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 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.
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Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An ‘empirical interpolation’ method: Application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris, Ser. I 339, 667–672 (2004)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)
Burman, E., Fernández, M.A., Hansbo, P.: Continuous interior penalty finite element method for Oseen’s equations. SIAM J. Numer. Anal. 44(3), 1248–1274 (2006)
Chen, Y., Hesthaven, J.S., Maday, Y.: A seamless reduced basis element method for 2d maxwell’s problem: An introduction. In: Barth, T.J., Griebel, M., Keyes, D.E., Nieminen, R.M., Roose, D., Schlick, T., Hesthaven, J.S., Rønquist, E.M. (eds.) Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 76, pp. 141–152. Springer, Berlin-Heidelberg (2011)
Deparis, S.: Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural norm approach. SIAM J. Numer. Anal. 46(4), 2039–2067 (2008)
Deparis, S., Rozza, G.: Reduced basis method for multi-parameter dependent steady Navier-Stokes equations: applications to natural convection in a cavity. J. Comput. Phys. 228(12), 4359–4378 (2009). EPFL-IACS report 12.2008
LifeV. http://www.lifev.org
Løvgren, A.E., Maday, Y., Rønquist, E.M.: A reduced basis element method for the steady Stokes problem. Modél. Math. Anal. Numér. 40(3), 529–552 (2006)
Løvgren, A.E., Maday, Y., Rønquist, E.M.: The reduced basis element method for fluid flows. In: Calgaro, C., Coulombel, J.-F., Goudon, T. (eds.) Analysis and Simulation of Fluid Dynamics. Advances in Mathematical Fluid Mechanics, pp. 129–154. Birkhäuser, Basel (2007)
Løvgren, A.E., Maday, Y., Rønquist, E.M.: The reduced basis element method: Offline-online decomposition in the nonconforming, nonaffine case. In: Barth, T.J., Griebel, M., Keyes, D.E., Nieminen, R.M., Roose, D., Schlick, T., Hesthaven, J.S., Rønquist, E.M. (eds.) Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 76, pp. 247–254. Springer, Berlin-Heidelberg (2011)
Maday, Y., Rønquist, E.M.: A reduced-basis element method. J. Sci. Comput. 17, 447–459 (2002)
Maday, Y., Rønquist, E.M.: The reduced-basis element method: Application to a thermal fin problem. SIAM J. Sci. Comput. 26(1), 240–258 (2004)
Noor, A.K., Peters, J.M.: Reduced basis technique for nonlinear analysis of structures. AIAA J. 18(4), 455–462 (1980)
Peterson, J.S.: The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10(4), 777–786 (1989)
Prud’homme, C., Rovas, D.V., Veroy, K., Machiels, L., Maday, Y., Patera, A.T., Turinici, G.: Reliable real-time solution of parametrized partial differential equations: Reduced basis output bound methods. J. Fluids Eng. 124, 70–80 (2002)
Rozza, G.: Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity. Appl. Numer. Math. 55, 403–424 (2005)
Rozza, G., Veroy, K.: On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Eng. 196(7), 1244–1260 (2007)
Rozza, G.: Reduced basis methods for Stokes equations in domains with non-affine parameter dependence. Comput. Vis. Sci. 12(1), 23–35 (2009)
Winkelmann, C.: Interior penalty finite element approximation of Navier-Stokes equations and application to free surface flows. Ph.D. Thesis, École Polytechnique Fédérale de Lausanne, December 2007
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Deparis, S., Løvgren, A.E. Stabilized Reduced Basis Approximation of Incompressible Three-Dimensional Navier-Stokes Equations in Parametrized Deformed Domains. J Sci Comput 50, 198–212 (2012). https://doi.org/10.1007/s10915-011-9478-2
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DOI: https://doi.org/10.1007/s10915-011-9478-2