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.

Classification: 35J25, 35J60, 35K15, 35K55

Keywords: reduced-basis methods, parametrized PDEs, non-affine parameter dependence, offine-online procedures, elliptic PDEs, parabolic PDEs, nonlinear PDEs

@article{M2AN_2007__41_3_575_0, author = {Grepl, Martin A. and Maday, Yvon and Nguyen, Ngoc C. and Patera, Anthony T.}, title = {Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {41}, number = {3}, year = {2007}, pages = {575-605}, doi = {10.1051/m2an:2007031}, zbl = {pre05289385}, mrnumber = {2355712}, language = {en}, url = {http://www.numdam.org/item/M2AN_2007__41_3_575_0} }

Grepl, Martin A.; Maday, Yvon; Nguyen, Ngoc C.; Patera, Anthony T. Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 3, pp. 575-605. doi : 10.1051/m2an:2007031. http://www.numdam.org/item/M2AN_2007__41_3_575_0/

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