We propose a novel universal construction of two-level overlapping Schwarz preconditioners for $2m$th-order elliptic boundary value problems, where $m$ is a positive integer. The word "universal" here signifies that the coarse space construction can be applied to any finite element discretization for any $m$ that satisfies some common assumptions. We present numerical results for conforming, nonconforming, and discontinuous Galerkin-type finite element discretizations for high-order problems to demonstrate the scalability of the proposed two-level overlapping Schwarz preconditioners.

翻译：针对$2m$阶椭圆边值问题（其中$m$为正整数），本文提出一种新颖的通用两水平重叠Schwarz预条件子构造方法。此处"通用"意指该粗空间构造方法可应用于任意满足若干常见假设的$m$值对应的有限元离散格式。我们针对高阶问题的协调、非协调及间断Galerkin型有限元离散格式给出了数值结果，以证明所提出的两水平重叠Schwarz预条件子具有可扩展性。