Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schr?dinger...Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schr?dinger equation with the finite element method. The error estimate and superconvergence property with order O(hk+1)in the H1norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(hk+1+ τ2) in the H1norm can be obtained in the Crank-Nicolson fully discrete scheme.展开更多
The effect of numerical integration in finite element methods applied to a class of nonlinear parabolic equations is considered and some sufficient conditions on the quadrature scheme to ensure that the order of conve...The effect of numerical integration in finite element methods applied to a class of nonlinear parabolic equations is considered and some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration are given. Optimal L 2 and H 1 estimates for the error and its time derivative are established.展开更多
In order to determine the projected coordinate origin in the cone-beam CT scanning system with respect to the Feldkamp-Davis-Kress （FDK） algorithm, we propose a simple yet feasible method to accurately measure the p...In order to determine the projected coordinate origin in the cone-beam CT scanning system with respect to the Feldkamp-Davis-Kress （FDK） algorithm, we propose a simple yet feasible method to accurately measure the projected coordinate origin. This method was established on the basis of the theory that the projection of a spherical object in the cone-beam field is an ellipse. We first utilized image processing and the least square estimation method to get each major axis of the elliptical Digital Radiography （DR） projections of a group of spherical objects. Then we determined the intersection point of the group of major axis by solving an over-determined equation set that was composed by the major axis equations of all the elliptical projections. Based on the experimental results, this new method was proved to be easy to implement in practical scanning systems with high accuracy and anti-noise capability.展开更多
The superconvergence of a two-dimensional time-independent nonlinear Schrodinger equation are analyzed with the rectangular Lagrange typefinite element of order k.Firstly,the error estimate and superclose property are ...The superconvergence of a two-dimensional time-independent nonlinear Schrodinger equation are analyzed with the rectangular Lagrange typefinite element of order k.Firstly,the error estimate and superclose property are given in H^(1)-norm with order O(h^(k+1))between thefinite element solution u_(h) and the interpolation func-tion uI by use of the elliptic projection operator.Then,the global superconvergence is obtained by the interpolation post-processing technique.In addition,some numerical examples with the order k=1 and k=2 are provided to demonstrate the theoretical analysis.展开更多
基金Project supported by the National Natural Science Foundation of China(No.11671157)
文摘Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schr?dinger equation with the finite element method. The error estimate and superconvergence property with order O(hk+1)in the H1norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(hk+1+ τ2) in the H1norm can be obtained in the Crank-Nicolson fully discrete scheme.
文摘The effect of numerical integration in finite element methods applied to a class of nonlinear parabolic equations is considered and some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration are given. Optimal L 2 and H 1 estimates for the error and its time derivative are established.
基金Supported by the National Natural Science Foundation of China (60872080)Peking Educational Committee Corporate Construction Project
文摘In order to determine the projected coordinate origin in the cone-beam CT scanning system with respect to the Feldkamp-Davis-Kress （FDK） algorithm, we propose a simple yet feasible method to accurately measure the projected coordinate origin. This method was established on the basis of the theory that the projection of a spherical object in the cone-beam field is an ellipse. We first utilized image processing and the least square estimation method to get each major axis of the elliptical Digital Radiography （DR） projections of a group of spherical objects. Then we determined the intersection point of the group of major axis by solving an over-determined equation set that was composed by the major axis equations of all the elliptical projections. Based on the experimental results, this new method was proved to be easy to implement in practical scanning systems with high accuracy and anti-noise capability.
基金The authors would like to express the sincere thanks to anonymous referees for their valuable comments.This research is supported by National Natural Science Foundation of China(No.11671340)Hunan Provincial Natural Science Foundation of China(Nos.2021JJ30209,2021JJ50108 and 2021JJ30178).
文摘The superconvergence of a two-dimensional time-independent nonlinear Schrodinger equation are analyzed with the rectangular Lagrange typefinite element of order k.Firstly,the error estimate and superclose property are given in H^(1)-norm with order O(h^(k+1))between thefinite element solution u_(h) and the interpolation func-tion uI by use of the elliptic projection operator.Then,the global superconvergence is obtained by the interpolation post-processing technique.In addition,some numerical examples with the order k=1 and k=2 are provided to demonstrate the theoretical analysis.
