This paper is devoted to the mixed Legendre spectral-finite element approximation of the three-dimensional, non-periodic, unsteady Navier-Stokes equations. A class of fully discrete schemes are constructed with artifi...This paper is devoted to the mixed Legendre spectral-finite element approximation of the three-dimensional, non-periodic, unsteady Navier-Stokes equations. A class of fully discrete schemes are constructed with artificial compression. The generalized stability and convergence are proved strictly on the assumption that the two-dimensional inf-sup condition of the finite element approximation is satisfied.展开更多
A combined Legendre spectral-finite element approximation is proposed for solving two-dimensional unsteady Navier-Stokes equation. The artificial compressibility is used. The generalized stability and convergence are ...A combined Legendre spectral-finite element approximation is proposed for solving two-dimensional unsteady Navier-Stokes equation. The artificial compressibility is used. The generalized stability and convergence are proved strictly. Some numerical results show the advantages of this method.展开更多
In this study, a classical spectral-finite difference scheme （SFDS） for the three-dimensional （3D） parabolic equation is reduced by using proper orthogonal decomposition （POD） and singular value decomposition （...In this study, a classical spectral-finite difference scheme （SFDS） for the three-dimensional （3D） parabolic equation is reduced by using proper orthogonal decomposition （POD） and singular value decomposition （SVD）. First, the 3D parabolic equation is discretized in spatial variables by using spectral collocation method and the discrete scheme is transformed into matrix formulation by tensor product. Second~ the classical SFDS is obtained by difference discretization in time-direction. The ensemble of data are comprised with the first few transient solutions of the classical SFDS for the 3D parabolic equation and the POD bases of ensemble of data are generated by using POD technique and SVD. The unknown quantities of the classical SFDS are replaced with the linear combination of POD bases and a reduced- order extrapolation SFDS with lower dimensions and sufficiently high accuracy for the 3D parabolic equation is established. Third, the error estimates between the classical SFDS solutions and the reduced-order extrapolation SFDS solutions and the implementation for solving the reduced-order extrapolation SFDS are provided. Finally, a numerical example shows that the errors of numerical computations are consistent with the theoretical results. Moreover, it is shown that the reduced-order extrapolation SFDS is effective and feasible to find the numerical solutions for the 3D parabolic equation.展开更多
In this paper, we construct a spectral-finite element scheme for solving semi-periodical two-dimensional vorticity equations. The error between the genuine solution and approximate solutionis estimated strictly. The n...In this paper, we construct a spectral-finite element scheme for solving semi-periodical two-dimensional vorticity equations. The error between the genuine solution and approximate solutionis estimated strictly. The numerical results show the advantages of such a method. The techniqueused in this paper can be easily generalized to three-dimensional problems.展开更多
Proposes a mixed Chebyshev spectral-finite element method for solving two-dimensional unsteady Navier-Stokes equation. Information on the spectral method; Discussion; Error estimations.
Combined Chebyshev spectral-finite element schemes are constructed for three-dimensionalunsteady vorticity equation. The generalized stability and convergence are proved strictly.
We present a fast Poisson solver on spherical shells.With a special change of variable,the radial part of the Laplacian transforms to a constant coefficient differential operator.As a result,the Fast Fourier Transform...We present a fast Poisson solver on spherical shells.With a special change of variable,the radial part of the Laplacian transforms to a constant coefficient differential operator.As a result,the Fast Fourier Transform can be applied to solve the Poisson equation with O(N^(3) logN)operations.Numerical examples have confirmed the accuracy and robustness of the new scheme.展开更多
文摘This paper is devoted to the mixed Legendre spectral-finite element approximation of the three-dimensional, non-periodic, unsteady Navier-Stokes equations. A class of fully discrete schemes are constructed with artificial compression. The generalized stability and convergence are proved strictly on the assumption that the two-dimensional inf-sup condition of the finite element approximation is satisfied.
文摘A combined Legendre spectral-finite element approximation is proposed for solving two-dimensional unsteady Navier-Stokes equation. The artificial compressibility is used. The generalized stability and convergence are proved strictly. Some numerical results show the advantages of this method.
基金This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11271127, 11361035）, the Doctoral Foundation of Guizhou Normal University, and the Science and Technology Fund of Guizhou Province （Grant No. 7052） in 2014.
文摘In this study, a classical spectral-finite difference scheme （SFDS） for the three-dimensional （3D） parabolic equation is reduced by using proper orthogonal decomposition （POD） and singular value decomposition （SVD）. First, the 3D parabolic equation is discretized in spatial variables by using spectral collocation method and the discrete scheme is transformed into matrix formulation by tensor product. Second~ the classical SFDS is obtained by difference discretization in time-direction. The ensemble of data are comprised with the first few transient solutions of the classical SFDS for the 3D parabolic equation and the POD bases of ensemble of data are generated by using POD technique and SVD. The unknown quantities of the classical SFDS are replaced with the linear combination of POD bases and a reduced- order extrapolation SFDS with lower dimensions and sufficiently high accuracy for the 3D parabolic equation is established. Third, the error estimates between the classical SFDS solutions and the reduced-order extrapolation SFDS solutions and the implementation for solving the reduced-order extrapolation SFDS are provided. Finally, a numerical example shows that the errors of numerical computations are consistent with the theoretical results. Moreover, it is shown that the reduced-order extrapolation SFDS is effective and feasible to find the numerical solutions for the 3D parabolic equation.
文摘In this paper, we construct a spectral-finite element scheme for solving semi-periodical two-dimensional vorticity equations. The error between the genuine solution and approximate solutionis estimated strictly. The numerical results show the advantages of such a method. The techniqueused in this paper can be easily generalized to three-dimensional problems.
基金the Chinese State Key Project of Basic ResearchN.G1999032804 and the Shanghai Natural Science Foundation N.00JC14057.
文摘Proposes a mixed Chebyshev spectral-finite element method for solving two-dimensional unsteady Navier-Stokes equation. Information on the spectral method; Discussion; Error estimations.
文摘Combined Chebyshev spectral-finite element schemes are constructed for three-dimensionalunsteady vorticity equation. The generalized stability and convergence are proved strictly.
基金The research of Liu was supported by the NSF grant DMS 10-11738The research of Wang was supported by National Science Council of Taiwan under grant 97-2115-M-007-005In addition,this work is also supported in part by National Center for Theoretical Sciences of Taiwan.
文摘We present a fast Poisson solver on spherical shells.With a special change of variable,the radial part of the Laplacian transforms to a constant coefficient differential operator.As a result,the Fast Fourier Transform can be applied to solve the Poisson equation with O(N^(3) logN)operations.Numerical examples have confirmed the accuracy and robustness of the new scheme.
