We suggest a strategy to consider discontinuities in the two-phase flow calculations. Our approach comprises enhancing the shape functions of the finite element method (FEM) with the moving least-squares (MLS) interpolation functions. The new shape functions correlate a parameter at any arbitrary point to its surrounding nodes instead of an element's nodes. We name this strategy the "moving least-squares" aided "finite element method" (MLS-FEM). In a previous paper Mostafaiyan et al., we have employed the concept of the MLS-FEM to develop the PMLS (pressure shape function enhanced by the MLS interpolation functions) method, which predicts pressure discontinuities due to the surface tension forces. We take another additional step to consider velocity and pressure discontinuities in a domain with two different fluids. Therefore, we use the MLS technique to enhance the pressure (P) and velocity (V) shape functions. We name this scheme PVMLS (pressure and velocity shape functions enhanced by the MLS technique). We validate the PVMLS method with a stationary drop problem (as a benchmark). We show that the previous PMLS method fails to predict an accurate pressure or smooth streamlines by increasing the Capillary number and deviating from the unit viscosity ratio. However, the PVMLS method provides more accurate results since it considers velocity as a discontinuous parameter. Also, we compare the results of the PVMLS method at very high viscosity ratios (undeformable fluid) with a boundary-fitted single-phase problem. The reported similarities between the predicted pressure values and streamlines in both strategies demonstrate the reliability of the PVMLS method. Finally, we discuss that by employing the PVMLS method to calculate the velocity field, the interface advection will be more mass-conservative.