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

doi:10.12286/jssx.j2021-0792

计算数学 ›› 2022, Vol. 44 ›› Issue (3): 396-421.DOI: 10.12286/jssx.j2021-0792

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

刘怡, 汪艳秋

南京师范大学数学科学学院, 南京 210023

收稿日期:2021-03-24 出版日期:2022-07-14 发布日期:2022-08-03

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刘怡, 汪艳秋. 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[J]. 计算数学, 2022, 44(3): 396-421.

Liu Yi, Wang Yanqiu. POLYGONAL DISCONTINUOUS GALERKIN METHODS ON CURVED REGIONS AND ITS MULTIGRID PRECONDITIONER[J]. Mathematica Numerica Sinica, 2022, 44(3): 396-421.

POLYGONAL DISCONTINUOUS GALERKIN METHODS ON CURVED REGIONS AND ITS MULTIGRID PRECONDITIONER

Liu Yi, Wang Yanqiu

School of Mathematics Science, Nanjing Normal University, Nanjing 210023, China

Received:2021-03-24 Online:2022-07-14 Published:2022-08-03

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\linebreakW-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.

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

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.

Key words: curved boundary, curved interface, polygonal mesh, interior penalty discontinuous Galerkin method, multigrid

MR(2010)主题分类:

65N30
65N55

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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

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