Accuracy of a hybrid finite-element method for solving a scattering Schrödinger equation

This hybrid method [finite-element discrete variable representation (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 they are orthogonal under the Gauss-Lobatto quadrature discretization approximation. Accuracy analyses of this method are performed for the case of a one-dimensional Schrödinger equation with various types of local and nonlocal potentials for scattering boundary conditions. The accuracy is ascertained by a 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), it is easy to program, and it can routinely achieve an accuracy of better than 1:10⁻⁸ for the numerical examples studied.