  Two-level domain decomposition (DD) methods are very powerful techniques for
the efficient numerical solution of partial differential equations (PDEs). A
two-level domain decomposition method requires two main components: a one-level
preconditioner (or its corresponding smoothing iterative method), which is
based on domain decomposition techniques, and a coarse correction step, which
relies on a coarse space. The coarse space must properly represent the error
components that the chosen one-level method is not capable to deal with. In the
literature most of the works introduced efficient coarse spaces obtained as the
span of functions defined on the entire space domain of the considered PDE.
Therefore, the corresponding two-level preconditioners and iterative methods
are defined in volume.
  In this paper, a new class of substructured two-level methods is
introduced,for which both domain decomposition smoothers and coarse correction
steps are defined on the interfaces (or skeletons). This approach has several
advantages. On the one hand, the required computational effort is cheaper than
the one required by classical volumetric two-level methods. On the other hand,
it allows one to use some of the well-known efficient coarse spaces proposed in
the literature. While analyzing in detail the new substructured methods, we
present a new convergence analysis for two-level iterative methods, which
covers the proposed substructured framework. Further, we study the asymptotic
optimality of coarse spaces both theoretically and numerically using deep
neural networks. Numerical experiments demonstrate the effectiveness of the
proposed new numerical framework.
