  In this paper we introduce a new Schwarz framework and theory, based on the
well-known idea of space decomposition, for nonsymmetric and indefinite linear
systems arising from continuous and discontinuous Galerkin approximations of
general nonsymmetric and indefinite elliptic partial differential equations.
The proposed Schwarz framework and theory are presented in a variational
setting in Banach spaces instead of Hilbert spaces which is the case for the
well-known symmetric and positive definite (SPD) Schwarz framework and theory.
Condition number estimates for the additive and hybrid Schwarz preconditioners
are established. The main idea of our nonsymmetric and indefinite Schwarz
framework and theory is to use weak coercivity (satisfied by the nonsymmetric
and indefinite bilinear form) induced norms to replace the standard bilinear
form induced norm in the SPD Schwarz framework and theory. Applications of the
proposed nonsymmetric and indefinite Schwarz framework and theory. Applications
of the proposed nonsymmetric and indefinite Schwarz framework to solutions of
discontinuous Galerkin approximations of convection-diffusion problems are also
discussed. Extensive 1-D numerical experiments are also provided to gauge the
performance of the proposed Schwarz methods.
