On the analysis of block smoothers for saddle point problems
For researchers working on multigrid methods for saddle point problems, this work offers a theoretical unification and a new smoother, though it is incremental in nature.
The paper introduces a new symmetric Uzawa smoother for saddle point problems and provides a unified analysis of smoothing properties for several Uzawa variants, demonstrating their effectiveness on the Stokes problem with numerical examples.
In this article, we discuss several classes of Uzawa smoothers for the application in multigrid methods in the context of saddle point problems. Beside commonly used variants, such as the inexact and block factorization version, we also introduce a new symmetric method, belonging to the class of Uzawa smoothers. For these variants we unify the analysis of the smoothing properties, which is an important part in the multigrid convergence theory. These methods are applied to the Stokes problem for which all smoothers are implemented as pointwise relaxation methods. Several numerical examples illustrate the theoretical results.