A parallel iterative Galerkin method based on domain decomposition technique with nonconforming quadrilateral finite elements will be analyzed for second-order elliptic equations subject to the Robin boundary condition, Optimal order error estimates are derived with respect to a broken H-1-norm and L-2-norm. Applications to time-dependent problems skill be considered. Some numerical experiments supporting the theoretical results will be given. This paper is to extend the work in [J. Douglas Jr., J.E. Santos. D. Sheen. X. Ye. Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. Mathematical Modelling and Numerical Analysis, RAIRO, Model. Math. Anal, Numer. 33 (4) (1999) 747] to the non-self-adjoint case of second-order equations including the term b . delu. We suppose that uniformly ellipticity holds, Hence the arguments in (loc. cit.) may be applied, word for word. So some proofs will be omitted. (C) 2002 Elsevier Science Inc. All rights reserved.