New local and parallel finite element algorithm based on the partition of unity

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• 作者：Haibiao Zheng^a ; ^b ; ^{hbzheng13@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Feng Shi^c ; ^{shifeng@hitsz.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Yanren Hou^d ; ^{yrhou@mail.xjtu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Jianping Zhao^e ; ^{zhaojianping@126.com" class="auth_mail" title="E-mail the corresponding author} ; Yong Cao^c ; ^{yongc@hitsz.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Ren Zhao^f ; ^{renzhao@wayne.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Local and parallel ; Oversampling ; Partition of unity ; Two-grid method
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：435
• 期：1
• 页码：1-19
• 全文大小：533 K

文摘

In this study, based on a combination of the two-grid method and the partition of unity-based domain decomposition method, we propose a new local and parallel finite element algorithm for the elliptic boundary value problem. The proposed method has three key features: (1) it inherits the flexibility and controllability of domain decomposition based on the partition of unity; (2) global fine grid correction is replaced by solving a series of locally defined approximate residual problems with homogeneous Dirichlet boundary conditions on some finer grids; (3) a global continuous finite element solution is constructed by solving a coarse grid correction problem and by assembling all the local solutions together using the partition of unity subordinate. Under appropriate assumptions, the optimal error estimates in he MathML source">L²$L^2$ and the energy norms are proved by new analytical results. In addition, several numerical simulations are presented to demonstrate the high efficiency and flexibility of the new algorithm.