  The $P_1$--nonconforming quadrilateral finite element space with periodic
boundary condition is investigated. The dimension and basis for the space are
characterized with the concept of minimally essential discrete boundary
conditions. We show that the situation is totally different based on the parity
of the number of discretization on coordinates. Based on the analysis on the
space, we propose several numerical schemes for elliptic problems with periodic
boundary condition. Some of these numerical schemes are related with solving a
linear equation consisting of a non-invertible matrix. By courtesy of the
Drazin inverse, the existence of corresponding numerical solutions is
guaranteed. The theoretical relation between the numerical solutions is
derived, and it is confirmed by numerical results. Finally, the extension to
the three dimensional is provided.
