Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-25T03:29:04.687Z Has data issue: false hasContentIssue false
  • English
  • Français

A reduced basis element methodfor the steady Stokes problem

Published online by Cambridge University Press:  22 July 2006

Alf Emil Løvgren ,
Yvon Maday  and
Einar M. Rønquist
Alf Emil Løvgren
Affiliation:
Norwegian University of Science and Technology, Department of Mathematical Sciences, 7491 Trondheim, Norway. [email protected]
Yvon Maday
Affiliation:
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, France.
Einar M. Rønquist
Affiliation:
Norwegian University of Science and Technology, Department of Mathematical Sciences, 7491 Trondheim, Norway. [email protected]
Get access

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. The basic idea is to first decomposethe computational domain into a series of subdomains that are deformationsof 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 deformations of the referencesubdomains and mapped onto the reference shape.The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mappedfrom the generic computational part to the actual computational part.We extend earlier work in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to satisfying the inf-sup condition,to a posteriori error estimation, and to “gluing” the local solutions together in the multidomain case.

Keywords

Stokes flowreduced basisreduced order modeldomain decompositionmortar methodoutput boundsa posteriori error estimators.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 3 , May 2006 , pp. 529 - 552
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R. Aris, Vectors, tensors and the basic equations of fluid mechanics. Dover Publications (1989).
Babuska, I., Error-bounds for finite element method. Numer. Math. 16 (1971) 322–333. CrossRef
Belgacem, B.F., The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173–197. CrossRef
Belgacem, B.F., Bernardi, C., Chorfi, N. and Maday, Y., Inf-sup conditions for the mortar spectral element discretization of the Stokes problem. Numer. Math. 85 (2000) 257–281. CrossRef
Bernardi, C. and Maday, Y., Polynomial approximation of some singular functions. Appl. Anal. 42 (1992) 1–32. CrossRef
Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Anal. Numér. 8 (1974) 129–151.
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Verlag (1991).
Fink, J.P. and Rheinboldt, W.C., On the error behavior of the reduced basis technique in nonlinear finite element approximations. Z. Angew. Math. Mech. 63 (1983) 21–28. CrossRef
Gordon, W.J. and Hall, C.A., Construction of curvilinear co-ordinate systems and applications to mesh generation. Int. J. Numer. Meth. Eng. 7 (1973) 461–477. CrossRef
Y. Maday and A.T. Patera, Spectral element methods for the Navier-Stokes equations. In Noor A. Ed., State of the Art Surveys in Computational Mechanics (1989) 71–143.
Maday, Y. and Rønquist, E.M., A reduced-basis element method. J. Sci. Comput. 17 (2002) 447–459. CrossRef
Maday, Y. and Rønquist, E.M., The reduced-basis element method: application to a thermal fin problem. SIAM J. Sci. Comput. 26 (2004) 240–258. CrossRef
Y. Maday, A.T. Patera, and E.M. Rønquist, The PN x PN-2 method for the approximation of the Stokes problem. Technical Report No. 92009, Department of Mechanical Engineering, Massachusetts Institute of Technology (1992).
Y. Maday, D. Meiron, A.T. Patera and E.M. Rønquist, Analysis of iterative methods for the steady and unsteady Stokes problem: Application to spectral element discretizations. SIAM J. Sci. Stat. Comp. (1993) 310–337. CrossRef
Noor, A.K. and Peters, J.M., Reduced basis technique for nonlinear analysis of structures. AIAA J. 19 (1980) 455–462.
Prud'homme, C., Rovas, D.V., Veroy, K., Machiels, L., Maday, Y., Patera, A.T. and Turinici, G., Reliable real-time solution of parametrized partial differential equations: Reduced basis output bound methods. J. Fluid Eng. 124 (2002) 70–80. CrossRef
P.A. Raviart and J.M. Thomas, A mixed finite element method for 2-nd order elliptic problems, in Mathematical Aspects of Finite Element Methodes, Lec. Notes Math. 606 I. Galligani and E. Magenes Eds., Springer-Verlag (1977).
D.V. Rovas, Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (October 2002).
K. Veroy, C. Prud'homme, D.V. Rovas and A.T. Patera, 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 2003).