A Posteriori Error Analysis of the Reduced Basis Method for Nonaffine Parametrized Nonlinear PDEs

Abstract

In this paper, we present the a posteriori error analysis for the reduced basis method (RBM) applied to nonlinear variational problems that depend on a parameter in a nonaffine manner. To this end, we generalize the analysis by Veroy and Patera [Int. J. Numer. Methods Fluids, 47 (2005), pp. 773-788] to nonaffine parametrized partial differential equations. We use the empirical interpolation method (EIM) in order to approximate the nonaffine parameter dependencies by a linear combination of affine functions. We also investigate a standard dual problem formulation, in particular for the computation of a general output functional, also in combination with the EIM. First, we study the well-posedness in terms of the Brezzi-Rappaz-Raviart theory. Then, we develop a posteriori error estimates and investigate offline/online decompositions. The a posteriori error analysis allows us to introduce an adaptive sampling procedure for the choice of the modes. Numerical experiments for a convection-diffusion problem around a rotating propeller show the effectivity of the scheme.