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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
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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2014

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