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 decompose the computational domain into a series of subdomains that are deformations of 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 reference subdomains 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, mapped from 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.
Mots-clés : Stokes flow, reduced basis, reduced order model, domain decomposition, mortar method, output bounds, a posteriori error estimators
@article{M2AN_2006__40_3_529_0,
author = {L{\o}vgren, Alf Emil and Maday, Yvon and R{\o}nquist, Einar M.},
title = {A reduced basis element method for the steady {Stokes} problem},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {529--552},
publisher = {EDP-Sciences},
volume = {40},
number = {3},
year = {2006},
doi = {10.1051/m2an:2006021},
mrnumber = {2245320},
zbl = {1129.76036},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2006021/}
}
TY - JOUR AU - Løvgren, Alf Emil AU - Maday, Yvon AU - Rønquist, Einar M. TI - A reduced basis element method for the steady Stokes problem JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2006 SP - 529 EP - 552 VL - 40 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2006021/ DO - 10.1051/m2an:2006021 LA - en ID - M2AN_2006__40_3_529_0 ER -
%0 Journal Article %A Løvgren, Alf Emil %A Maday, Yvon %A Rønquist, Einar M. %T A reduced basis element method for the steady Stokes problem %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 529-552 %V 40 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2006021/ %R 10.1051/m2an:2006021 %G en %F M2AN_2006__40_3_529_0
Løvgren, Alf Emil; Maday, Yvon; Rønquist, Einar M. A reduced basis element method for the steady Stokes problem. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 3, pp. 529-552. doi : 10.1051/m2an:2006021. http://www.numdam.org/articles/10.1051/m2an:2006021/
[1] , Vectors, tensors and the basic equations of fluid mechanics. Dover Publications (1989). | Zbl
[2] , Error-bounds for finite element method. Numer. Math. 16 (1971) 322-333. | EuDML | Zbl
[3] , The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173-197. | Zbl
[4] ,, and, Inf-sup conditions for the mortar spectral element discretization of the Stokes problem. Numer. Math. 85 (2000) 257-281. | Zbl
[5] and, Polynomial approximation of some singular functions. Appl. Anal. 42 (1992) 1-32. | Zbl
[6] , On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Anal. Numér. 8 (1974) 129-151. | EuDML | Numdam | Zbl
[7] and, Mixed and Hybrid Finite Element Methods. Springer Verlag (1991). | MR | Zbl
[8] and, On the error behavior of the reduced basis technique in nonlinear finite element approximations. Z. Angew. Math. Mech. 63 (1983) 21-28. | Zbl
[9] and, Construction of curvilinear co-ordinate systems and applications to mesh generation. Int. J. Numer. Meth. Eng. 7 (1973) 461-477. | Zbl
[10] and, Spectral element methods for the Navier-Stokes equations. In Noor A. Ed., State of the Art Surveys in Computational Mechanics (1989) 71-143. | Zbl
[11] and, A reduced-basis element method. J. Sci. Comput. 17 (2002) 447-459. | Zbl
[12] and, The reduced-basis element method: application to a thermal fin problem. SIAM J. Sci. Comput. 26 (2004) 240-258. | Zbl
[13] ,, and, The method for the approximation of the Stokes problem. Technical Report No. 92009, Department of Mechanical Engineering, Massachusetts Institute of Technology (1992).
[14] ,, and, Analysis of iterative methods for the steady and unsteady Stokes problem: Application to spectral element discretizations. SIAM J. Sci. Stat. Comp. (1993) 310-337. | Zbl
[15] and, Reduced basis technique for nonlinear analysis of structures. AIAA J. 19 (1980) 455-462.
[16] ,,,,, and, Reliable real-time solution of parametrized partial differential equations: Reduced basis output bound methods. J. Fluid Eng. 124 (2002) 70-80.
[17] and, 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). | MR | Zbl
[18] , Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (October 2002).
[19] ,, and, 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).
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