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

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

Received: 21 February 2013
Revised: 28 June 2013

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.