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

Haasdonk B, Ohlberger M, Rozza G

Research article (journal)

Abstract

During the last decades, reduced basis (RB) methods have been developed to a wide methodology formodel reduction of problems that are governed by parametrized partial differential equations (PDEs ). In particularequations of elliptic and parabolic type for linear, low degree polynomial or monotonic nonlinearities have beentreated 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 thecurrent study, we address a certain class of time-dependent evolution schemes with explicit discretization operatorsthat 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 acollateral 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 evolutionoperator. We formulate an a-posteriori error estimator for quantification of the resulting reduced simulation error.Numerical experiments on a parametrized convection problem, discretized with a finite volume scheme, demonstratethe applicability of the model reduction technique. We obtain a parametrized reduced model, which enables parametervariation with fast simulation response. We quantify the computational gain with respect to the non-reducedmodel and investigate the error convergence.

Details zur Publikation

Release year: 2008
Language in which the publication is written: English