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Double greedy algorithms: Reduced basis methods for transport dominated problems∗

Published online by Cambridge University Press:  20 January 2014

Wolfgang Dahmen ,
Christian Plesken  and
Gerrit Welper
Wolfgang Dahmen
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany.. [email protected]; [email protected]; [email protected]
Christian Plesken
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany.. [email protected]; [email protected]; [email protected]
Gerrit Welper
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany.. [email protected]; [email protected]; [email protected]
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Abstract

The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov n-widths of the solution sets. The central ingredient is the construction of computationally feasible “tight” surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters.

Keywords

Tight surrogatesstable variational formulationssaddle point problemsdouble greedy schemesgreedy stabilizationrate-optimalitytransport equationsconvection-diffusion equations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014 , pp. 623 - 663
Copyright
© EDP Sciences, SMAI, 2014

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References

Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G. and Wojtaszczyk, P., Convergence Rates for Greedy Algorithms in Reduced Basis Methods. SIAM J. Math. Anal. 43 (2011) 1457–1472. Google Scholar
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, in vol. 15 of Springer Ser. Comput. Math. Springer-Verlag (1991).
Buffa, A., Maday, Y., Patera, A.T., Prud’homme, C. and Turinici, G., A Priori convergence of the greedy algorithm for the parameterized reduced basis. ESAIM: M2AN 46 (2012) 595–603. Google Scholar
Cascon, J.M., Kreuzer, C., Nochetto, R.H. and Siebert, K.G., Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (2008) 2524–2550. Google Scholar
Cohen, A., Dahmen, W. and Welper, G., Adaptivity and Variational Stabilization for Convection-Diffusion Equations. ESAIM: M2AN 46 (2012) 1247–1273. Google Scholar
W. Dahmen, Parameter dependent transport equations, in Workshop J.L.L.-SMP: Reduced Basis Methods in High Dimensions. Available at http://www.ljll.math.upmc.fr/fr/archives/actualites/2011/workshop˙ljll˙smp˙rbihd.html
Dahmen, W., Huang, C., Schwab, C. and Welper, G., Adaptive Petrov−Galerkin methods for first order transport equations. SIAM J. Numer. Anal. 50 (2012) 2420–2445. Google Scholar
Demkowicz, L.F. and Gopalakrishnan, J., A class of discontinuous Petrov−Galerkin Methods I: The transport equation. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1558–1572. Google Scholar
Demkowicz, L. and Gopalakrishnan, J., A class of discontinuous Petrov−Galerkin methods. Part II: Optimal test functions. Numer. Methods for Partial Differ. Equ. 27 (2011) 70–105. Google Scholar
Deparis, S., Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural norm approach. SIAM J. Numer. Anal. 46 (2008) 2039–2067. Google Scholar
DeVore, R., Petrova, G. and Wojtaszczyk, P., Greedy algorithms for reduced bases in Banach spaces, Constructive Approximation 37 (2013) 455–466. Google Scholar
A. Ern and J.-L. Guermond, Theory and practice of finite elements. Springer (2004).
A. Gerner and K. Veroy-Grepl, Certified reduced basis methods for parametrized saddle point problems, preprint (2012). To appear in SIAM J. Sci. Comput.
Gerner, A.-L. and Veroy, K., Reduced basis a posteriori error bounds for the Stokes equations in parameterized domains: A penalty approach. M3AS: Math. Models Methods Appl. Sci. 21 (2011) 2103–2134. Google Scholar
Grepl, M.A., Certified Reduced Basis Methods for Nonaffine Linear Time-Varying and Nonlinear Parabolic Partial Differential Equations. M3AS: Math. Models Methods Appl. Sci. 22 (2012) 40. Google Scholar
Grepl, M.A. and Patera, A.T., A Posteriori Error Bounds for Reduced-Basis Approximations of Parametrized Parabolic Partial Differential Equations. ESAIM: M2AN 39 (2005) 157–181. Google Scholar
Haasdonk, B., Convergence rates for the POD-greedy method. ESAIM: M2AN 47 (2013) 859–873. Google Scholar
Hughes, T. and Sangalli, G., Variational Multiscale Analysis: the Fine-scale Green’s Function, Projection, Optimization, Localization, and Stabilized Methods. SIAM J. Numer. Anal. 45 (2007) 539–557. Google Scholar
G. Kanschat, E. Meinköhn, R. Rannacher and R. Wehrse, Numerical methods in multidimensional radiative transfer, Springer (2009).
G.G. Lorentz, M. von Golitschek and Yu. Makovoz, Constructive approximation: Advanced problems, vol. 304. Springer Grundlehren, Berlin (1996).
Maday, Y., Patera, A.T. and Turinici, G., A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations. J. Sci. Comput. 17 (2002) 437–446. Google Scholar
Manteuffel, T., McCormick, S., Ruge, J. and Schmidt, J.G., First-order system ℒℒ∗ (FOSLL)∗ for general scalar elliptic problems in the plane. SIAM J. Numer. Anal. 43 (2005) 2098–2120. Google Scholar
Nguyen, N.-C., Rozza, G. and Patera, A.T., Reduced basis approximation for the time-dependent viscous Burgers’ equation. Calcolo 46 (2009) 157–185. Google Scholar
T. Patera and K. Urban, An improved error bound for reduced basis approximation of linear parabolic problems, submitted to Mathematics of Computation (in press 2013).
A.T. Patera and G. Rozza, Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations, Version 1.0, Copyright MIT 2006–2007, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering.
H.-J. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, in vol. 24 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2nd Edition (2008).
G. Rozza and D.B.P. Huynh and A. Manzoni, Reduced basis approximation and a posteriori error estiamtion for Stokes flows in parametrized geometries: roles of the inf-sup stability constants, Numer. Math. DOI: 10.1007/s00211-013-0534-8.
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
Rozza, G. and Veroy, K., On the stability of reduced basis techniques for Stokes equations in parametrized domains. Comput. Methods Appl. Mechanics Engrg. 196 (2007) 1244–1260. Google Scholar
Sangalli, G., A uniform analysis of non-symmetric and coercive linear operators. SIAM J. Math. Anal. 36 (2005) 2033–2048. Google Scholar
M. Schlottbom, On Forward and Inverse Models in Optical Tomography, Ph.D. Thesis. RWTH Aachen (2011).
Sen, S., Veroy, K., Huynh, D.B.P., Deparis, S., Nguyn, 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
Verfürth, R., Robust a posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal. 43 (2005) 1766–1782. Google Scholar
G. Welper, Infinite dimensional stabilization of convection-dominated problems, Ph.D. Thesis. RWTH Aachen (2012).
Zitelli, J., Muga, I., Demkowicz, L., Gopalakrishnan, J., Pardo, D. and Calo, V., A class of discontinuous Petrov−Galerkin methods. Part IV: Wave propagation. J. Comput. Phys. 230 (2011) 2406–2432. Google Scholar