  In this contribution, we discuss the finite-element discrete variable
representation (FE-DVR) of the nonequilibrium Green's function and its
implications on the description of strongly inhomogeneous quantum systems. In
detail, we show that the complementary features of FEs and the DVR allows for a
notably more efficient solution of the two-time Schwinger/Keldysh/Kadanoff-Baym
equations compared to a general basis approach. Particularly, the use of the
FE-DVR leads to an essential speedup in computing the self-energies.
  As atomic and molecular examples we consider the He atom and the linear
version of H$_3^+$ in one spatial dimension. For these closed-shell models we,
in Hartree-Fock and second Born approximation, compute the ground-state
properties and compare with the exact findings obtained from the solution of
the few-particle time-dependent Schr\"odinger equation.
