杂交间断有限元方法多边形网格上解二阶椭圆问题  

A Hybridizable Discontinuous Finite Element Method for Solving Second Order Elliptic Problems on Polygonal Meshes

作　　者：张玲丽 王慧娟 徐世鹏 Zhang Lingli;Wang Huijuan;Xu Shipeng(School of Mathematics and Computer Science,Jiangxi Science and Technology Normal University,Nanchang 330038,Jiangxi,P.R.China)

机构地区：[1]江西科技师范大学数学与计算机科学学院,江西南昌330038

出　　处：《江西科技师范大学学报》2022年第6期106-111,123,共7页Journal of Jiangxi Science & Technology Normal University

基　　金：江西省教育厅项目(GJJ201124);江西科技师范大学博士启动基金项目(2018BSQD009)。

摘　　要：本文提出一种杂交间断有限元方法,用于解二阶椭圆问题。首先通过网格单元内部的自由度以及单元边界上的自由度,重构一个新的多项式基函数,然后采用间断有限元加罚方法进行加罚以保证数值格式的稳定性。该方法不仅能够降低间断伽辽金方法的计算量,而且能够适应多边形网格。同时,本文进行了理论分析,首先证明了能量范数下的误差估计,然后通过Aubin-Nitsche论证给出了L2范数下的误差估计。结果表明,两种范数下的收敛阶均是最优的。In this paper, a hybridizable discontinuous finite element method(Hybridizable FEM) is proposed to solve second-order elliptic problems. Firstly, a new polynomial basis function is reconstructed through the degrees of freedom inside the grid elements and the degrees of freedom on the element boundaries, and then the discontinuous finite element penalty method is used to ensure the stability of the numerical scheme. This method can not only reduce the computational complexity of the discontinuous Galerkin method, but also adapt to the polygonal meshes. At the same time, this paper makes a theoretical analysis, first proves the error estimation under the energy norm, and then gives the error estimation under the L~2norm through Aubin-Nitsche arguement. The results show that both the orders of convergence under the two norms are optimal.

关 键 词：弱伽辽金 弱梯度 杂交有限元方法 椭圆问题 误差估计 多边形网格 

分 类 号：O242.21[理学—计算数学]