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

TLDR

The study extends reduced‑basis approximations for linear elliptic and parabolic PDEs with affine parameters to handle nonaffine parameter dependence and nonlinear field dependence. This is achieved by approximating nonaffine and nonlinear terms with a coefficient function that enables an efficient offline‑online decomposition, employing a collateral reduced‑basis space and stable interpolation, and applying the approach to linear nonaffine and nonlinear elliptic and parabolic equations. Numerical experiments confirm the viability of the proposed method.

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.