  In this work, a local Fourier analysis is presented to study the convergence
of multigrid methods based on additive Schwarz smoothers. This analysis is
presented as a general framework which allows us to study these smoothers for
any type of discretization and problem. The presented framework is crucial in
practice since it allows one to know a priori the answer to questions such as
what is the size of the patch to use within these relaxations, the size of the
overlapping, or even the optimal values for the weights involved in the
smoother. Results are shown for a class of additive and restricted additive
Schwarz relaxations used within a multigrid framework applied to high-order
finite-element discretizations and saddle point problems, which are two of the
contexts in which these type of relaxations are widely used.
