A reduced basis method for evolution schemes with parameter-dependent explicit operators

Haasdonk, B. ; Ohlberger, M. ; Rozza, G.

Electronic transactions on numerical analysis, Tome 32 (2008), pp. 145-161.

Résumé

Summary: During the last decades, reduced basis (RB) methods have been developed to a wide methodology for model reduction of problems that are governed by parametrized partial differential equations (P DEs ). In particular $$###$$ equations of elliptic and parabolic type for linear, low degree polynomial or monotonic nonlinearities have been treated successfully by RB methods using finite element schemes. Due to the characteristic offline-online decomposition, the reduced models often become suitable for a multi-query or real-time setting, where simulation results, such as field-variables or output estimates, can be approximated reliably and rapidly for varying parameters. In the current study, we address a certain class of time-dependent evolution schemes with explicit discretization operators that are arbitrarily parameter dependent. We extend the RB methodology to these cases by applying the $empiricalinterpolation$ method to localized discretization operators. The main technical ingredients are: (i) generation of a $collateral reduced basis$ modelling the effects of the discretization operator under parameter variations in the offlinephase and (ii) an online simulation scheme based on a numerical subgrid and localized evaluations of the evolution operator. We formulate an a-posteriori error estimator for quantification of the resulting reduced simulation error.

Détail

Classification : 76M12, 65M15, 35L90, 35K90, 76R99
Keywords: model reduction, reduced basis methods, parameter dependent explicit operators, empirical interpolation, a-posteriori error estimates