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This paper extends earlier research in numerical analysis and computational fluid dynamics (CFD) to obtain a novel finite element method for transient 3-D incompressible Navier-Stokes equations, along with efficient, parallelizable algorithms to cary out an implementation of the method in such a fashion as to be useful in mainstream industrial settings. The approach is based on a flexible operator-splitting technique due to one of the authors that allows the introduction of increasingly complex physics one step at a time. The approach should also allow treating potential flow, Euler flow, and Navier-Stokes flow simultaneously in different parts of the flow region. Parallelization is achieved through domain decomposition techniques. A new type of Eulerian time discretization is employed to increase accuracy and maintain computational efficiency. This new finite element procedure employs alternating-direction operator splittings in conjuction with Strang-type splittings to model problems of increasing complexity in a step-by-step and natural manner. Later, it can be extended to include additional operators e.g. for turbulence modeling, chemical reactions, etc. In addition to being practical from the standpoint of software design and of engineering analysis, its development and implementation allows the use of specialized numerical methods for the solution of each phys