  Schwarz methods are attractive parallel solution techniques for solving
large-scale linear systems obtained from discretizations of partial
differential equations (PDEs). Due to the iterative nature of Schwarz methods,
convergence rates are an important criterion to quantify their performance.
Optimized Schwarz methods (OSM) form a class of Schwarz methods that are
designed to achieve faster convergence rates by employing optimized
transmission conditions between subdomains. It has been shown recently that for
a two-subdomain case, OSM is a natural solver for hybridizable discontinuous
Galerkin (HDG) discretizations of elliptic PDEs. In this paper, we generalize
the preceding result to the many-subdomain case and obtain sharp convergence
rates with respect to the mesh size and polynomial degree, the subdomain
diameter, and the zeroth-order term of the underlying PDE, which allows us for
the first time to give precise convergence estimates for OSM used to solve
parabolic problems by implicit time stepping. We illustrate our theoretical
results with numerical experiments.
