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Abstract 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.

Abstract This article mainly studies a collocation spectral method for two-dimensional (2D) Sobolev equations. To this end, a collocation spectral model based on the Chebyshev polynomials for the 2D Sobolev equations is first established. And then, the existence, uniqueness, stability, and convergence of the collocation spectral numerical solutions are discussed. Finally, some numerical experiments are provided to verify the corrections of theoretical results. This implies that the collocation spectral model is very effective for solving the 2D Sobolev equations.

Abstract In this article, we devote ourselves to establishing a natural boundary element (NBE) method for the Sobolev equation in the 2D unbounded domain. To this end, we first constitute the time semi-discretized super-convergence format for the Sobolev equation by means of the Newmark method. Then, using the principle of natural boundary reduction, we establish a fully discretized NBE format based on the natural integral equation and the Poisson integral formula of this problem and analyze the errors between the exact solution and the fully discretized NBE solutions. Finally, we use some numerical experiments to verify that the NBE method is effective and feasible for solving the Sobolev equation in the 2D unbounded domain.

Abstract In this article, we devote ourselves to building a stabilized finite volume element (SFVE) method with a non-dimensional real together with two Gaussian quadratures of the nonlinear incompressible viscoelastic flow equation in a two-dimensional (2D) domain, analyzing the existence, stability, and error estimates of the SFVE solutions and verifying the validity of the preceding theoretical conclusions by numerical simulations.

Abstract In this study, we first establish a stabilized mixed finite element (SMFE) model based on parameter-free and two local Gauss integrals for the two-dimensional (2D) nonlinear incompressible viscoelastic fluid system. And then, we prove the existence, uniqueness, and convergence of the SMFE solutions. Finally, we use a numerical example to verify the correctness of the previous theoretical results.