Geometric multigrid algorithms for elliptic interface problems using structured grids
Geometric multigrid algorithms for elliptic interface problems using structured grids
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DOI：
10.1007/s11075-018-0544-9
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发表时间：
2018-06
影响因子：
2.1
通讯作者：
Gwanghyun Jo;D. Kwak
中科院分区：
文献类型：
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作者：
Gwanghyun Jo;D. Kwak
在这项工作中，我们开发了几何多重网格算法的浸入式有限元方法的椭圆问题的接口（周等。33，149-168 2010; Kwak和Lee，Int. J. Pure Appl.Math.104，471-494 2015; Li等人，Numer. 96，61-98 2003，2004; Lin等人，SIAM J. Numer. Anal.53，1121-1144 2015）。我们需要仔细设计各级之间的传递算子，因为更精细的网格问题的残差一旦投影到更粗糙的网格上就不满足通量条件。因此，我们必须修改投影残差，以便满足通量条件。同样，延长后必须修改校正。本文提出了两种算法：一种适用于具有顶点自由度的有限元空间，另一种适用于边平均自由度。对于第二种情况，我们使用了用于P1情形的一致子空间校正的思想（Lee 1993）。数值实验表明，最佳的可扩展性方面的算术运算，即，for-cycle和用-cycle预处理的CG算法。在循环中，我们只使用了一个高斯-赛德尔平滑。还报告了CPU时间。
In this work, we develop geometric multigrid algorithms for the immersed finite element methods for elliptic problems with interface (Chou et al. Adv. Comput. Math.33, 149–168 2010; Kwak and Lee, Int. J. Pure Appl. Math.104, 471–494 2015; Li et al. Numer. Math.96, 61–98 2003, 2004; Lin et al. SIAM J. Numer. Anal.53, 1121–1144 2015). We need to design the transfer operators between levels carefully, since the residuals of finer grid problems do not satisfy the flux condition once projected onto coarser grids. Hence, we have to modify the projected residuals so that the flux conditions are satisfied. Similarly, the correction has to be modified after prolongation. Two algorithms are suggested: one for finite element spaces having vertex degrees of freedom and the other for edge average degrees of freedom. For the second case, we use the idea of conforming subspace correction used forP1nonconforming case (Lee 1993). Numerical experiments show the optimal scalability in terms of number of arithmetic operations, i.e.,for-cycle and CG algorithms preconditioned with-cycle. In-cycle, we used only one Gauss-Seidel smoothing. The CPU times are also reported.