Analysis of the SORAS domain decomposition preconditioner for non-self-adjoint or indefinite problems

Marcella Bonazzoli (1) , Xavier Claeys (2, 3) , Frédéric Nataf (2, 3) , Pierre-Henri Tournier (2, 3)

1 DeFI - Shape reconstruction and identification
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 ALPINES - Algorithms and parallel tools for integrated numerical simulations

We analyze the convergence of the one-level overlapping domain decomposition preconditioner SORAS (Symmetrized Optimized Restricted Additive Schwarz) applied to a general linear system whose matrix is not necessarily symmetric/self-adjoint nor positive definite. By generalizing the theory for the Helmholtz equation developed in [I.G. Graham, E.A. Spence, and J. Zou, preprint arXiv:1806.03731, 2019], we identify a list of assumptions and estimates that are sufficient to obtain an upper bound on the norm of the preconditioned matrix, and a lower bound on the distance of its field of values from the origin. As an illustration of this framework, we prove new estimates for overlapping domain decomposition methods with Robin-type transmission conditions for the heterogeneous reaction-convection-diffusion equation.

Mots clés: Preconditioners; Field of values; Reaction-convection-diffusion equation; Non-self-adjoint problems; Domain decomposition; Indefinite problems

Domaines: Analyse numérique [math.NA]

Marcella Bonazzoli, Xavier Claeys, Frédéric Nataf, Pierre-Henri Tournier. Analysis of the SORAS domain decomposition preconditioner for non-self-adjoint or indefinite problems. 2020. ⟨hal-02513123v1⟩