  For nonlinear reduced-order models, especially for those with non-polynomial
nonlinearities, the computational complexity still depends on the dimension of
the original dynamical system. As a result, the reduced-order model loses its
computational efficiency, which, however, is its the most significant
advantage. Nonlinear dimensional reduction methods, such as the discrete
empirical interpolation method, have been widely used to evaluate the nonlinear
terms at a low cost. But when the finite element method is utilized for the
spatial discretization, nonlinear snapshot generation requires inner products
to be fulfilled, which costs lots of off-line time. Numerical integrations are
also needed over elements sharing the selected interpolation points during the
simulation, which keeps on-line time high. In this paper, we extend the finite
element method with interpolated coefficients to nonlinear reduced-order
models. It approximates the nonlinear function in the reduced-order model by
its finite element interpolation, which makes coefficient matrices of the
nonlinear terms pre-computable and, thus, leads to great savings in the
computational efforts. Due to the separation of spatial and temporal variables
in the finite element interpolation, the discrete empirical interpolation
method can be directly applied on the nonlinear functions. Therefore, the main
computational hurdles when applying the discrete empirical interpolation method
in the finite element 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 finite element method under certain
smoothness assumptions of the nonlinear functions. Several numerical tests are
presented to validate the proposed method and verify the theoretical results.
