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

Finite elements and the discrete variable representation in nonequilibrium Green's function calculations. Atomic and molecular models