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A preconditioner for linearized Navier-Stokes problem in exterior domains

Delphine Jennequin 1
1 SIMPAF - SImulations and Modeling for PArticles and Fluids
LPP - Laboratoire Paul Painlevé - UMR 8524, Inria Lille - Nord Europe
Abstract : We aim to approach the solution of the stationary incompressible Navier-Stokes equations in a three-dimensional exterior domain. Therefore, we cut the exterior domain by a sphere of radius $R$ and we impose some suitable approximate boundary conditions (ABC) to the truncation boundary of the computational domain: the minimal requirement of these conditions is to ensure the solvability of the truncated system and the decay of the truncation error if $R$ grows. We associate to the truncated problem a mesh made of homothetic layers, called exponential mesh, such that the number of degrees of freedom only grows logarithmically with $R$ and such that the optimal error estimate holds. In order to reduce the storage, we are interested in discretizations by equal-order velocity-pressure finite elements with additional stabilization terms. Therefore, the linearisation inside a quasi-Newton or fixed-point method leads to a generalized saddle-npoint problem, that may be solved by a Krylov method applied on the preconditioned complete system matrix. We introduce a bloc-triangular preconditioner such that the decay rate of the Krylov method is independent of the mesh size $h$ and we give an estimate for this rate in function of the truncation radius and of the Reynolds number. Some three-dimensional numerical results well confirm the theory and show the robustness of our method.
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https://hal.inria.fr/inria-00070165
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Submitted on : Friday, May 19, 2006 - 7:18:32 PM
Last modification on : Friday, November 27, 2020 - 2:18:02 PM
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Delphine Jennequin. A preconditioner for linearized Navier-Stokes problem in exterior domains. [Research Report] RR-5861, INRIA. 2006, pp.22. ⟨inria-00070165⟩

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