We give here an overview of the orbital-flee density functional theory that is used for modeling atoms and molecules. We review typical approximations to the kinetic energy, exchange-correlation corrections to the k...We give here an overview of the orbital-flee density functional theory that is used for modeling atoms and molecules. We review typical approximations to the kinetic energy, exchange-correlation corrections to the kinetic and Hartree energies, and constructions of the pseudopotentials. We discuss numerical discretizations for the orbital-free methods and include several numerical results for illustrations.展开更多
To obtain convergent numerical approximations without using any orthogonalization operations is of great importance in electronic structure calculations.In this paper,we propose and analyze a class of iteration scheme...To obtain convergent numerical approximations without using any orthogonalization operations is of great importance in electronic structure calculations.In this paper,we propose and analyze a class of iteration schemes for the discretized Kohn-Sham Density Functional Theory model,with which the iterative approximations are guaranteed to converge to the Kohn-Sham orbitals without any orthogonalization as long as the initial orbitals are orthogonal and the time step sizes are given properly.In addition,we present a feasible and efficient approach to get suitable time step sizes and report some numerical experiments to validate our theory.展开更多
Based on two-grid discretizations, in this paper, some new local and parallel finite element algorithms are proposed and analyzed for the stationary incompressible Navier- Stokes problem. These algorithms are motivate...Based on two-grid discretizations, in this paper, some new local and parallel finite element algorithms are proposed and analyzed for the stationary incompressible Navier- Stokes problem. These algorithms are motivated by the observation that for a solution to the Navier-Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One major technical tool for the analysis is some local a priori error estimates that are also obtained in this paper for the finite element solutions on general shape-regular grids.展开更多
In this paper, both the standard finite element discretization and a two-scale finite element discretization for SchrSdinger equations are studied. The numerical analysis is based on the regularity that is also obtain...In this paper, both the standard finite element discretization and a two-scale finite element discretization for SchrSdinger equations are studied. The numerical analysis is based on the regularity that is also obtained in this paper for the Schroedinger equations. Very satisfying applications to electronic structure computations are provided, too.展开更多
The Hohenberg-Kohn theorem plays a fundamental role in density functional theory, which has become the most popular and powerful computational approach to study the electronic structure of matter.In this article, we s...The Hohenberg-Kohn theorem plays a fundamental role in density functional theory, which has become the most popular and powerful computational approach to study the electronic structure of matter.In this article, we study the Hohenberg-Kohn theorem for a class of external potentials based on a unique continuation principle.展开更多
In this paper, a two-scale higher-order finite element discretization scheme is proposed and analyzed for a Schroedinger equation on tensor product domains. With the scheme, the solution of the eigenvalue problem on a...In this paper, a two-scale higher-order finite element discretization scheme is proposed and analyzed for a Schroedinger equation on tensor product domains. With the scheme, the solution of the eigenvalue problem on a fine grid can be reduced to an eigenvalue problem on a much coarser grid together with some eigenvalue problems on partially fine grids. It is shown theoretically and numerically that the proposed two-scale higher-order scheme not only significantly reduces the number of degrees of freedom but also produces very accurate approximations.展开更多
Based on the Boolean sum technique, we introduce and analyze in this paper a class of multi-level iterative corrections for finite dimensional approximations. This type of multi-level corrections is adaptive and can p...Based on the Boolean sum technique, we introduce and analyze in this paper a class of multi-level iterative corrections for finite dimensional approximations. This type of multi-level corrections is adaptive and can produce highly accurate approximations. For illustration, we present some old and new finite element correction schemes for an elliptic boundary value problem.展开更多
In this paper,a two-scale finite element approach is proposed and analyzed for approximationsof Green's function in three-dimensions.This approach is based on a two-scale finite elementspace defined,respectively,o...In this paper,a two-scale finite element approach is proposed and analyzed for approximationsof Green's function in three-dimensions.This approach is based on a two-scale finite elementspace defined,respectively,on the whole domain with size H and on some subdomain containing singularpoints with size h (h << H).It is shown that this two-scale discretization approach is very efficient.In particular,the two-scale discretization approach is applied to solve Poisson-Boltzmann equationssuccessfully.展开更多
