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Efficient reduced-basis treatment of nonaffineand nonlinear partial differential equations

Published online by Cambridge University Press:  02 August 2007

Martin A. Grepl ,
Yvon Maday ,
Ngoc C. Nguyen  and
Anthony T. Patera
Martin A. Grepl
Affiliation:
Massachusetts Institute of Technology, Room 3-264, Cambridge, MA, USA.
Yvon Maday
Affiliation:
Université Pierre et Marie Curie-Paris6, UMR 7598 Laboratoire Jacques-Louis Lions, B.C. 187, 75005 Paris, France. maday@ann.jussieu.fr Division of Applied Mathematics, Brown University.
Ngoc C. Nguyen
Affiliation:
Massachusetts Institute of Technology, Room 37-435, Cambridge, MA, USA.
Anthony T. Patera
Affiliation:
Massachusetts Institute of Technology, Room 3-266, Cambridge, MA, USA.
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Abstract

In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameterdependence to problems involving (a) nonaffine dependence on theparameter, and (b) nonlinear dependence on the field variable.The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computationaldecomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateralreduced-basis approximation space, and (ii) a stable and inexpensiveinterpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in eachinstance, we discuss the reduced-basis approximation and the associated offline-online computationalprocedures. Numerical results are presented to assess our approach.

Keywords

Reduced-basis methodsparametrized PDEsnon-affine parameter dependenceoffine-online procedureselliptic PDEsparabolic PDEsnonlinear PDEs.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 3 , May 2007 , pp. 575 - 605
Copyright
© EDP Sciences, SMAI, 2007

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