We analyze the convergence of the one-level overlapping domain decomposition preconditioner SORAS (Symmetrized Optimized Restricted Additive Schwarz) applied to a generic linear system whose matrix is not necessarily symmetric/self-adjoint nor positive definite. By generalizing the theory for the Helmholtz equation developed in [I.G. Graham, E.A. Spence, and J. Zou, SIAM J.Numer.Anal., 2020], we identify a list of assumptions and estimates that are sufficient to obtain an upper bound on the norm of the preconditioned matrix, and a lower bound on the distance of its field of values from the origin. We stress that our theory is general in the sense that it is not specific to one particular boundary value problem. Moreover, it does not rely on a coarse mesh whose elements are sufficiently small. As an illustration of this framework, we prove new estimates for overlapping domain decomposition methods with Robin-type transmission conditions for the heterogeneous reaction-convection-diffusion equation.
翻译:我们分析了单层重叠的域分解先决条件SORAS(优化优化限制Additive Schwarz)的趋同性,它适用于一个通用线性系统,其矩阵不一定对称/自我连接,也不确定。我们通过概括[I.G. Graham, E.A.Spence, 和J.Zou, SIAM J.Numer.Anal., 2020] 所开发的赫尔姆霍尔茨方程式的理论,我们确定了一个假设和估计清单,这些假设和估计足以获得对先决条件矩阵规范的上限,而对于其数值领域与源的距离,限制较低。我们强调,我们的理论是一般性的,因为它不是某个特定的边界价值问题所特有的。此外,它并不依赖其元素足够小的粗糙的中间线。作为这个框架的例证,我们证明,对于与罗宾型的分解传输条件相重叠的域方法与多种反应-凝固变异方形等的新的估计。