“A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems”

Authors: Alf E. Løvgren, Yvon Maday and Einar M. Rønquist,
Affiliation: NTNU and Université Pierre et Marie Curie
Reference: 2006, Vol 27, No 2, pp. 79-94.

Keywords: Stokes flow, reduced basis, reduced order model, domain decomposition, mortar method, output bounds, a posteriori error estimators

Abstract: The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations (Fink and Rheinboldt (1983), Noor and Peters (1980), Prud'homme et al. (2002)), and its applications extend to, for example, control and optimization problems. The basic idea is to first decompose the computational domain into a series of subdomains that are similar to a few reference domains (or generic computational parts). Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation, but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the reference domain for the part to the actual domain. We extend earlier work (Maday and Rønquist (2002), Maday and Rønquist (2004)) in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to constructing the basis functions, to the mapping of the velocity fields, to satisfying the inf-sup condition, and to ´gluing´ the local solutions together in the multidomain case (Belgacem et al. (2000)). We also demonstrate an algorithm for choosing the most efficient precomputed solutions. Two-dimensional examples are presented for pipes, bifurcations, and couplings of pipes and bifurcations in order to simulate hierarchical flow systems.

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DOI forward links to this article:
[1] G. Rozza, D. B. P. Huynh and A. T. Patera (2007), doi:10.1007/BF03024948
[2] G. Rozza, D. B. P. Huynh and A. T. Patera (2008), doi:10.1007/s11831-008-9019-9
[3] Otmane Souhar and Christophe Prud homme (2014), doi:10.1007/s10825-014-0630-8
[4] Paolo Pacciarini, Paola Gervasio and Alfio Quarteroni (2016), doi:10.1016/j.camwa.2016.01.030
[5] Jan S. Hesthaven, Gianluigi Rozza and Benjamin Stamm (2016), doi:10.1007/978-3-319-22470-1_1
[6] Paola F. Antonietti, Paolo Pacciarini and Alfio Quarteroni (2016), doi:10.1051/m2an/2015045
[1] ARIS, R. (1989). Vectors, Tensors and the Basic Equations of Fluid Mechanics, Dover Publications.
[2] BABUSKA, I. (1971). Error-bounds for finite element method, Numer. Math., 16, pp. 322-333 doi:10.1007/BF02165003
[3] BARRAULT, M., MADAY, Y., NGUYEN, N. C. PATERA, A. T. (2004). An empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Acad. Sci. Paris, Serie I, 339, pp. 667-672.
[4] BELGACEM, B. F., BERNARDI, C., CHORFI, N. MADAY, Y. (2000). Inf-sup conditions for the mortar spectral element discretization of the Stokes problem, Numer. Math., 85, pp. 257-281 doi:10.1007/PL00005388
[5] BERKOOZ, G., HOLMES, P. LUMLEY, J. L. (1993). The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech., 25, pp. 539-575 doi:10.1146/annurev.fl.25.010193.002543
[6] BREZZI, F. (1974). On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrange multipliers, Rairo Anal. Numer., 8 R2, pp. 129-151.
[7] BURKHARDT, J., GUNZBURGER, M. LEE, H. (2005). Centroidal Voronoi tessellations-based reduced-order modeling of complex systems, Submitted to SIAM J. Sci. Comput.
[8] FINK, J.P. RHEINBOLDT, W. C. (1983). On the error behavior of the reduced basis technique in nonlinear finite element approximations, Z. Angew. Math. Mech., 63, pp. 21-28 doi:10.1002/zamm.19830630105
[9] LØVGREN, A. E., MADAY, Y. RØNQUIST, E. M. (2004). A reduced basis element method for the steady Stokes problem, Submitted to M2AN.
[10] LØVGREN, A. E., MADAY, Y. RØNQUIST, E. M. (2005). A reduced basis element method for the steady Stokes problem: Application to flow in bifurcations, In progress, to be submitted in June.
[11] MACHIELS, L., MADAY, Y., OLIVERA, I. B., PATERA, A. T. ROVAS, D. V. (2000). Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, C. R. Acad. Sci. Paris, Serie I, 331, pp. 153-158.
[12] MADAY, Y. PATERA, A. T. (1989). Spectral element methods for the Navier-Stokes equations, In Noor A..ed State of the Art Surveys in Computational Mechanics, pp. 71-143.
[13] MADAY, Y., PATERA, A. T. RØNQUIST, E. M. (1992). The PNxPN-2 method for the approximation of the Stokes problem, Technical Report No. 92009, Department of Mechanical Engineering, Massachusetts Institute of Technology.
[14] MADAY, Y. RØNQUIST, E. M. (2002). A reduced-basis element method, J. Sci. Comput., 17, pp. 447-459 doi:10.1023/A:1015197908587
[15] MADAY, Y. RØNQUIST, E. M. (2004). The reduced-basis element method: Application to a thermal fin problem, SIAM J. Sci. Comput., 2.1, pp. 240-258 doi:10.1137/S1064827502419932
[16] NOOR, A. K. PETERS, J. M. (1980). Reduced basis technique for nonlinear analysis of structures, AIAA J., 1.4, pp. 455-462 doi:10.2514/3.50778
[17] PRUD'HOMME, C., ROVAS, D. V., VEROY, K., MACHIELS, L., MADAY, Y., PATERA, A. T. TURINICI, G. (2002). Reliable real-time solution of parametrized partial differential equations: Reduced basis output bound methods, J. Fluids Engineering, 124, pp. 70-80 doi:10.1115/1.1448332
[18] ROVAS, D. V. (2002). Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations, PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, October.
[19] VEROY, K., PRUD'HOMME, C., ROVAS, D. V. PATERA, A.T. (2003). A Posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, AIAA Paper 2003-3847. In Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, June.

  title={{A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems}},
  author={Løvgren, Alf E. and Maday, Yvon and Rønquist, Einar M.},
  journal={Modeling, Identification and Control},
  publisher={Norwegian Society of Automatic Control}