  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.
