A highly accurate 2-spinor fully relativistic solution of the two-center Coulomb problem utilizes the finite-element method (FEM). Like in previous work on the 4-spinor Dirac equation we use elliptic-hyperbolic coordinates, global factors, further one-dimensional singular coordinate transformations and extrapolation techniques. In the same grid 2-spinor energies for H-2(+) are up to four orders more accurate than 4-spinor values, for the super-heavy Th-2(179+) up to one order and one generally finds, high convergence orders similar to a non-relativistic approach. The 2-spinor values converge from above, exhibit remarkable properties and the calculations are considerably faster and more stable. (C) 2003 Elsevier B.V. All rights reserved.