In this paper,we study an adaptive finite element method for a class of nonlinear eigenvalue problems resulting from quantum physics that may have a nonconvex energy functional.We prove the convergence of adaptive fin...In this paper,we study an adaptive finite element method for a class of nonlinear eigenvalue problems resulting from quantum physics that may have a nonconvex energy functional.We prove the convergence of adaptive finite element approximations and present several numerical examples of micro-structure of matter calculations that support our theory.展开更多
The Teter,Payne,and Allan“preconditioning”function plays a significant role in planewave DFT calculations.This function is often called the TPA preconditioner.We present a detailed study of this“preconditioning”fu...The Teter,Payne,and Allan“preconditioning”function plays a significant role in planewave DFT calculations.This function is often called the TPA preconditioner.We present a detailed study of this“preconditioning”function.We develop a general formula that can readily generate a class of“preconditioning”functions.These functions have higher order approximation accuracy and fulfill the two essential“preconditioning”purposes as required in planewave DFT calculations.Our general class of functions are expected to have applications in other areas.展开更多
The finite element method is a promising method for electronic structure calculations.In this paper,a new parallelmesh refinementmethod for electronic structure calculations is presented.Some properties of the method ...The finite element method is a promising method for electronic structure calculations.In this paper,a new parallelmesh refinementmethod for electronic structure calculations is presented.Some properties of the method are investigated to make itmore efficient andmore convenient for implementation.Several practical issues such as distributed memory parallel computation,less tetrahedra prototypes,and the assignment of the mesh elements carried out independently in each sub-domain will be discussed.The numerical experiments on the periodic system,cluster and nano-tube are presented to demonstrate the effectiveness of the proposed method.展开更多
基金supported by the National Science Foundation of China under the grant 10425105the National Basic Research Program under the grant 2005CB321704.
文摘We give here an overview of the orbital-flee density functional theory that is used for modeling atoms and molecules. We review typical approximations to the kinetic energy, exchange-correlation corrections to the kinetic and Hartree energies, and constructions of the pseudopotentials. We discuss numerical discretizations for the orbital-free methods and include several numerical results for illustrations.
基金This work was supported by the National Key R&D Program of China under grants 2019YFA0709600,2019YFA0709601the National Natural Science Foundation of China under grant 12021001.
文摘To obtain convergent numerical approximations without using any orthogonalization operations is of great importance in electronic structure calculations.In this paper,we propose and analyze a class of iteration schemes for the discretized Kohn-Sham Density Functional Theory model,with which the iterative approximations are guaranteed to converge to the Kohn-Sham orbitals without any orthogonalization as long as the initial orbitals are orthogonal and the time step sizes are given properly.In addition,we present a feasible and efficient approach to get suitable time step sizes and report some numerical experiments to validate our theory.
基金The first author was partially subsidized by the NSF of China 10371095. The third author was partially supported by the National Science Foundation of China under the grant 10425105 and the National Basic Research Program under the grant 2005CB321704.
文摘Based on two-grid discretizations, in this paper, some new local and parallel finite element algorithms are proposed and analyzed for the stationary incompressible Navier- Stokes problem. These algorithms are motivated by the observation that for a solution to the Navier-Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One major technical tool for the analysis is some local a priori error estimates that are also obtained in this paper for the finite element solutions on general shape-regular grids.
基金the National Science Founda-tion of China under grant 10425105the National Basic Research Program under grant 2005CB321704
文摘In this paper, both the standard finite element discretization and a two-scale finite element discretization for SchrSdinger equations are studied. The numerical analysis is based on the regularity that is also obtained in this paper for the Schroedinger equations. Very satisfying applications to electronic structure computations are provided, too.
