Introduction
Navier-Stokes equations. Simplified models. Type of the equations. Boundary and initial conditions.

Elliptic one-dimensional equation
Weak formulation. Linear finite elements. Mass and stiffness matrices.

Unsteady equations
The standard time integration schemes. Heat parabolic equation. Temporal stability. Advection hyperbolic equation. Numerical dissipation and dispersion. Upwinding. Non linear Burgers equations.

Multi-dimensional equations
Elliptic equations : tensor product of the bases. Solution of the linear Algebraic systems : conjugate gradient, preconditioning techniques. Parabolic and hyperbolic equations.
Advection-diffusion type equations : stability, SUPG.

Stokes equation
Variational formulation for the steady and transient Stokes equations. Velocity-pressure discretization. Inf-sup condition. Uzawa algorithm. Velocity-pressure decoupling for 3D problems.

Navier-Stokes equations
Incorporation of the non linear terms. One step Newton method. Examples of the flow around a circular cylinder. Blood flow in a stenosed artery.