  We analyze the convergence of the one-level overlapping domain decomposition
preconditioner SORAS (Symmetrized Optimized Restricted Additive Schwarz)
applied to a generic linear system whose matrix is not necessarily
symmetric/self-adjoint nor positive definite. By generalizing the theory for
the Helmholtz equation developed in [I.G. Graham, E.A. Spence, and J. Zou, SIAM
J.Numer.Anal., 2020], we identify a list of assumptions and estimates that are
sufficient to obtain an upper bound on the norm of the preconditioned matrix,
and a lower bound on the distance of its field of values from the origin. We
stress that our theory is general in the sense that it is not specific to one
particular boundary value problem. Moreover, it does not rely on a coarse mesh
whose elements are sufficiently small. As an illustration of this framework, we
prove new estimates for overlapping domain decomposition methods with
Robin-type transmission conditions for the heterogeneous
reaction-convection-diffusion equation (to prove the stability assumption for
this equation we consider the case of a coercive bilinear form, which is
non-symmetric, though).
