摘要
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,u_(h),p_(h),u_(h)and p_(h),where u_(h)and p_(h)are for approximation of u and p=-α▽u inside each element,and u_(h)and p_(h)are for approximation of the 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.
基金
supported by Center for Computational Mathematics and Applications,The Pennsylvania State University
supported by National Natural Science Foundation of China(Grant No.11901016)
the startup grant from Peking University
supported by National Science Foundation of USA(Grant No.DMS-1522615)。
