Reduced basis method for finite volume approximations of parametrized linear evolution equations

¹ Institute of Mathematics, University of Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg, Germany. haasdonk@mathematik.uni-freiburg.de
² Institute of Numerical and Applied Mathematics, University of Münster, Einsteinstr. 62, 48149 Münster, Germany. mario.ohlberger@math.uni-muenster.de

Received: 15 November 2006
Revised: 23 October 2007

Abstract

The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P²DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for these general evolution problems and the derivation of rigorous a-posteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized in case of affine parameter-dependence of the problem. This is the basis for a rapid online computation in case of multiple simulation requests. We introduce a new offline basis-generation algorithm based on our a-posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convection-diffusion problem demonstrate the efficient applicability of the approach.