# Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations

Martin A. Grepl; Yvon Maday; Ngoc C. Nguyen; Anthony T. Patera

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

- Volume: 41, Issue: 3, page 575-605
- ISSN: 0764-583X

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topGrepl, Martin A., et al. "Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations." ESAIM: Mathematical Modelling and Numerical Analysis 41.3 (2007): 575-605. <http://eudml.org/doc/250042>.

@article{Grepl2007,

abstract = {
In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter
dependence to problems involving (a) nonaffine dependence on the
parameter, 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 computational
decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral
reduced-basis approximation space, and (ii) a stable and inexpensive
interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each
instance, we discuss the reduced-basis approximation and the associated offline-online computational
procedures. Numerical results are presented to assess our approach.
},

author = {Grepl, Martin A., Maday, Yvon, Nguyen, Ngoc C., Patera, Anthony T.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Reduced-basis methods; parametrized PDEs; non-affine parameter dependence; offine-online procedures; elliptic PDEs; parabolic PDEs; nonlinear PDEs.; Galerkin method; numerical results; reduced-basis methods; offline-online procedures; nonlinear PDEs},

language = {eng},

month = {8},

number = {3},

pages = {575-605},

publisher = {EDP Sciences},

title = {Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations},

url = {http://eudml.org/doc/250042},

volume = {41},

year = {2007},

}

TY - JOUR

AU - Grepl, Martin A.

AU - Maday, Yvon

AU - Nguyen, Ngoc C.

AU - Patera, Anthony T.

TI - Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2007/8//

PB - EDP Sciences

VL - 41

IS - 3

SP - 575

EP - 605

AB -
In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter
dependence to problems involving (a) nonaffine dependence on the
parameter, 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 computational
decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral
reduced-basis approximation space, and (ii) a stable and inexpensive
interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each
instance, we discuss the reduced-basis approximation and the associated offline-online computational
procedures. Numerical results are presented to assess our approach.

LA - eng

KW - Reduced-basis methods; parametrized PDEs; non-affine parameter dependence; offine-online procedures; elliptic PDEs; parabolic PDEs; nonlinear PDEs.; Galerkin method; numerical results; reduced-basis methods; offline-online procedures; nonlinear PDEs

UR - http://eudml.org/doc/250042

ER -

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