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In this paper, we mainly use a proper orthogonal decomposition (POD) to reduce the order of the coefficient vectors of the solutions for the classical collocation spectral (CS) method of two-dimensional (2D) Sobolev equations. We first establish a reduced-order extrapolating collocation spectral (ROECS) method for 2D Sobolev equations so that the ROECS method includes the same basis functions as the classic CS method and the superiority of the classic CS method. Then we use the matrix means to discuss the existence, stability, and convergence of the ROECS solutions so that the procedure of theoretical analysis becomes very concise. Lastly, we present two set of numerical examples to validate the effectiveness of theoretical conclusions and to illuminate that the ROECS method is far superior to the classic CS method, which shows that the ROECS method is quite valid to solve Sobolev equations. Therefore, both theory and method of this paper are completely different from the existing reduced-order methods.

A reduced-order extrapolating collocation spectral method based on POD for the 2D Sobolev equations