The Restricted Additive Schwarz method with impedance transmission conditions, also known as the Optimised Restricted Additive Schwarz (ORAS) method, is a simple overlapping one-level parallel domain decomposition method, which has been successfully used as an iterative solver and as a preconditioner for discretized Helmholtz boundary-value problems. In this paper, we give, for the first time, a convergence analysis for ORAS as an iterative solver -- and also as a preconditioner -- for nodal finite element Helmholtz systems of any polynomial order. The analysis starts by showing (for general domain decompositions) that ORAS as an unconventional finite element approximation of a classical parallel iterative Schwarz method, formulated at the PDE (non-discrete) level. This non-discrete Schwarz method was recently analysed in [Gong, Gander, Graham, Lafontaine, Spence, arXiv 2106.05218], and the present paper gives a corresponding discrete version of this analysis. In particular, for domain decompositions in strips in 2-d, we show that, when the mesh size is small enough, ORAS inherits the convergence properties of the Schwarz method, independent of polynomial order. The proof relies on characterising the ORAS iteration in terms of discrete `impedance-to-impedance maps', which we prove (via a novel weighted finite-element error analysis) converge as $h\rightarrow 0$ in the operator norm to their non-discrete counterparts.
翻译:限制Additive Schwarz 和阻塞传输条件的方法,也称为优化限制Additive Schwarz (ORAS) 方法,是一个简单的重叠的单级平行域分解方法,它已被成功用作迭代解答器,并且是分解的 Helmholtz 边界价值问题的前提条件。在本文中,我们首次对ORAS作为迭代解答器的混合解析器 -- -- 也作为任何复合顺序的节点定要素 Helmholtz 对应方 -- -- 的前提条件。分析开始时显示(一般域解密)ORAS是传统平行迭代Schwarz 方法的非常规性定解析元素,在 PDE (非分解) 级别上,这种非分解的Schwarz 方法最近在[Gong, Gander, Graham, Graham, Lafontaine, Swarpentrial, arXiv.05218], 以及本文在本次分析中给出了一种相应的离析版本。在轨分析中,运行者已经足够分解了其缩的缩的顺序, 也就是的递缩缩缩缩缩缩缩缩缩的解, 当我们在磁系中显示了中显示了它的位置, 的缩缩的缩的缩缩缩缩缩的缩的缩的缩的缩缩缩缩缩的缩的缩缩缩的缩的顺序。