  We demonstrate that a small modification of the multiplicative, additive and
restricted additive Schwarz preconditioner at the algebraic level, motivated by
optimized Schwarz methods defined at the continuous level, leads to a
significant reduction in the iteration count of the iterative solver. Numerical
experiments using finite difference and spectral element discretizations of the
positive definite Helmholtz problem and an idealized climate simulation
illustrate the effectiveness of the new approach.
