The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P${}^{2}$DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for these general evolution problems and the derivation of rigorous a-posteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized in case of affine parameter-dependence of the problem. This is the basis for a rapid online computation in case of multiple simulation requests. We introduce a new offline basis-generation algorithm based on our a-posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convection-diffusion problem demonstrate the efficient applicability of the approach.

Keywords: model reduction, reduced basis methods, finite volume methods, a-posteriori error estimates

@article{M2AN_2008__42_2_277_0, author = {Haasdonk, Bernard and Ohlberger, Mario}, title = {Reduced basis method for finite volume approximations of parametrized linear evolution equations}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {277--302}, publisher = {EDP-Sciences}, volume = {42}, number = {2}, year = {2008}, doi = {10.1051/m2an:2008001}, mrnumber = {2405149}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2008001/} }

TY - JOUR AU - Haasdonk, Bernard AU - Ohlberger, Mario TI - Reduced basis method for finite volume approximations of parametrized linear evolution equations JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 277 EP - 302 VL - 42 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2008001/ DO - 10.1051/m2an:2008001 LA - en ID - M2AN_2008__42_2_277_0 ER -

%0 Journal Article %A Haasdonk, Bernard %A Ohlberger, Mario %T Reduced basis method for finite volume approximations of parametrized linear evolution equations %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 277-302 %V 42 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2008001/ %R 10.1051/m2an:2008001 %G en %F M2AN_2008__42_2_277_0

Haasdonk, Bernard; Ohlberger, Mario. Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: Modélisation mathématique et analyse numérique, Volume 42 (2008) no. 2, pp. 277-302. doi : 10.1051/m2an:2008001. http://www.numdam.org/articles/10.1051/m2an:2008001/

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