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

• Published in: Journal of scientific computing 2012, Vol.50 (1), p.198-212
• Main Authors: Deparis, Simone, Løvgren, A. Emil
• Format: Article
• Language: English
• Subjects: Algorithms; Approximation; Computational Mathematics and Numerical Analysis; Finite element method; Fluid flow; Incompressible flow; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Navier-Stokes equations; Parameters; Pipes; Theoretical
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-011-9478-2

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 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.