A domain decomposition preconditioner of Neumann-Neumann type for the Stokes equations

Victorita Dolean (1) , Frédéric Nataf (2) , Gerd Rapin (3)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 LJLL - Laboratoire Jacques-Louis Lions
3 NAM - Mathematics Department [Gottingen]

In this paper we recall a new domain decomposition method for the Stokes problem obtained via the Smith factorization. From the theoretical point of view, this domain decomposition method is optimal in the sense that it converges in two iterations for a decomposition into two equal domains. Previous results illustrated the fast convergence of the proposed algorithm in some cases. Our algorithm has shown a more robust behavior than Neumann- Neumann or FETI type methods for particular decompositions; as far as general decompositions are concerned, the performances of the three algorithms are similar. Nevertheless, the computations of the singular values of the interface preconditioned problem have shown that one needs a coarse space whose dimension is less than the one needed for the Neumann-Neumann algorithm. In this work we present a new strategy in order to improve the convergence of the new algorithm in the presence of cross points.

Domaines: Analyse numérique [math.NA]

Victorita Dolean, Frédéric Nataf, Gerd Rapin. A domain decomposition preconditioner of Neumann-Neumann type for the Stokes equations. Bercovier, M., Gander, M.J., Kornhuber, R., Widlund, O. Domain decomposition methods in science and engineering XVIII, Springer Verlag, pp.161-168, 2009, LNCSE. ⟨hal-00413482⟩