In the present paper the edge stabilization technique is applied to a convection-diffusion problem with exponential boundary layers on the unit square, using a Shishkin mesh with bilinear finite elements in the layer ...In the present paper the edge stabilization technique is applied to a convection-diffusion problem with exponential boundary layers on the unit square, using a Shishkin mesh with bilinear finite elements in the layer regions and linear elements on the coarse part of the mesh. An error bound is proved for ‖πu-u^h‖Е, where πu is some interpolant of the solution u and uh the discrete solution. This supercloseness result implies an optimal error estimate with respect to the L2 norm and opens the door to the application of postprocessing for improving the discrete solution.展开更多
In this paper we propose a uniformly convergent numerical method for discretizing singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation and quantum chemis...In this paper we propose a uniformly convergent numerical method for discretizing singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation and quantum chemistry.We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint.Matched asymptotic approximations for the problem are presented to locate the positions and characterize the widths of boundary layers and/or interior layers in the solution.A uniformly convergent numerical method is proposed by using the normalized gradient flow and piecewise uniform mesh techniques based on the asymptotic approximations for the problem.Extensive numerical results are reported to demonstrate the effectiveness of our numerical method for the problems.Finally,the method is applied to compute ground and excited states of Bose-Einstein condensation in the semiclassical regime and some conclusive findings are reported.展开更多
This paper is concerned with the superconvergent points of the continuous Galerkin solutions for delay differential equations of pantograph type. We prove the local nodal superconvergence of continuous Galerkin soluti...This paper is concerned with the superconvergent points of the continuous Galerkin solutions for delay differential equations of pantograph type. We prove the local nodal superconvergence of continuous Galerkin solutions under uniform meshes and locate all the superconvergent points based on the supercloseness between the continuous Galerkin solution U and the interpolation Hhu of the exact solution u. The theoretical results are illustrated by numerical examples.展开更多
A singularly perturbed linear convection-diffusion problem for heat transfer in two dimensions with a parabolic boundary layer is solved numerically The numerical method consists of a special piecewise uniform mesh co...A singularly perturbed linear convection-diffusion problem for heat transfer in two dimensions with a parabolic boundary layer is solved numerically The numerical method consists of a special piecewise uniform mesh condensing in a neighbourhood of the parabolic layer and a standard finite difference operator satisfying a discrete maximum principle. The numerical computations demonstrate numerically that the method is epsilon-uniform in the sense that the Fate of convergence and error constant of the method are independent of the singular perturbation parameter epsilon. This means that no matter how small the singular perturbation parameter epsilon is, the numerical method produces solutions with guaranteed accuracy depending solely on the number of mesh points used.展开更多
文摘In the present paper the edge stabilization technique is applied to a convection-diffusion problem with exponential boundary layers on the unit square, using a Shishkin mesh with bilinear finite elements in the layer regions and linear elements on the coarse part of the mesh. An error bound is proved for ‖πu-u^h‖Е, where πu is some interpolant of the solution u and uh the discrete solution. This supercloseness result implies an optimal error estimate with respect to the L2 norm and opens the door to the application of postprocessing for improving the discrete solution.
基金Singapore Ministry of Education grant No.R-146-000-083-112 and would like to thank Professor Tao Tang for very helpful discussion on the subject.
文摘In this paper we propose a uniformly convergent numerical method for discretizing singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation and quantum chemistry.We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint.Matched asymptotic approximations for the problem are presented to locate the positions and characterize the widths of boundary layers and/or interior layers in the solution.A uniformly convergent numerical method is proposed by using the normalized gradient flow and piecewise uniform mesh techniques based on the asymptotic approximations for the problem.Extensive numerical results are reported to demonstrate the effectiveness of our numerical method for the problems.Finally,the method is applied to compute ground and excited states of Bose-Einstein condensation in the semiclassical regime and some conclusive findings are reported.
基金Acknowledgments. The second author is supported by NSFC （Nos. 11571027, 91430215）, by Beijing Nova Program （No. 2151100003150140） and by the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions （No. CIT＆TCD201504012）. The third author is supported by the Natural Science Foundation of Fujian Province of China （No.2013J05015）, by NSFC （No.11301437）, and by the Fundamental Research ~nds for the Central Universities （No. 20720150004）.
文摘This paper is concerned with the superconvergent points of the continuous Galerkin solutions for delay differential equations of pantograph type. We prove the local nodal superconvergence of continuous Galerkin solutions under uniform meshes and locate all the superconvergent points based on the supercloseness between the continuous Galerkin solution U and the interpolation Hhu of the exact solution u. The theoretical results are illustrated by numerical examples.
文摘A singularly perturbed linear convection-diffusion problem for heat transfer in two dimensions with a parabolic boundary layer is solved numerically The numerical method consists of a special piecewise uniform mesh condensing in a neighbourhood of the parabolic layer and a standard finite difference operator satisfying a discrete maximum principle. The numerical computations demonstrate numerically that the method is epsilon-uniform in the sense that the Fate of convergence and error constant of the method are independent of the singular perturbation parameter epsilon. This means that no matter how small the singular perturbation parameter epsilon is, the numerical method produces solutions with guaranteed accuracy depending solely on the number of mesh points used.
