A self-adaptive multilevel finite element method for the stationary Schrödinger equation in three space dimensions

Ackermann, Jörg; Erdmann, Bodo; Roitzsch, Rainer

doi:10.1063/1.468257

Cited by 39 publications

(27 citation statements)

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“…Inverse iteration was used to solve the large-dimension eigenvalue problem for the two-dimensional harmonic oscillator and the linear H 2+ 3 molecule; accuracies comparable or even superior to the previous study were reported. Subsequently, they extended their method to three dimensions (Ackerman et al, 1994); in this work, conjugate-gradient techniques were employed to solve the eigenproblem. Results were presented for the three-dimensional harmonic oscillator and H 2+ 3 in the equilateral triangle geometry.…”

Section: Applicationsmentioning

confidence: 99%

“…Test calculations were performed on the linear H 2+ 3 molecule, and highly accurate results (to 10 −7 au) were obtained. Ackerman and Roitzsch (1993) proposed an adaptive multilevel FE approach which utilized high-order shape functions. Inverse iteration was used to solve the large-dimension eigenvalue problem for the two-dimensional harmonic oscillator and the linear H 2+ 3 molecule; accuracies comparable or even superior to the previous study were reported.…”

Section: Applicationsmentioning

confidence: 99%

“…White et al (1989) employed a cubic-polynomial basis and constructed an orthogonal basis from the nonorthogonal set. Ackermann et al (1994) used a tetrahedral discretization with orders p = 1 − 5. Pask et al (1999) utilized piecewise cubic functions (termed "serendipity" elements).…”

Section: Finite-element Basesmentioning

confidence: 99%

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Real-space mesh techniques in density-functional theory

2000

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This review discusses progress in efficient solvers which have as their foundation a representation in real space, either through finite-difference or finite-element formulations. The relationship of real-space approaches to linear-scaling electrostatics and electronic structure methods is first discussed. Then the basic aspects of real-space representations are presented. Multigrid techniques for solving the discretized problems are covered; these numerical schemes allow for highly efficient solution of the grid-based equations. Applications to problems in electrostatics are discussed, in particular numerical solutions of Poisson and Poisson-Boltzmann equations. Next, methods for solving self-consistent eigenvalue problems in real space are presented; these techniques have been extensively applied to solutions of the Hartree-Fock and Kohn-Sham equations of electronic structure, and to eigenvalue problems arising in semiconductor and polymer physics. Finally, real-space methods have found recent application in computations of optical response and excited states in time-dependent density functional theory, and these computational developments are summarized. Multiscale solvers are competitive with the most efficient available plane-wave techniques in terms of the number of self-consistency steps required to reach the ground state, and they require less work in each self-consistency update on a uniform grid. Besides excellent efficiencies, the decided advantages of the real-space multiscale approach are 1) the near-locality of each function update, 2) the ability to handle global eigenfunction constraints and potential updates on coarse levels, and 3) the ability to incorporate adaptive local mesh refinements without loss of optimal multigrid efficiencies.

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Section: Applicationsmentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

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Real-space mesh techniques in density-functional theory

2000

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“…The approximation of the wave function as a linear combination of these local polynomials is called a finite-element description. FEM treatment for the helium atom in the infinite nuclear mass approximation [46,47,48,49,50], and the hydrogen molecular ion H + 2 in the Born-Oppenheimer approximation, have been presented by several authors [51,52,53,54,55,56]. An adaptable FEM approach was used by Ackermann [57] and co-workers to obtain the energy values to a precision of 10 −11 with moderate computational effort.…”

Section: Perturbation Hartree-fock and The Finite Element Methodsmentioning

confidence: 99%

Kinetic energy operator approach to the quantum three-body problem with Coulomb interactions

