A reduced-order two-grid method based on POD technique for the semilinear parabolic equation

In the conventional two-grid (TG) method, the nonlinear system on the fine grid is transformed into a nonlinear subsystem on the coarse grid and a linear subsystem on the fine grid to reduce computational costs. It has been successfully applied in various fields. Nonetheless, its computational efficiency remains relatively low. For this, we develop a novel reduced-order two-grid (ROTG) method with less degrees of freedom for solving the semilinear parabolic equation. For the two subsystems mentioned, the proper orthogonal decomposition (POD) technique is utilized to substantially reduce degrees of freedom. An a priori error estimate for the ROTG scheme is derived. Finally, we conduct several numerical tests to observe the ROTG method's behavior and verify the theoretical analysis.