Starting with a short introduction of the investigated nonconforming linear quadrilateral ~P1-finite element which has been recently developed by Park, we examine in detail the numerical behaviour of this element with special emphasis on the treatment of Dirichlet boundary conditions, efficient matrix assembly, solver aspects and the use as Stokes element in CFD. Furthermore, we compare the numerical characteristics of ~P1 with other low order finite elements. Moreover, we derive a dual weighted residual based a-posteriori error estimation procedure in the sense of Becker and Rannacher for ~P1. Several test examples show the efficiency and reliability of the proposed method for elliptic 2nd order problems.