The Local and Parallel Finite Element Scheme for Electric Structure Eigenvalue Problems

Fubiao Lin,Junying Cao,Zhixin Liu,Qiang Guo

doi:10.1155/2021/1049917

Abstract

In this paper, an efficient multiscale finite element method via local defect-correction technique is developed. This method is used to solve the Schrödinger eigenvalue problem with three-dimensional domain. First, this paper considers a three-dimensional bounded spherical region, which is the truncation of a three-dimensional unbounded region. Using polar coordinate transformation, we successfully transform the three-dimensional problem into a series of one-dimensional eigenvalue problems. These one-dimensional eigenvalue problems also bring singularity. Second, using local refinement technique, we establish a new multiscale finite element discretization method. The scheme can correct the defects repeatedly on the local refinement grid, which can solve the singularity problem efficiently. Finally, the error estimates of eigenvalues and eigenfunctions are also proved. Numerical examples show that our numerical method can significantly improve the accuracy of eigenvalues.

Highlights

As a matter of fact, due to the influence of Coulomb potentials, the convergence order of three-dimensional numerical methods and the computational efficiency of numerical methods will further deteriorate [26]. erefore, one of the most direct and effective methods is to transform the three-dimensional problem into one-dimensional problem
Inspired by [27,28,29] and others references, it is necessary to further study the high-precision numerical method for singular problems. erefore, in this paper, we turn to discuss finite element multiscale discretization based on local defect correction
(1) We first extend local and parallel three-scale finite element discretizations for symmetric eigenvalue problems established by Dai and Zhou [22] to solve Schrodinger eigenvalue problem

Summary

Introduction

As an important equation in quantum mechanics, Schrodinger eigenvalue problems have important physical background modern electronic structure computations [1, 2]. us, finite element methods for solving this problem become an important topic which has attracted the attention of mathematical and physical fields: a priori error estimate is discussed in [3], some posteriori error estimates and adaptive algorithms have been studied in [4,5,6,7], and, in addition, it includes twoscale method [8,9,10,11,12] and the extrapolation methods [13,14,15,16]. For elliptic boundary value problem, Xu and Zhou [18] combined two-grid finite element discretization scheme with the local defect correction to propose a general and powerful parallel-computing technique. Is technique has been used and developed by many scholars, for instance, it can be used to solve Stokes equation (see [19, 20]), Especially, Xu and Zhou [21], Dai and Zhou [22], and Bi et al [23,24,25] developed this method and established local and parallel three-scale finite element discretizations for symmetric elliptic singular eigenvalue problems. (1) We first extend local and parallel three-scale finite element discretizations for symmetric eigenvalue problems established by Dai and Zhou [22] to solve Schrodinger eigenvalue problem.

Dimension Reduction Scheme

Multiscale Discretizations Based on Local Defect Correction

Theoretical Analysis

Numerical Experiments

Findings

Conclusion