Double greedy algorithms: Reduced basis methods for transport dominated problems

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 c...

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Bibliographic Details
Published in:ESAIM. Mathematical modelling and numerical analysis 2014-05, Vol.48 (3), p.623-663
Main Authors: Dahmen, Wolfgang, Plesken, Christian, Welper, Gerrit
Format: Article
Language:English
Subjects:
35B30
65J10
65N12
65N15
Accuracy
Algorithms
convection-diffusion equations
double greedy schemes
greedy stabilization
Hilbert space
Mathematical models
Numerical analysis
Optimization
rate-optimality
saddle point problems
stable variational formulations
Studies
Tight surrogates
transport equations
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Summary: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.
ISSN:0764-583X
1290-3841
DOI:10.1051/m2an/2013103