Core Concepts
The paper proposes a novel universal construction of two-level overlapping Schwarz preconditioners for 2mth-order elliptic boundary value problems, where m is a positive integer. The coarse space construction can be applied to any finite element discretization for any m that satisfies some common assumptions, ensuring scalable convergence rates.
Abstract
The paper introduces two-level overlapping Schwarz preconditioners for solving 2mth-order elliptic boundary value problems, where m is a positive integer. The key aspect is the construction of a universal coarse space that can be applied to any finite element discretization for any m, without depending on the specific discretization method.
The main highlights are:
The coarse space is constructed using conforming finite element spaces defined on a coarse mesh, with a universal approach that is independent of both m and the type of finite element discretization.
The proposed preconditioners ensure uniformly bounded condition numbers of the preconditioned operators when the ratio of subdomain size to overlap is fixed, regardless of the problem size.
Numerical results are presented for fourth- and sixth-order problems discretized using various conforming, nonconforming, and discontinuous Galerkin-type finite elements, demonstrating the scalability of the proposed preconditioners.