Summary.
In this paper we consider additive Schwarz-type iteration methods for saddle point problems as smoothers in a multigrid method. Each iteration step of the additive Schwarz method requires the solutions of several small local saddle point problems. This method can be viewed as an additive version of a (multiplicative) Vanka-type iteration, well-known as a smoother for multigrid methods in computational fluid dynamics. It is shown that, under suitable conditions, the iteration can be interpreted as a symmetric inexact Uzawa method. In the case of symmetric saddle point problems the smoothing property, an important part in a multigrid convergence proof, is analyzed for symmetric inexact Uzawa methods including the special case of the additive Schwarz-type iterations. As an example the theory is applied to the Crouzeix-Raviart mixed finite element for the Stokes equations and some numerical experiments are presented.
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Mathematics Subject Classification (1991): 65N22, 65F10, 65N30
Supported by the Austrian Science Foundation (FWF) under the grant SFB F013}\and Walter Zulehner
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Schöberl, J., Zulehner, W. On Schwarz-type Smoothers for Saddle Point Problems. Numer. Math. 95, 377–399 (2003). https://doi.org/10.1007/s00211-002-0448-3
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DOI: https://doi.org/10.1007/s00211-002-0448-3