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This study focuses on reduced-order modeling (ROM) for solving parabolic partial differential equations by applying the Fourier transform in the time variable. Each time-step solution is represented by a combination of Fourier coefficients that consist of the solutions of the frequency-dependent elliptic systems. To recover the time-space approximations, we use the Gaussian quadrature rule based on Legendre-Gauss-Lobatto points for the discrete inversion of the Fourier transformed approximate solutions. In this study, in order to extend the scope of ROM and its utility to the frequency-domain (FD) methodologies, we propose a ROM, based on the finite element approximation, by employing proper orthogonal decomposition to the frequency-dependent elliptic solutions delivered in the process of the FD method. We have verified the performance of our ROM by analyzing its errors and provide the relevant experimental results.

Reduced order modeling of the elliptic systems of the frequency-domain method and its error estimation