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Parallel fully coupled Schwarz preconditioners for saddle point problems. (English) Zbl 1112.65026

Summary: We study some parallel overlapping Schwarz preconditioners for solving Stokes-like problems arising from finite element discretization of incompressible flow problems. Most of the existing methods are based on the splitting of the velocity and pressure variables. With the splitting, fast solution methods are often constructed using various fast Poisson solvers for one of the variables. More recently, several papers have investigated the so-catted fully coupled approaches in which the variables are not separated. The fully coupled approach has some advantages over the variable splitting method when solving Stokes-like equations with many variables, where a good splitting may be hard to obtain.
In this paper we systematically study the parallel scalability of several versions of the fully coupled Schwarz method for both symmetric and nonsymmetric Stokes-like problems. We show numerically that the performance of a two-level method with a multiplicative iterative coarse solver is superior to the other variants of Schwarz preconditioners.

MSC:

65F10 Iterative numerical methods for linear systems
65Y05 Parallel numerical computation
76D07 Stokes and related (Oseen, etc.) flows
65F30 Other matrix algorithms (MSC2010)
35Q30 Navier-Stokes equations
76M10 Finite element methods applied to problems in fluid mechanics
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

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