曲边区域上的多边形网格间断有限元离散及其多重网格算法  

POLYGONAL DISCONTINUOUS GALERKIN METHODS ON CURVED REGIONS AND ITS MULTIGRID PRECONDITIONER

作　　者：刘怡 汪艳秋 Liu Yi;Wang Yanqiu(School of Mathematics Science,Nanjing Normal University,Nanjing 210023,China)

机构地区：[1]南京师范大学数学科学学院,南京210023

出　　处：《计算数学》2022年第3期396-421,共26页Mathematica Numerica Sinica

摘　　要：本文利用多边形网格上的间断有限元方法离散二阶椭圆方程,在曲边区域上,采用多条直短边逼近曲边的以直代曲的策略,实现了高阶元在能量范数下的最优收敛.本文还将这一方法用于带曲边界面问题的求解,同样得到高阶元的最优收敛.此外我们还设计并分析了这一方法的W-cycle和Variable V-cycle多重网格预条件方法,证明当光滑次数足够多时,多重网格预条件算法一致收敛.最后给出了数值算例,证实该算法的可行性并验证了理论分析的结果.The main purpose of this paper is to study the discontinuous Galerkin discretization of second-order elliptic partial differential equations on curved regions.We use multiple short edges to approximate the curved boundary and achieve the optimal convergence order in H1 norm for high-order elements.This method is also applied to the interface problem with curved interfaces and obtains the optimal convergence of high-order elements.Furthermore,we prove that the W-cycle and V-cycle multigrid preconditioners converge uniformly provided that the number of smoothing is sufficiently large.Finally,numerical results are presented to verify the correctness of the theoretical results.

关 键 词：曲边边界 曲界面 多边形网格 间断有限元方法 多重网格算法 

分 类 号：O241.82[理学—计算数学]