The main objective of the research is to develop a research code solving transient incompressible Navier-Stokes equation. In this research code, Adams-Bashforth method was applied to the convective terms of the navier stokes equation and the splitted equations were discretized spatially by finite element methods to solve the complex geometry problems easily. To reduce the divergence on the boundaries of pressure poisson equation due to the unsuitable pressure boundary conditions, multi step approximation pressure boundary conditions derived from the boundary linear momentum equations were used. Simulations of Lid Driven Flow and Flow over Cylinder were conducted to prove the accuracy by means of the comparison with results of the previous workers.