  We consider the lowest--degree nonconforming finite element methods for the
approximation of elliptic problems in high dimensions. The $P_1$--nonconforming
polyhedral finite element is introduced for any high dimension. Our finite
element is simple and cheap as it is based on the triangulation of domains into
parallelotes, which are combinatorially equivalent to $d$--dimensional cube,
rather than the triangulation of domains into simplices. Our nonconforming
element is nonparametric, and on each polytope it contains only linear
polynomials, but it is sufficient to give optimal order convergence for
second--order elliptic problems.
