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An overlapping Schwarz method for spectral element solution of the incompressible Navier-Stokes equations. (English) Zbl 0904.76057
This paper considers problems encountered in large-scale spectral element simulations of unsteady incompressible flows. An accurate simulation of even two-dimensional flows can require hundreds of thousands of grid points if the Reynolds number is of the order of $$10^4$$. The author has developed an additive overlapping Schwarz preconditioner for the computationally challenging pressure operator which arises when an Uzawa decoupling procedure is applied to the $$P_N- P_{N-2}$$ spectral element formulation of the incompressible Navier-Stokes equations. The pressure preconditioner is derived from local finite element Laplacians based on a triangulation of the Gauss (pressure) points, coupled with a global coarse grid operator of the spectral element vertices. The Schwarz procedure yields significantly improved convergence rates over previously employed deflation/block-Jacobi-based schemes. The overall Navier-Stokes solution times for several production runs have been reduced by a factor of 5 with the development of this preconditioner.
Reviewer: J.Siekmann (Essen)

##### MSC:
 76M25 Other numerical methods (fluid mechanics) (MSC2010) 76D05 Navier-Stokes equations for incompressible viscous fluids 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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