GenEO谱粗空间在SPD区域分解中的应用
GenEO spectral coarse spaces in SPD domain decomposition
AI总结：
本文在抽象Schwarz框架中定义GenEO粗空间，统一并扩展其理论，证明与问题参数无关的条件数界限，并通过二维弹性问题验证。
摘要：
两级区域分解方法是预条件Krylov求解器。一级和两级区域分解方法的区别在于后者存在粗空间。抽象Schwarz框架是一种形式化方法，允许定义和研究多种两级方法。本文的目标是在抽象Schwarz框架中定义一族称为GenEO粗空间（即重叠中的广义特征值）的粗空间。具体而言，本工作是对几种方法的推广，每种方法针对特定的区域分解方法而存在。本文既统一了GenEO理论，又将其扩展到新的设置。证明基于抽象Schwarz理论，该理论现在适用于通过投影进行粗空间修正，并已扩展以考虑奇异局部求解。证明了预条件算子条件数的界限，这些界限与问题中的参数（例如底层PDE中的任何系数或子域数量）无关。通过在每个子域中寻找一些精心选择的广义特征值问题的低频或高频空间来计算粗空间。通过为二维线性弹性问题定义两级加性Schwarz、Neumann-Neumann和Inexact Schwarz预条件器来说明抽象框架。为此示例提供了显式理论界限以及数值结果。
英文摘要：
Two-level domain decomposition methods are preconditioned Krylov solvers. What separates one- and two-level domain decomposition methods is the presence of a coarse space in the latter. The abstract Schwarz framework is a formalism that allows to define and study a large variety of two-level methods. The objective of this article is to define, in the abstract Schwarz framework, a family of coarse spaces called the GenEO coarse spaces (for Generalized Eigenvalues in the Overlaps). In detail, this work is a generalization of several methods, each of which exists for a particular choice of domain decomposition method. The article both unifies the GenEO theory and extends it to new settings. The proofs are based on an abstract Schwarz theory which now applies to coarse space corrections by projection, and has been extended to consider singular local solves. Bounds for the condition numbers of the preconditioned operators are proved that are independent of the parameters in the problem (e.g., any coefficients in an underlying PDE or the number of subdomains). The coarse spaces are computed by finding low- or high-frequency spaces of some well-chosen generalized eigenvalue problems in each subdomain. The abstract framework is illustrated by defining two-level Additive Schwarz, Neumann-Neumann and Inexact Schwarz preconditioners for a two-dimensional linear elasticity problem. Explicit theoretical bounds as well as numerical results are provided for this example.
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