Finite Element Method of High-Order Accuracy for Solving Two Dimensional Elliptic Boundary-Value Problems of Two and Three Identical Atoms in a Line

A.A Gusev

doi:10.22363/2312-9735-2018-26-3-226-243

Abstract

We considered models of three identical atoms in a line with molecular pair interactions and diatomic molecule scattered by an atom or tunneling through potential barriers. The models are formulated as 2D elliptic boundary-value problems (BVPs) in the Jacobi and polar coordinates. The BVP in Jacobi coordinates solved by finite element method of high-order accuracy for discrete spectrums of models under consideration. To solve the scattering problems the BVP in polar coordinates are reduced by means of Kantorovich method to a system of second-order ordinary differential equations with respect to the radial variable using the expansion of the desired solutions in the set of angular basis functions that depend on the radial variable as a parameter. The efficiency of the elaborated method, algorithms and programs is demonstrated by benchmark calculations of the resonance scattering, metastable and bound states of the considered models and also by a comparison of results for bound states of the three atomic system in the framework of direct solving the BVP by FEM and Kantorovich reduction.

Highlights

With the development of modern computing power, there are more possibilities for setting and numerically solving multidimensional boundary-value problems with high accuracy
The aim of this paper is to demonstrate the efficiency of the elaborated algorithms and program complexes KANTBP 4M, KANTBP 3, ODPEVP [8,9,10] by benchmark calculations of the resonance scattering below the dissociation threshold, metastable and bound states of the considered models and by a comparison of results for bound states of the three atomic system in the framework of direct solving boundary-value problems (BVPs) by 2D finite element method (FEM) [3,7] and Kantorovich method (KM)
The model for three atomic beryllium system in a straight line was formulated as a 2D boundary-value problem for the Schrodinger equation in Jacobi and polar coordinates

Summary

Introduction

With the development of modern computing power, there are more possibilities for setting and numerically solving multidimensional boundary-value problems with high accuracy. The aim of this paper is to demonstrate the efficiency of the elaborated algorithms and program complexes KANTBP 4M, KANTBP 3, ODPEVP [8,9,10] by benchmark calculations of the resonance scattering below the dissociation threshold, metastable and bound states of the considered models and by a comparison of results for bound states of the three atomic system in the framework of direct solving BVP by 2D FEM [3,7] and Kantorovich method (KM). We elaborated algorithms for calculating the asymptotic parametric angular functions, the effective potentials and the fundamental solutions of the SODEs in the form of expansions by inverse powers of radial variable [11] and apply them to the construction of the asymptotic states of the triatomic scattering problem, because in three-atomic systems at large values of the hyperradial variable the effective potentials of SODEs have.

Setting of the Problem

Boundary-Value Problems

Metastable and Scattering States of the Dimer Tunneling

Conclusion

High-Accuracy Finite Element Method