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Mathematical analysis of robustness of two-level domain decomposition methods with respect to inexact coarse solves
Numerische Mathematik ( IF 2.1 ) Pub Date : 2020-02-05 , DOI: 10.1007/s00211-020-01102-6
Frédéric Nataf

Convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. The GenEO coarse space has been shown to lead to a robust two-level Schwarz preconditioner which scales well over multiple cores [19, 2]. The robustness is due to its good approximation properties for problems with highly heterogeneous material parameters. It is available in the finite element packages FreeFem++ [9], Feel++ [17], Dune [1] and is implemented as a standalone library in HPDDM [10] and as such is available as well as a PETSc preconditioner. But the coarse component of the preconditioner can ultimately become a bottleneck if the number of subdomains is very large and exact solves are used. It is therefore interesting to consider the effect of inexact coarse solves. In this paper, robustness of GenEO methods is analyzed with respect to inexact coarse solves. Interestingly, the GenEO-2 method introduced in [7] has to be modified in order to be able to prove its robustness in this context.

中文翻译:

两级域分解方法对不精确粗解的鲁棒性数学分析

域分解方法的收敛在很大程度上依赖于第二级使用的粗空间的效率。GenEO 粗空间已被证明可以产生一个健壮的两级 Schwarz 预处理器,它可以很好地扩展到多个内核 [19, 2]。鲁棒性是由于其对高度异质材料参数的问题具有良好的近似特性。它在有限元包 FreeFem++ [9]、Feel++ [17]、Dune [1] 中可用,并在 HPDDM [10] 中作为独立库实现,因此与 PETSc 预处理器一样可用。但是,如果子域的数量非常大并且使用精确求解,则预处理器的粗略部分最终会成为瓶颈。因此,考虑不精确粗求解的影响是很有趣的。在本文中,针对不精确的粗求解分析了 GenEO 方法的鲁棒性。有趣的是,必须修改 [7] 中介绍的 GenEO-2 方法,以便能够证明其在这种情况下的鲁棒性。
更新日期:2020-02-05
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