  Schwarz methods are attractive parallel solvers for large scale linear
systems obtained when partial differential equations are discretized. For
hybridizable discontinuous Galerkin (HDG) methods, this is a relatively new
field of research, because HDG methods impose continuity across elements using
a Robin condition, while classical Schwarz solvers use Dirichlet transmission
conditions. Robin conditions are used in optimized Schwarz methods to get
faster convergence compared to classical Schwarz methods, and this even without
overlap, when the Robin parameter is well chosen. We present in this paper a
rigorous convergence analysis of Schwarz methods for the concrete case of
hybridizable interior penalty (IPH) method. We show that the penalization
parameter needed for convergence of IPH leads to slow convergence of the
classical additive Schwarz method, and propose a modified solver which leads to
much faster convergence. Our analysis is entirely at the discrete level, and
thus holds for arbitrary interfaces between two subdomains. We then generalize
the method to the case of many subdomains, including cross points, and obtain a
new class of preconditioners for Krylov subspace methods which exhibit better
convergence properties than the classical additive Schwarz preconditioner. We
illustrate our results with numerical experiments.
