Nonlinear model reduction based on the finite element method with interpolated coefficients: Semilinear parabolic equations

Abstract

For nonlinear reduced‐order models (ROMs), especially for those with high‐order polynomial nonlinearities or nonpolynomial nonlinearities, the computational complexity still depends on the dimension of the original dynamical system. To overcome this issue, we develop an efficient finite element (FE) discretization algorithm for nonlinear ROMs. The proposed approach approximates the nonlinear function by its FE interpolant, which makes the inner product evaluations in generating the nonlinear terms computationally cheaper than that in the standard FE discretization. Due to the separation of spatial and temporal variables in the FE interpolation, the discrete empirical interpolation method (DEIM) can be directly applied on the nonlinear functions in the same manner as that in the finite difference setting. Therefore, the main computational hurdles for applying the DEIM in the FE context are conquered. We also establish a rigorous asymptotic error estimation, which shows that the proposed approach achieves the same accuracy as that of the standard FE method under certain smoothness assumptions of the nonlinear functions. Several numerical tests are presented to validate the proposed method and verify the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1713–1741, 2015