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The $P_1$--nonconforming quadrilateral finite element space with periodicboundary condition is investigated. The dimension and basis for the space arecharacterized with the concept of minimally essential discrete boundaryconditions. We show that the situation is totally different based on the parityof the number of discretization on coordinates. Based on the analysis on thespace, we propose several numerical schemes for elliptic problems with periodicboundary condition. Some of these numerical schemes are related with solving alinear equation consisting of a non-invertible matrix. By courtesy of theDrazin inverse, the existence of corresponding numerical solutions isguaranteed. The theoretical relation between the numerical solutions isderived, and it is confirmed by numerical results. Finally, the extension tothe three dimensional is provided.

$P_1$--Nonconforming Quadrilateral Finite Element Space with Periodic  Boundary Conditions: Part I. Fundamental results on dimensions, bases,  solvers, and error analysis