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
In this paper,we study the adaptive planewave discretization for a cluster of eigenvalues of second-order elliptic partial differential equations.We first design an a posteriori error estimator and prove both the uppe...In this paper,we study the adaptive planewave discretization for a cluster of eigenvalues of second-order elliptic partial differential equations.We first design an a posteriori error estimator and prove both the upper and lower bounds.Based on the a posteriori error estimator,we propose an adaptive planewave method.We then prove that the adaptive planewave approximations have the linear convergence rate and quasi-optimal complexity.展开更多
We investigate the ultra weak variational formulation(UWVF)of the 2-D Helmholtz equation using a new choice of basis functions.Traditionally the UWVF basis functions are chosen to be plane waves.Here,we instead use fi...We investigate the ultra weak variational formulation(UWVF)of the 2-D Helmholtz equation using a new choice of basis functions.Traditionally the UWVF basis functions are chosen to be plane waves.Here,we instead use first kind Bessel functions.We compare the performance of the two bases.Moreover,we show that it is possible to use coupled plane wave and Bessel bases in the same mesh.As test cases we shall consider propagating plane and evanescent waves in a rectangular domain and a singular 2-D Helmholtz problem in an L-shaped domain.展开更多
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supported by the National Key R&D Program of China under grants Nos.2019YFA0709600 and 2019YFA0709601the National Natural Science Foundation of China under grants Nos.92270206,12021001 and 11671389the National Center for Mathematics and Interdisciplinary Sciences,CAS.
文摘In this paper,we study the adaptive planewave discretization for a cluster of eigenvalues of second-order elliptic partial differential equations.We first design an a posteriori error estimator and prove both the upper and lower bounds.Based on the a posteriori error estimator,we propose an adaptive planewave method.We then prove that the adaptive planewave approximations have the linear convergence rate and quasi-optimal complexity.
文摘We investigate the ultra weak variational formulation(UWVF)of the 2-D Helmholtz equation using a new choice of basis functions.Traditionally the UWVF basis functions are chosen to be plane waves.Here,we instead use first kind Bessel functions.We compare the performance of the two bases.Moreover,we show that it is possible to use coupled plane wave and Bessel bases in the same mesh.As test cases we shall consider propagating plane and evanescent waves in a rectangular domain and a singular 2-D Helmholtz problem in an L-shaped domain.
