Certified reduced basis methods for parametrized parabolic partial differential equations with non-affine source terms

Abstract

We present rigorous a posteriori output error bounds for reduced basis approximations of parametrized parabolic partial differential equations with non-affine source terms. The method employs the empirical interpolation method in order to construct affine coefficient-function approximations of the non-affine parametrized functions. Our a posteriori error bounds take both error contributions explicitly into account—the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation. To this end, we employ recently developed rigorous error bounds for the empirical interpolation method and develop error estimation and primal–dual formulations to provide rigorous bounds for the error in specific outputs of interest. We present an efficient offline–online computational procedure for the calculation of the reduced basis approximation and associated error bound. The method is thus ideally suited for many-query or real-time contexts. As a specific motivational example we consider a three-dimensional mathematical model of a welding process. Our numerical results show that we obtain efficient and reliable mathematical models which may be gainfully employed in manufacturing and product development.