Finite Element Methods for Flow Problems
Unsteady Transport  
  Steady Transport
  Unsteady Transport
Ex 1: Cosine profile
Ex 2: Wave package
Ex 3: Rotating cone
  Compressible Flow
  Unsteady Convection-Diffusion
  Incompressible Flow
Equations governing transient convection problem are
Note that boundary conditions are only imposed on the inflow boundary.
We will now only consider linear problems, so that the convective term can be written as   f(u) = a u  with a independent of the solution u.

When the convection velocity a is divergence-free the problem can be rewritten as

The main feature of this equation is that space and time are linked by the characteristics. To compute a numerical solution, a double discretization has to be performed (in space and in time) and to obtain accurate solutions it cannot be done anyhow. Stability of numerical methods depends on the Courant number , that links spatial and time discretization.

It is also important the order in which discretizations are performed. We will first carry out time discretization using one of the following time-stepping algorithms:
- family methods (such as Euler, Crank-Nicolson...)
Lax-Wendroff method
Leap-Frog method
Taylor-Galerkin methods (higher order)

Then, we use the finite element method for solving the steady problem posed in each time step.

Examples on unsteady transport problems
 Propagation of a cosine profile
 Traveling wave package
 The rotating cone problem

 Steady Transport
Compressible Flow