Abstract: In virtue of an additional variable in the framework ofthe Finite Increment Calculus(FIC) theory, a pressure stabilized fractionalstep algorithm is developed in this paper with enhanced pressure stabilityin comparison with the classic one. In addition, the calculation of the highorder spatial derivatives which exists in the standard FIC procedure is alsoavoided. To ensure superior overall performance of the proposed numericalscheme in accuracy, efficiency and robustness, a coupled finite element andmeshfree method is developed for the spatial discretization andinterpolation approximation, in which the meshfree approximation is adoptedin the region where the mesh is distorted to preserve the accuracy androbustness of numerical solutions from the deterioration of the meshquality, while the finite element approximation is employed in the regionwhere the quality of the mesh is acceptable and on the boundaries whereessential boundary conditions of flow problems are imposed to ensure highcomputational efficiency and proper imposition of the essential boundaryconditions. Numerical results for the lid-driven cavity flow problemdemonstrate the better pressure stability of the proposed pressurestabilized fractional step algorithm than that of the classic one, and itscapability in removing the spurious oscillations in the resulting pressurefield induced by the incompatible interpolation approximations for thevelocity and pressure fields violating the LBB condition. The two exampleproblems, i.e. the plane Poisseuille flow and the injection molding problemsare illustrated to prominently demonstrate the superiority of the proposedcoupled finite element and meshfree method over the independent finiteelement and meshfree methods in the overall performance.