Accuracy of a new hybrid finite element method for solving a scattering Schroedinger equation

TL;DR

This paper evaluates a hybrid finite element-discrete variable representation method for solving the one-dimensional scattering Schrödinger equation, demonstrating high accuracy and computational efficiency compared to traditional methods.

Contribution

The study provides a detailed accuracy analysis of the FE-DVR method for scattering problems, showing it is faster and more accurate than conventional finite difference approaches.

Findings

FE-DVR achieves better than 1:10^(-8) accuracy in phase shifts.

FE-DVR is 100 times faster than Numerov's finite difference method.

The method is easy to implement and highly accurate for various potentials.

Abstract

This hybrid method (FE-DVR), introduced by Resigno and McCurdy, Phys. Rev. A 62, 032706 (2000), uses Lagrange polynomials in each partition, rather than "hat" functions or Gaussian functions. These polynomials are discrete variable representation functions, and are.orthogonal under the Gauss-Lobatto quadrature discretization approximation. Accuracy analyses of this method are performed for the case of a one dimensional Schroedinger equation with various types of local and nonlocal potentials for scattering boundary conditions. The accuracy is ascertained by comparison with a spectral Chebyshev integral equation method, accurate to 1:10^(-11). For an accuracy of the phase shift of 1:10^(-8) The FE-DVR method is found to be 100 times faster than a sixth order finite difference method (Numerov), is easy to program, and can routinely achieve an accuracy of better than 1:10^(-8) for the numerical examples studied.