In this paper, a-posteriori error estimators are proposed for the Legendre spectral Galerkin method for two-point boundary value problems. The key idea is to postprocess the Galerkin approximation, and the analysis sh...In this paper, a-posteriori error estimators are proposed for the Legendre spectral Galerkin method for two-point boundary value problems. The key idea is to postprocess the Galerkin approximation, and the analysis shows that the postproeess improves the order of convergence. Consequently, we obtain asymptotically exact aposteriori error estimators based on the postprocessing results. Numerical examples are included to illustrate the theoretical analysis.展开更多
In this paper,we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations.A spectral approximation scheme for the parabolic optimal control problem...In this paper,we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations.A spectral approximation scheme for the parabolic optimal control problem is presented.We obtain a posteriori error estimates of the approximated solutions for both the state and the control.展开更多
In this paper, a first order semi-discrete method of a partial integro-differential equation with a weakly singular kernel is considered. We apply Galerkin spectral method in one direction, and the inversion technique...In this paper, a first order semi-discrete method of a partial integro-differential equation with a weakly singular kernel is considered. We apply Galerkin spectral method in one direction, and the inversion technique for the Laplace transform in another direction, the result of the numerical experiment proves the accuracy of this method.展开更多
A primitive variable spectral method for simulating incompressible viscous flows inside a finite cylinder is presented.One element of originality of the proposed method is that the radial discretization of the Fourier...A primitive variable spectral method for simulating incompressible viscous flows inside a finite cylinder is presented.One element of originality of the proposed method is that the radial discretization of the Fourier coefficients depends on the Fourier mode,its dimension decreasing with the increase of the azimuthal modal number.This principle was introduced independently by Matsushima and Marcus and by Verkley in polar coordinates and is adopted here for the first time to formulate a 3D cylindrical Galerkin projection method.A second element of originality is the use of a special basis of Jacobi polynomials introduced recently for the radial dependence in the solution of Dirichlet problems.In this basis the radial operators are represented by matrices of minimal sparsity-diagonal stiffness and tridiagonal mass-provided here in closed form for the first time,and lead to a Helmholtz operator characterized by a favorable condition number.Finally,a new method is presented for eliminating the singular behaviour of the solution originated by the rotation of the lid with respect to the cylindrical wall.Thanks to these elements,the resulting Navier-Stokes spectral solver guarantees the differentiability to any order of the solution in the entire computational domain and does not suffer from the time-step stability restriction occurring in spectral methods with a point clustering close to the axis.Several test examples are offered that demonstrate the spectral accuracy of the solution method under different representative conditions.展开更多
基金supported partially by the innovation fund of Shanghai Normal Universitysupported partially by NSERC of Canada under Grant OGP0046726.
文摘In this paper, a-posteriori error estimators are proposed for the Legendre spectral Galerkin method for two-point boundary value problems. The key idea is to postprocess the Galerkin approximation, and the analysis shows that the postproeess improves the order of convergence. Consequently, we obtain asymptotically exact aposteriori error estimators based on the postprocessing results. Numerical examples are included to illustrate the theoretical analysis.
基金the National Basic Research Programthe National Natural Science Foundation of China(Grant No.2005CB321703)+2 种基金Scientific Research Fund of Hunan Provincial Education Departmentthe Outstanding Youth Scientist of the National Natural Science Foundation of China(Grant No.10625106)the National Basic Research Program of China(Grant No.2005CB321701)
文摘In this paper,we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations.A spectral approximation scheme for the parabolic optimal control problem is presented.We obtain a posteriori error estimates of the approximated solutions for both the state and the control.
基金Supported by the Natural Science Foundation of Fujian Province (2001J009, Z0511015).
文摘In this paper, a first order semi-discrete method of a partial integro-differential equation with a weakly singular kernel is considered. We apply Galerkin spectral method in one direction, and the inversion technique for the Laplace transform in another direction, the result of the numerical experiment proves the accuracy of this method.
文摘A primitive variable spectral method for simulating incompressible viscous flows inside a finite cylinder is presented.One element of originality of the proposed method is that the radial discretization of the Fourier coefficients depends on the Fourier mode,its dimension decreasing with the increase of the azimuthal modal number.This principle was introduced independently by Matsushima and Marcus and by Verkley in polar coordinates and is adopted here for the first time to formulate a 3D cylindrical Galerkin projection method.A second element of originality is the use of a special basis of Jacobi polynomials introduced recently for the radial dependence in the solution of Dirichlet problems.In this basis the radial operators are represented by matrices of minimal sparsity-diagonal stiffness and tridiagonal mass-provided here in closed form for the first time,and lead to a Helmholtz operator characterized by a favorable condition number.Finally,a new method is presented for eliminating the singular behaviour of the solution originated by the rotation of the lid with respect to the cylindrical wall.Thanks to these elements,the resulting Navier-Stokes spectral solver guarantees the differentiability to any order of the solution in the entire computational domain and does not suffer from the time-step stability restriction occurring in spectral methods with a point clustering close to the axis.Several test examples are offered that demonstrate the spectral accuracy of the solution method under different representative conditions.
