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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.. dahmen@igpm.rwth-aachen.de; plesken@igpm.rwth-aachen.de; welper@igpm.rwth-aachen.de
Christian Plesken
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany.. dahmen@igpm.rwth-aachen.de; plesken@igpm.rwth-aachen.de; welper@igpm.rwth-aachen.de
Gerrit Welper
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany.. dahmen@igpm.rwth-aachen.de; plesken@igpm.rwth-aachen.de; welper@igpm.rwth-aachen.de
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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

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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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