A comprehensive and structured introduction to finite element methods for incompressible fluid dynamics, covering both spatial and temporal discretization, from the Stokes equations to the direct numerical simulation of the Navier-Stokes equations. This book provides a rigorous yet practical introduction to the finite element method applied to incompressible fluid mechanics. Designed for graduate engineering students, researchers, and industry professionals, it guides readers from the physical foundations of the Navier-Stokes equations to advanced numerical implementations. Combining mathematical clarity with a progressive pedagogical approach, the authors introduce key concepts in numerical linear algebra, discretization techniques, and time-dependent simulations, before moving toward multidimensional problems and real-world applications. Emphasis is placed on understanding algorithms and developing computational autonomy rather than relying on black-box software solutions. At the intersection of physics, applied mathematics, and computational science, this volume offers both a structured learning path and a valuable reference for anyone working with the numerical simulation of incompressible flows.