Abstract
The reduced basis element approximation is a discretization method for solving partial differential equations that has inherited features from the domain decomposition method and the reduced basis approximation paradigm in a similar way as the spectral element method has inherited features from domain decomposition methods and spectral approximations. We present here a review of the method directed to the application of fluid flow simulations in hierarchical geometries. We present the rational and the basics of the method together with details on the implementation. We illustrate also the rapid convergence with numerical results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
B.O. Almroth, P. Stern, F.A. Brogan — Automatic choice of global shape functions in structural analysis. AIAA Journal, 16 (1978) 525–528.
L. Baffico, C. Grandmont, Y. Maday, and A. Osses — Homogenization of an elastic media with gaseous bubbles, in preparation.
F. Brezzi and M. Fortin — Mixed and Hybrid Finite Element Methods. Springer Verlag, 1991.
M. Briane, Y. Maday, and F. Madigou — Homogenization of a two-dimensional fractal conductivity, in preparation.
C. Fetita, S.M., D. Perchet, F. Prêteux, M. Thiriet, and L. Vial — An image-based computational model of oscillatory flow in the proximal part of tracheobronchial trees, Computer Methods in Biomechanics and Biomedical Engineering, 8(4), 279–293, (2005).
J.P. Fink and W.C. Rheinboldt — On the error behavior of the reduced basis technique for nonlinear finite element approximations. Zeitschrift für Angewandte Mathematik und Mechanik, 63(1), (1983) 21–28.
R.L. Fox and H. Miura — An approximate analysis technique for design calculations. AIAA Journal, 9(1), (1971) 177–179.
M.B. Giles and E. Süli — Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numerica, Vol. 11, 145–236, Cambridge University Press, 2002.
V. Girault, P.A. Raviart — Finite element methods for Navier-Stokes equations: Theory and algorithms, (Springer Series in Computational Mathematics. Volume 5), Berlin and New York, Springer-Verlag (1986)
W. Gordon and C. Hall — Transfinite element methods: blending-function interpolation over arbitrary curved element domains, Numer. Math. (21), 1973/74, pp. 109–129.
C. Grandmont, Y. Maday, and B. Maury — A multiscale/multimodel approach of the respiration tree, in New Trends in Continuum Mechanics — M. Mihailescu-Suliciu, Ed. — Theta Foundation, Bucharest, Romania (2005).
M.A. Grepl, Y. Maday, N.C. Nguyen, and A.T. Patera — Efficient reduced basis treatment of non-affine and nonlinear partial differential equations, submitted to M2AN (2006).
A.E. Løvgren, Y. Maday, and E.M. Rønquist — A reduced basis element method for the steady Stokes problem, to appear in M2AN.
A.E. Løvgren, Y. Maday, and E.M. Rønquist — in progress.
L. Machiels, Y. Maday, I.B Oliveira, A.T. Patera, and D.V. Rovas — Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, CR Acad Sci Paris Series I 331:(2000) 153–158.
Y. Maday, A.T. Patera, and G. Turinici — A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations, J. Sci. Comput., 17 (2002), no. 1–4, 437–446.
Y. Maday, L. Machiels, A.T. Patera, and D.V. Rovas — Blackbox reduced-basis output bound methods for shape optimization, in Proceedings 12th International Domain Decomposition Conference, Chiba, Japan (2000) 429–436.
Y. Maday and E.M. Rønquist — The reduced-basis element method: Application to a thermal fin problem. SIAM Journal on Scientific Computing, 2004.
A.K. Noor and J.M. Peters — Reduced basis technique for nonlinear analysis of structures. AIAA Journal, 18(4) (1980) 455–462.
V. Milisic and A. Quarteroni. Analysis of lumped parameter models for blood flow simulations and their relation with 1D models, to appear in M2AN, 2004.
F. Murat and J. Simon, Sur le Contrôle par un Domaine Géometrique, Publication of the Laboratory of Numerical Analysis, University Paris VI, 1976.
Pinkus, A. — n-Widths in Approximation Theory, Springer-Verlag, Berlin (1985).
T.A. Porsching — Estimation of the error in the reduced basis method solution of nonlinear equations. Mathematics of Computation. 45(172) (1985) 487–496.
C. Prud’homme, D.V. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera, G. Turinici — Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods, J. Fluids Engineering, 124, (2002) 70–80.
C. Prud’homme, D.V. Rovas, K. Veroy, and A.T. Patera — A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. Programming. M2AN Math. Model. Numer. Anal. 36 (2002), no. 5, 747–771.
A. Quarteroni and A. Valli — Domain decomposition methods for partial differential equations, Numerical Mathematics and Scientific Computation, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, (1999).
P.A. Raviart and J.M. Thomas. — A mixed finite element method for 2nd order elliptic problems. In I. Galligani and E. Magenes, editors, Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics, Vol. 606. Springer-Verlag, 1977.
D.V. Rovas. — Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, October 2002.
M. Thiriet — http://www-rocq.inria.fr/who/Marc.Thiriet/Vitesv/
A. Toselli and O. Widlund — Domain decomposition methods — algorithms and theory, Springer Series in Computational Mathematics, 34, Springer-Verlag, Berlin, (2005).
R. Verfurth — A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, 1996.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Løvgren, A.E., Maday, Y., Rønquist, E.M. (2006). The Reduced Basis Element Method for Fluid Flows. In: Calgaro, C., Coulombel, JF., Goudon, T. (eds) Analysis and Simulation of Fluid Dynamics. Advances in Mathematical Fluid Mechanics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7742-7_8
Download citation
DOI: https://doi.org/10.1007/978-3-7643-7742-7_8
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7741-0
Online ISBN: 978-3-7643-7742-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)