A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this bound is extended to the fine level by adding a proper correction term. Because the approximate problems are affine, an efficient offline/online computational scheme can be developed for the certified solution (error bounds and stability factors) of the parametric equations considered. We experiment with different correction terms suited for a posteriori error estimation of the reduced basis solution of elliptic coercive and noncoercive problems.
Mots-clés : parametric model reduction, a posteriori error estimation, stability factors, coercivity constant, inf-sup condition, parametrized PDEs, reduced basis method, successive constraint method, empirical interpolation
@article{M2AN_2012__46_6_1555_0,
author = {Lassila, Toni and Manzoni, Andrea and Rozza, Gianluigi},
title = {On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1555--1576},
publisher = {EDP-Sciences},
volume = {46},
number = {6},
year = {2012},
doi = {10.1051/m2an/2012016},
mrnumber = {2996340},
zbl = {1276.65069},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2012016/}
}
TY - JOUR AU - Lassila, Toni AU - Manzoni, Andrea AU - Rozza, Gianluigi TI - On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 1555 EP - 1576 VL - 46 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012016/ DO - 10.1051/m2an/2012016 LA - en ID - M2AN_2012__46_6_1555_0 ER -
%0 Journal Article %A Lassila, Toni %A Manzoni, Andrea %A Rozza, Gianluigi %T On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 1555-1576 %V 46 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012016/ %R 10.1051/m2an/2012016 %G en %F M2AN_2012__46_6_1555_0
Lassila, Toni; Manzoni, Andrea; Rozza, Gianluigi. On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 6, pp. 1555-1576. doi : 10.1051/m2an/2012016. http://www.numdam.org/articles/10.1051/m2an/2012016/
[1] and , Is the pollution effect of the FEM avoidable for Helmholtz equation considering high wave numbers? SIAM Rev. 42 (2000) 451-484. | MR | Zbl
[2] , , and , An ‘empirical interpolation' method : application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris, Sér. I Math. 339 (2004) 667-672. | MR | Zbl
[3] and , The mathematical theory of finite element methods, 2nd edition. Springer (2002). | MR | Zbl
[4] , and , Mixed and Hybrid Finite Element Methods. Springer Series in Comput. Math. 15 (1991). | MR | Zbl
[5] , , and , A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations. C. R. Acad. Sci. Paris, Sér. I Math. 346 (2008) 1295-1300. | MR | Zbl
[6] , and , A posteriori error bounds for the empirical interpolation method. C. R. Acad. Sci. Paris, Sér. I Math. 348 (2010) 575-579. | MR | Zbl
[7] , and , An “hp” certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Sci. Comput. 32 (2010) 3170-3200. | MR | Zbl
[8] , and , An “hp” certified reduced basis method for parametrized parabolic partial differential equations. Math. Comput. Model. Dyn. 17 (2011) 395-422. | MR
[9] , , and , A two-step certified reduced basis method. J. Sci. Comput. 51 (2012) 28-58. | MR | Zbl
[10] and , Theory and practice of finite elements. Springer-Verlag, New York (2004). | MR | Zbl
[11] , Partial Differential Equations. Amer. Math. Soc. (1998). | JFM
[12] and , Reduced basis a posteriori error bounds for the stokes equations in parametrized domains : a penalty approach. Math. Mod. Methods Appl. Sci. 21 (2011) 2103-2134. | MR
[13] and , The failure of the Tacoma bridge : a physical model. Am. J. Phys. 74 (2006) 706-716. | MR | Zbl
[14] , , and , Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM : M2AN 41 (2007) 575-605. | Numdam | MR | Zbl
[15] and , Aerodynamics of wings and bodies. Dover New York (1985). | MR | Zbl
[16] and , Reduced basis method and a posteriori error estimation : application to linear elasticity problems (2011). Submitted.
[17] , , and , A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability costants. C. R. Acad. Sci. Paris, Sér. I Math. 345 (2007) 473-478. | MR | Zbl
[18] , , , and , A natural-norm successive constraint method for inf-sup lower bounds. Comput. Methods Appl. Mech. Eng. 199 (2010) 1963-1975. | MR | Zbl
[19] , , and , Rapid reliable solution of the parametrized partial differential equations of continuum mechanics and transport. Available on http://augustine.mit.edu.
[20] and , Parametric free-form shape design with PDE models and reduced basis method. Comput. Methods Appl. Mech. Eng. 199 (2010) 1583-1592. | MR | Zbl
[21] and , Model reduction of semiaffinely parametrized partial differential equations by two-level affine approximation. C. R. Math. Acad. Sci. Paris, Ser. I 349 (2011) 61-66. | MR | Zbl
[22] , and , A reduced basis model with parametric coupling for fluid-structure interaction problems. SIAM J. Sci. Comput. 34 (2012) A1187-A1213. | MR
[23] , , and , A general multipurpose interpolation procedure : the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404. | MR | Zbl
[24] , and , Model reduction techniques for fast blood flow simulation in parametrized geometries. Int. J. Numer. Methods Biomed. Eng. (2011). In press, DOI: 10.1002/cnm.1465. | MR
[25] , and , Shape optimization of cardiovascular geometries by reduced basis methods and free-form deformation techniques. Int. J. Numer. Methods Fluids (2011). In press, DOI: 10.1002/fld.2712.
[26] , Theoretical aerodynamics. Dover (1973). | Zbl
[27] and , Applied shape optimization for fluids. Oxford University Press (2001). | MR | Zbl
[28] , A posteriori error estimation and basis adaptivity for reduced-basis approximation of nonaffine-parametrized linear elliptic partial differential equations. J. Comput. Phys. 227 (2007) 983-1006. | MR | Zbl
[29] , and , Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers equation. Calcolo 46 (2009) 157-185. | MR | Zbl
[30] and , Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equation. Version 1.0, Copyright MIT (2006), to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering (2009).
[31] , , and , A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. ESAIM : M2AN 36 (2002) 747-771. | Numdam | Zbl
[32] , and , Certified reduced basis approximation for parametrized partial differential equations in industrial applications. J. Math. Ind. 1 (2011). | MR | Zbl
[33] , Reduced basis approximation and error bounds for potential flows in parametrized geometries. Commun. Comput. Phys. 9 (2011) 1-48. | MR | Zbl
[34] , and , Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15 (2008) 229-275. | MR
[35] , and , Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries : roles of the inf-sup stability constants. Technical Report 22.2010, MATHICSE (2010). Online version available at : http://cmcs.epfl.ch/people/manzoni.
[36] , and , Reduced basis approximation for shape optimization in thermal flows with a parametrized polynomial geometric map, in Spectral and High Order Methods for Partial Differential Equations. Selected papers from the ICOSAHOM'09 Conference, Trondheim, Norway, edited by J.S. Hesthaven and E.M. Rønquist. Lect. Notes Comput. Sci. Eng. 76 (2011) 307-315. | MR
[37] , , , , and , “Natural norm” a posteriori error estimators for reduced basis approximations. J. Comput. Phys. 217 (2006) 37-62. | MR | Zbl
[38] , , and , A successive constraint method with minimal offline constraints for lower bounds of parametric coercivity constant. C. R. Acad. Sci. Paris, Sér. I Math. (2011). Submitted.
[39] and , Some observation on Babuška and Brezzi theories. Numer. Math. 94 (2003) 195-202. | MR | Zbl
[40] , Efficient greedy algorithms for successive constraints methods with high-dimensional parameters. C. R. Acad. Sci. Paris, Sér. I Math. (2011). Submitted.
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