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

¹ Massachusetts Institute of Technology, Room 3-264, Cambridge, MA, USA.
² Université Pierre et Marie Curie-Paris6, UMR 7598 Laboratoire Jacques-Louis Lions, B.C. 187, 75005 Paris, France. maday@ann.jussieu.fr
³ Division of Applied Mathematics, Brown University.
⁴ Massachusetts Institute of Technology, Room 37-435, Cambridge, MA, USA.
⁵ Massachusetts Institute of Technology, Room 3-266, Cambridge, MA, USA.

Received: 5 April 2006
Revised: 1 November 2006

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.