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

Deparis, Simone ; Løvgren, A.

In: Journal of Scientific Computing, 2012, vol. 50, no. 1, p. 198-212

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    Title
    • Stabilized Reduced Basis Approximation of Incompressible Three-Dimensional Navier-Stokes Equations in Parametrized Deformed Domains
    Author
    • Deparis, Simone. CMCS, MATHICSE, EPFL, Station 8, 1015, Lausanne, Switzerland
    • Løvgren, A.. SIMULA Research Laboratory, P.O. Box 134, 1325, Lysaker, Norway
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Journal of Scientific Computing, 2012, vol. 50, no. 1, p. 198-212. Springer US; http://www.springer-ny.com
    Other Version
    OAI-PMH Identifier
    • oai:doc.rero.ch:313129
    Summary
    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 1−P 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