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

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 non-affine manner. To this end, we generalize the analysis by Veroy and Patera ([16]) to non-affine parametrized partial differential equations. We use the Empirical Interpolation Method (EIM) in order to approximate the non-affine 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 of all involved problems in terms of the Brezzi-Rappaz-Raviart theory. Then, we develop a-posteriori error estimates for all problems and investigate offline/online decompositions. The a-posteriori error analysis allows us to introduce an adaptive sampling procedure for the choice of the snapshots. Numerical experiments for a convection-diffusion problem around a rotating propeller show the effectivity of the scheme.