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On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition

Published online by Cambridge University Press:  01 August 2012

Toni Lassila ,
Andrea Manzoni  and
Gianluigi Rozza
Toni Lassila
Affiliation:
Modelling and Scientific Computing, Mathematics Institute of Computational Science and Engineering, École Polytechnique Fédérale de Lausanne, Station 8, EPFL, 1015 Lausanne, Switzerland. toni.lassila@epfl.ch; andrea.manzoni@epfl.ch; gianluigi.rozza@epfl.ch
Andrea Manzoni
Affiliation:
Modelling and Scientific Computing, Mathematics Institute of Computational Science and Engineering, École Polytechnique Fédérale de Lausanne, Station 8, EPFL, 1015 Lausanne, Switzerland. toni.lassila@epfl.ch; andrea.manzoni@epfl.ch; gianluigi.rozza@epfl.ch
Gianluigi Rozza
Affiliation:
Modelling and Scientific Computing, Mathematics Institute of Computational Science and Engineering, École Polytechnique Fédérale de Lausanne, Station 8, EPFL, 1015 Lausanne, Switzerland. toni.lassila@epfl.ch; andrea.manzoni@epfl.ch; gianluigi.rozza@epfl.ch
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Abstract

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.

Keywords

Parametric model reductiona posteriori error estimationstability factorscoercivity constantinf-sup conditionparametrized PDEsreduced basis methodsuccessive constraint methodempirical interpolation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 6 , November 2012 , pp. 1555 - 1576
Copyright
© EDP Sciences, SMAI, 2012

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References

Babuška, I. and Sauter, S.A., Is the pollution effect of the FEM avoidable for Helmholtz equation considering high wave numbers? SIAM Rev. 42 (2000) 451–484. Google Scholar
Barrault, M., Maday, Y., Nguyen, N.C. and Patera, A.T., 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. Google Scholar
S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 2nd edition. Springer (2002).
F. Brezzi, and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Series in Comput. Math. 15 (1991).
Chen, Y., Hesthaven, J., Maday, Y. and Rodriguez, J., 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. Google Scholar
Eftang, J.L., Grepl, M.A. and Patera, A.T., A posteriori error bounds for the empirical interpolation method. C. R. Acad. Sci. Paris, Sér. I Math. 348 (2010) 575–579. Google Scholar
Eftang, J.L., Patera, A.T. and Rønquist, E.M., An “hp” certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Sci. Comput. 32 (2010) 3170–3200. Google Scholar
Eftang, J.L., Knezevic, D.J. and Patera, A.T., An “hp” certified reduced basis method for parametrized parabolic partial differential equations. Math. Comput. Model. Dyn. 17 (2011) 395–422. Google Scholar
Eftang, J.L., Huynh, D.B.P., Knezevic, D.J. and Patera, A.T., A two-step certified reduced basis method. J. Sci. Comput. 51 (2012) 28–58. Google Scholar
A. Ern and J.-L. Guermond, Theory and practice of finite elements. Springer-Verlag, New York (2004).
L.C. Evans, Partial Differential Equations. Amer. Math. Soc. (1998).
Gerner, A.L. and Veroy, K., 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. Google Scholar
Green, D. and Unruh, W.G., The failure of the Tacoma bridge : a physical model. Am. J. Phys. 74 (2006) 706–716. Google Scholar
Grepl, M.A., Maday, Y., Nguyen, N.C. and Patera, A.T., Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM : M2AN 41 (2007) 575–605. Google Scholar
A. Holt and M. Landahl, Aerodynamics of wings and bodies. Dover New York (1985).
D.B.P. Huynh and G. Rozza, Reduced basis method and a posteriori error estimation : application to linear elasticity problems (2011). Submitted.
Huynh, D.B.P, Rozza, G., Sen, S. and Patera, A.T., 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. Google Scholar
Huynh, D.B.P., Knezevic, D., Chen, Y., Hesthaven, J. and Patera, A.T., A natural-norm successive constraint method for inf-sup lower bounds. Comput. Methods Appl. Mech. Eng. 199 (2010) 1963–1975. Google Scholar
D.B.P. Huynh, N.C. Nguyen, A.T. Patera and G. Rozza, Rapid reliable solution of the parametrized partial differential equations of continuum mechanics and transport. Available on http://augustine.mit.edu.
Lassila, T. and Rozza, G., Parametric free-form shape design with PDE models and reduced basis method. Comput. Methods Appl. Mech. Eng. 199 (2010) 1583–1592. Google Scholar
Lassila, T. and Rozza, G., 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. Google Scholar
Lassila, T., Quarteroni, A. and Rozza, G., A reduced basis model with parametric coupling for fluid-structure interaction problems. SIAM J. Sci. Comput. 34 (2012) A1187–A1213. Google Scholar
Maday, Y., Nguyen, N.C., Patera, A.T. and Pau, G.S.H., A general multipurpose interpolation procedure : the magic points. Commun. Pure Appl. Anal. 8 (2009) 383–404. Google Scholar
A. Manzoni, A. Quarteroni and G. Rozza, 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.
A. Manzoni, A. Quarteroni and G. Rozza, 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.
L.M. Milne-Thomson, Theoretical aerodynamics. Dover (1973).
B. Mohammadi and O. Pironneau, Applied shape optimization for fluids. Oxford University Press (2001).
Nguyen, N.C., 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. Google Scholar
Nguyen, N.C., Rozza, G. and Patera, A.T., Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers equation. Calcolo 46 (2009) 157–185. Google Scholar
A.T. Patera and G. Rozza, 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).
Prud’homme, C., Rovas, D.V., Veroy, K. and Patera, A.T., A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. ESAIM : M2AN 36 (2002) 747–771. Google Scholar
A. Quarteroni, G. Rozza and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations in industrial applications. J. Math. Ind. 1 (2011).
Rozza, G., Reduced basis approximation and error bounds for potential flows in parametrized geometries. Commun. Comput. Phys. 9 (2011) 1–48. Google Scholar
Rozza, G., Huynh, D.B.P. and Patera, A.T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15 (2008) 229–275. Google Scholar
G. Rozza, D.B.P. Huynh and A. Manzoni, 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.
G. Rozza, T. Lassila and A. Manzoni, 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.
Sen, S., Veroy, K., Huynh, P., Deparis, S., Nguyen, N.C. and Patera, A.T., “Natural norm” a posteriori error estimators for reduced basis approximations. J. Comput. Phys. 217 (2006) 37–62. Google Scholar
S. Vallaghe, A. Le-Hyaric, M. Fouquemberg and C. Prud’homme, 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.
Xu, J. and Zikatanov, L., Some observation on Babuška and Brezzi theories. Numer. Math. 94 (2003) 195–202. Google Scholar
S. Zhang, Efficient greedy algorithms for successive constraints methods with high-dimensional parameters. C. R. Acad. Sci. Paris, Sér. I Math. (2011). Submitted.