Two-level domain decomposition methods are preconditioned Krylov solvers. What separates one and two-level domain decomposition method 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). This is a generalization of existing methods for particular choices of domain decomposition methods. 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.
翻译:两种水平域分解法是Krylov解析法的先决条件。 区分一种和两种水平域分解法的方法是后者中存在粗糙的空间。 抽象的Schwarz框架是一种形式主义,它允许定义和研究大量的两种层次的方法。 本条的目的是在抽象的Schwarz框架内,定义一个叫做GenEO 共解空间的粗糙空间组( 重叠时通用电子价值组) 。 这是对特定域分解方法选择的现有方法的概括化。 前提条件操作者条件号的界点被证明独立于问题参数( 例如, 基础PDE中的任何系数或子域数) 。 粗略空间的计算方法是在每个子域中找到一些精心选择的通用电子值问题的低或高频空间。