Two numerical methods based on the finite-element and finite-volume scheme have been developed for the unsteady Navier-Stokes equations of incompressible flow in two dimensions. The momentum equations combined with a pressure correction equation are solved employing a non-staggered grid. The solution is advanced in time with an explicit/implicit marching scheme. A fourth-order smoothing operator has been employed to stabilize the solutions. An adaptive algorithm has been implemented, which refines the grid locally in order to resolve detected flow features. A combination of quadrilateral, as well as triangular cells provides flexibility in forming the adaptive grids. Applications of the employed adaptive algorithms include both steady and unsteady flows, with low and high Reynolds numbers. Comparisons with analytical, as well as experimental data evaluate accuracy and robustness of the two methods. Results show that the method based on the finite-element scheme is more accurate and stable.