-019-1809-7
M3 - Article
AN - SCOPUS:85096954897
JO - Science China Mathematics
JF - Science China Mathematics
SN - 1674-7283
ER -