Chi

Fang

Hsiang

et al. 2007

Solid State Communications

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Abstract. We present a review of the quantum three-body problem, with emphasis on the different methodologies, different three-body atomic systems and their historical interest. With the review as the background, a more recently proposed non-variational, kinetic energy operator approach to the solution of quantum three-body problem is presented, based on the utilization of symmetries intrinsic to the kinetic energy operator, i.e., the three-body Laplacian operator with the respective masses. Through a four-step reduction process, the nine dimensional problem is reduced to a one dimensional coupled system of ordinary differential equations, amenable to accurate numerical solution as an infinite-dimensional algebraic eigenvalue problem. A key observation in this reduction process is that in the functional subspace of the kinetic energy operator where all the rotational degrees of freedom have been projected out, there is an intrinsic symmetry which can be made explicit through the introduction of Jacobi-spherical coordinates. A numerical scheme is presented whereby the Coulomb matrix elements are calculated to a high degree of accuracy with minimal effort, and the truncation of the linear equations is carried out through a systematic procedure. The resulting matrix equations are solved through an iteration process. Numerical results are presented for (1) the negative hydrogen ion H − , (2) the helium and heliumlike ions (Z = 3 ∼ 6), (3) the hydrogen molecular ion H + 2 , and (4) the positronium negative ion Ps − . Up to thirteen-significant-figure accuracy is achieved for the ground state eigenvalues when double precision programming is used. Comparison with the variational and other approaches shows our ground state eigenvalues to be comparable, generally with less decimal digits than the variational results, but can yield highly accurate wavefunctions as by-products. Results on low-lying excited states and their wavefunctions are obtained simultaneously with the ground state properties, some at accuracies not achieved by other methods. In particular, for the doubly excited state 3 P e of H − and the 1,3 P e states of helium, some results are obtained for the first time. Also, we have calculated fourteen H + 2 excited states, up to its dissociation level. Analysis of the wavefunction characteristics, especially in relation to the electronelectron correlation effects, are presented. A significant advantage of the kinetic energy operator approach is its general applicability to different three-body systems, with only the charges, masses, and the symmetry of the desired state as the required inputs. Potential applications of the present approach to scattering and other problems are noted.

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“…The problem of using mixed basis functions was solved by Yamakawa and Hyodo [2005] who addressed the problem of treating core electrons in molecular orbits. The use of mixed basis functions as a local error estimate by Ackermann and Roitzsch [1993]; Ackermann et al [1994] then leads to the so-called "multilevel method" and the "hierarchical method" investigated by Sugawara [1998].…”

Section: Introductionmentioning

confidence: 99%

Parallel Implementation

Trobec¹,

Kosec²

2015

Parallel Scientific Computing

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ResumenEl problema de valores propios (también llamado de autovalores) está presente en multitud dé areas científicas a través de, por ejemplo, la resolución de ecuaciones en derivadas parciales, reducción de modelos y cálculo de funciones matriciales, entre otras muchas aplicaciones. Si los problemas son de dimensión moderada (menor a 10 6 ), pueden ser abordados mediante los llamados métodos directos, como el algoritmo iterativo QR o el método de divide y vencerás. Sin embargo, si el problema es de gran dimensión y sólo se requieren unas pocas soluciones, comparado con el tamaño del problema, los métodos iterativos pueden resultar más eficientes. Además, los métodos iterativos pueden ofrecer mejor rendimiento en arquitecturas de altas prestaciones, como las plataformas paralelas de memoria distribuida, en las que existe un cierto número de nodos computacionales con espacio de memoria propio y sólo pueden compartir información y sincronizarse mediante el paso de mensajes.Esta tesis aborda la implementación de métodos de tipo Davidson, destacando Generalized Davidson y Jacobi-Davidson, una clase de métodos iterativos que pueden ser competitivos en casos especialmente difíciles como calcular valores propios en el interior del espectro o cuando la factorización de matrices es prohibitiva o ineficiente, y sólo es posible una factorización aproximada. La implementación se desarrolla en SLEPc (Scalable Library for Eigenvalue Problem Computations), librería de software libre para la resolución de problemas de gran tamaño de valores propios, problemas cuadráticos de valores propios y problemas de valores singulares, entre otros.Como resultado de esta tesis, SLEPc incorpora una implementación de Generalized Davidson y Jacobi-Davidson para problemas estándares y generalizados, tanto hermitianos como no hermitianos, característica a destacar ya que los problemas no hermitianos no están soportados en otras librerías libres y paralelas con métodos de Davidson. Además de incorporar mejoras en la convergencia de los métodos, como la extracción de pares propios mediante el método de Rayleigh-Ritz armónico, se han realizado otras optimizaciones para mejorar el rendimiento, como evitar la aritmética compleja cuando las matrices son reales (aunque los pares propios puedan tener parte imaginaria) o agrupar las operaciones a bloques para aprovechar la memoria caché. Además se ha presentado un método nuevo para expandir el subespacio, llamado GD2, que ofrece mejores resultados en comparación con Generalized Davidson cuando el precondicionador está muy alejado del ideal.La estabilidad numérica y el rendimiento computacional de la implementación se han evaluado mediante una batería de problemas procedentes de aplicaciones reales, y comparado con otras librerías como PRIMME o Anasazi.Finalmente se presenta la integración de la implementación en dos aplicaciones científicas. Por un lado, Jacobi-Davidson ha mejorado las prestaciones de los cálculos de valores propios y los estudios multiparamétricos que aparecen en GENE, un código que re...

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A self-adaptive multilevel finite element method for the stationary Schrödinger equation in three space dimensions

Cited by 39 publications

References 40 publications

Real-space mesh techniques in density-functional theory

Real-space mesh techniques in density-functional theory

Kinetic energy operator approach to the quantum three-body problem with Coulomb interactions

Parallel Implementation

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