Starting with a short introduction of the new nonconforming linear quadrilateral $/tilde{P}_1$-finite element which has been recently proposed by Park (/cite{Park2002,ParkSheen2003}), 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 $/tilde{P}_1$ 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 (c.f.~/cite{BeckerRannacher1996}) for $/tilde{P}_1$. Several test examples show the efficiency and reliability of the proposed method for elliptic
2nd order problems.