Simultaneous Empirical Interpolation and Reduced Basis Method: Application to Non-linear Multi-Physics Problem

Abstract

This paper focuses on the reduced basis method in the case of non-linear and non-affinely parametrized partial differential equations where affine decomposition is not obtained. In this context, Empirical Interpolation Method (EIM) (Barrault et al. C R Acad Sci Paris Ser I 339(9):667–672, 2004) is commonly used to recover the affine decomposition necessary to deploy the Reduced Basis (RB) methodology. The build of each EIM approximation requires many finite element solves which increases significantly the computational cost hence making it inefficient on large problems (Daversin et al. ESAIM proceedings, EDP Sciences, Paris, vol. 43, pp. 225–254, 2013). We propose a Simultaneous EIM and RB method (ser) whose principle is based on the use of reduced basis approximations into the EIM building step. The number of finite element solves required by ser can drop to N + 1 where N is the dimension of the RB approximation space, thus providing a huge computational gain. The ser method has already been introduced in Daversin and Prud’homme (C R Acad Sci Paris Ser I 353:1105–1109, 2015) through which it is illustrated on a 2D benchmark itself introduced in Grepl et al. (Modél Math Anal Numér 41(03):575–605, 2007). This paper develops the ser method with some variants and in particular a multilevel ser, ser(ℓ) which improves significantly ser at the cost of ℓN + 1 finite element solves. Finally we discuss these extensions on a 3D multi-physics problem.