基金supported by National Natural Science Foundation of China (Grant Nos. 91730302,91330202 and 11671389)the Key Research Program of Frontier Sciences of the Chinese Academy of Sciences(Grant No. QYZDJ-SSW-SYS010)
文摘The Hohenberg-Kohn theorem plays a fundamental role in density functional theory, which has become the most popular and powerful computational approach to study the electronic structure of matter.In this article, we study the Hohenberg-Kohn theorem for a class of external potentials based on a unique continuation principle.
基金supported by the National Natural Science Foundation of China (10701083 and 10425105)the National Basic Research Program of China (2005CB321704).
文摘In this paper, a two-scale higher-order finite element discretization scheme is proposed and analyzed for a Schroedinger equation on tensor product domains. With the scheme, the solution of the eigenvalue problem on a fine grid can be reduced to an eigenvalue problem on a much coarser grid together with some eigenvalue problems on partially fine grids. It is shown theoretically and numerically that the proposed two-scale higher-order scheme not only significantly reduces the number of degrees of freedom but also produces very accurate approximations.
基金supported by the National Science Foundation of China under grant 10425105the National Basic Research Program under grant 2005CB321704
文摘Based on the Boolean sum technique, we introduce and analyze in this paper a class of multi-level iterative corrections for finite dimensional approximations. This type of multi-level corrections is adaptive and can produce highly accurate approximations. For illustration, we present some old and new finite element correction schemes for an elliptic boundary value problem.
基金partially supported by the National Science Foundation of China under Grant Nos. 10425105 and 10871198the National Basic Research Program under Grant No. 2005CB321704
文摘In this paper,a two-scale finite element approach is proposed and analyzed for approximationsof Green's function in three-dimensions.This approach is based on a two-scale finite elementspace defined,respectively,on the whole domain with size H and on some subdomain containing singularpoints with size h (h << H).It is shown that this two-scale discretization approach is very efficient.In particular,the two-scale discretization approach is applied to solve Poisson-Boltzmann equationssuccessfully.
基金This work was partially supported by the National Science Foundation of China under grants 10871198 and 10971059the National Basic Research Program of China under grant 2005CB321704the National High Technology Research and Development Program of China under grant 2009AA01A134。
文摘In this paper,we study an adaptive finite element method for a class of nonlinear eigenvalue problems resulting from quantum physics that may have a nonconvex energy functional.We prove the convergence of adaptive finite element approximations and present several numerical examples of micro-structure of matter calculations that support our theory.
基金Y.Zhou is supported in part by the NSF under grant DMS-1228271 and by a J.T.Oden fellowship from the University of Texas at Austin.J.R.Chelikowsky acknowledges support provided by the Scientific Discovery through Advanced Computing(SciDAC)program funded by U.S.Department of Energy,Office of Science,Advanced Scientific Computing Research and Basic Energy Sciences under award number DE-SC0008877X.Gao is supported in part by the NSF of China under grant 61300012 and the Defense Industrial Technology Development Program+2 种基金A.Zhou is supported in part by the Funds for Creative Research Groups of China under grant 11321061the National Basic Research Program of China under grant 2011CB309703the National Center for Mathematics and Interdisciplinary Sciences,Chinese Academy of Sciences.
文摘The Teter,Payne,and Allan“preconditioning”function plays a significant role in planewave DFT calculations.This function is often called the TPA preconditioner.We present a detailed study of this“preconditioning”function.We develop a general formula that can readily generate a class of“preconditioning”functions.These functions have higher order approximation accuracy and fulfill the two essential“preconditioning”purposes as required in planewave DFT calculations.Our general class of functions are expected to have applications in other areas.
基金This work is partially supported by NSF of China,the National Basic Research Program of China,MOE and Shanghai basic research project.
文摘The finite element method is a promising method for electronic structure calculations.In this paper,a new parallelmesh refinementmethod for electronic structure calculations is presented.Some properties of the method are investigated to make itmore efficient andmore convenient for implementation.Several practical issues such as distributed memory parallel computation,less tetrahedra prototypes,and the assignment of the mesh elements carried out independently in each sub-domain will be discussed.The numerical experiments on the periodic system,cluster and nano-tube are presented to demonstrate the effectiveness of the proposed method.
