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 Graham et al. (SIAM J Numer Anal 58(5):2515–2543, 2020. https://doi.org/10.1137/19M1272512), 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 (to prove the stability assumption for this equation we consider the case of a coercive bilinear form, which is non-symmetric, though).

我们分析了一级重叠域分解预处理器 SORAS（对称优化受限加法施瓦茨）的收敛性，该系统应用于矩阵不一定是对称/自伴随或正定矩阵的通用线性系统。通过概括格雷厄姆等人开发的亥姆霍兹方程的理论。（SIAM J Numer Anal 58(5):2515–2543, 2020. https://doi.org/10.1137/19M1272512），我们确定了一系列假设和估计，这些假设和估计足以获得预处理矩阵，以及其值域与原点的距离的下限。我们强调我们的理论是通用的，因为它并不特定于某个特定的边界值问题。此外，它不依赖于单元足够小的粗网格。