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Journal of Scientific Computing

, Volume 50, Issue 1, pp 198–212 | Cite as

Stabilized Reduced Basis Approximation of Incompressible Three-Dimensional Navier-Stokes Equations in Parametrized Deformed Domains

  • Simone Deparis
  • A. Emil Løvgren
Article

Abstract

In this work we are interested in the numerical solution of the steady incompressible Navier-Stokes equations for fluid flow in pipes with varying curvatures and cross-sections. We intend to compute a reduced basis approximation of the solution, employing the geometry as a parameter in the reduced basis method. This has previously been done in a spectral element \(P_{{ \mathcal{N}}} - P_{{ \mathcal{N}}-2}\) setting in two dimensions for the steady Stokes equations. To compute the necessary basis-functions in the reduced basis method, we propose to use a stabilized P 1P 1 finite element method for solving the Navier-Stokes equations on different geometries. By employing the same stabilization in the reduced basis approximation, we avoid having to enrich the velocity basis in order to satisfy the inf-sup condition. This reduces the complexity of the reduced basis method for the Navier-Stokes problem, while keeping its good approximation properties. We prove the well posedness of the reduced problem and present numerical results for selected parameter dependent three dimensional pipes.

Keywords

Reduced basis methods Steady incompressible Navier-Stokes equations Stabilization 

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.CMCS, MATHICSEEPFLLausanneSwitzerland
  2. 2.SIMULA Research LaboratoryLysakerNorway

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