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.
Mots-clés : tight surrogates, stable variational formulations, saddle point problems, double greedy schemes, greedy stabilization, rate-optimality, transport equations, convection-diffusion equations
@article{M2AN_2014__48_3_623_0,
author = {Dahmen, Wolfgang and Plesken, Christian and Welper, Gerrit},
title = {Double greedy algorithms: {Reduced} basis methods for transport dominated problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {623--663},
publisher = {EDP-Sciences},
volume = {48},
number = {3},
year = {2014},
doi = {10.1051/m2an/2013103},
mrnumber = {3177860},
zbl = {1291.65339},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2013103/}
}
TY - JOUR AU - Dahmen, Wolfgang AU - Plesken, Christian AU - Welper, Gerrit TI - Double greedy algorithms: Reduced basis methods for transport dominated problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 623 EP - 663 VL - 48 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013103/ DO - 10.1051/m2an/2013103 LA - en ID - M2AN_2014__48_3_623_0 ER -
%0 Journal Article %A Dahmen, Wolfgang %A Plesken, Christian %A Welper, Gerrit %T Double greedy algorithms: Reduced basis methods for transport dominated problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 623-663 %V 48 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013103/ %R 10.1051/m2an/2013103 %G en %F M2AN_2014__48_3_623_0
Dahmen, Wolfgang; Plesken, Christian; Welper, Gerrit. Double greedy algorithms: Reduced basis methods for transport dominated problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 3, pp. 623-663. doi : 10.1051/m2an/2013103. http://www.numdam.org/articles/10.1051/m2an/2013103/
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