A finite element implementation of the dual variable method for the Navier-Stokes equations

The numerical solution of two-dimensional, transient, incompressible Navier-Stokes problems on rather arbitrary bounded regions is considered. A finite element implementation of the method is presented, using quadrilateral piecewise bilinear velocity/constant pressure elements. The primary tool employed is combinatoric graph theory. A class of finite element decompositions is described, and the pertinent properties of such decompositions are discussed. Decompositions consisting of rectangular or square elements are considered as a special sub-class, as are decompositions of flow regions which are not simply connected. Algorithms for the determination of a basis for N(A) and a particular solution are presented and discussed with examples.