Accurate solution of the Dirac equation on Lagrange meshes
  The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. With a basis of Lagrange functions involving associated Laguerre polynomials related to the Gauss quadrature, the method is applied to the Dirac equation. The potential may possess a $1/r$ singularity. For hydrogenic atoms, numerically exact energies and wave functions are obtained with small numbers $n+1$ of mesh points, where $n$ is the principal quantum number. Numerically exact mean values of powers $-2$ to 3 of the radial coordinate $r$ can also be obtained with $n+2$ mesh points. For the Yukawa potential, a 15-digit agreement with benchmark energies of the literature is obtained with 50 mesh points or less. 
Daniel Baye, Livio Filippin and Michel Godefroid
physics.atom-ph