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This paper is mainly concerned with developing and establishing the reduced-order extrapolated format about the unknown coefficient vectors in numerical solutions to the finite element (FE) method for the hyperbolic type equation. To this end, the functional-form FE format, the existence, stability, and error estimates of the FE solutions, the matrix-form FE format for the hyperbolic type equation are first proposed. Afterwards, a reduced-order extrapolated FE (ROEFE) format is established by means of a proper orthogonal decomposition (POD) technique, and the existence and stability as well as error estimates of the ROEFE solutions are demonstrated by matrix analysis, leading to an elegant theoretical development. Particularly, our work reveals that the ROEFE format possesses the same basis functions and accuracy as the FE method. Finally, some numerical tests are illustrated to computationally confirm the validity and correctness of the ROEFE format.

In this paper, by using proper orthogonal decomposition (POD) to reduce the order of the coefficient vector of the classical Crank–Nicolson finite spectral element (CCNFSE) method for the two-dimensional (2D) non-stationary Navier-Stokes equations about vorticity-stream functions, we first establish a reduced-order extrapolated Crank–Nicolson finite spectral element (ROECNFSE) method for the 2D non-stationary Navier-Stokes equations so that the ROECNFSE method has the same basis functions as the CCNFSE method and maintains all the advantages of the CCNFSE method. Then, by means of matrix analysis, we analyze the existence, stability, and convergence of the ROECNFSE solutions such that the theoretical analysis becomes very simple and convenient. Finally, we utilize some numerical experiments to validate that the numerical computational consequences are consistent with the theoretical ones such that the effectiveness and feasibility of the ROECNFSE method are further verified. Therefore, both theory and method of this paper is new and completely different from the existing reduced-order methods.