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：我们针对$2m$阶椭圆边值问题提出了一种新颖的通用双层重叠Schwarz预处理器构造方法，其中$m$为正整数。此处的"通用"意指粗空间构造可应用于满足某些常见假设的、针对任意$m$的任何有限元离散化方案。我们通过高阶问题的协调、非协调及间断Galerkin型有限元离散化的数值算例，验证了所提出的双层重叠Schwarz预处理器的可扩展性。