Prima settimana 
5 Marzo  Lez 
Second order elliptic partial differential equation. Weak formulation.
Existence, uniqueness and stability of the solution. Approximation by finite
dimensional subspace. Galerkin method. Céa's Lemma.

6 Marzo  Lez 
Finite elements for 1D elliptic equation. Piecewise linear finite
element. Interpolation error estimates. Higher degree finite elements.

8 Marzo  Lab 
Coding finite elements in 1D. Construction of the stiffness matrix and of the
right hand side vector. Use of the reference element. Quadrature formula.
Computation of the solution for simple academic examples.
Rate of convergence.

Seconda settimana 
12 Marzo  Lez 
Finite elements for 2D elliptic problems with homogeneous boundary conditions.
Interpolation error. Regularity of the solution. Error estimates. Condition
number of the stiffness matrix.

13 Marzo  Lez 
Generalization to other boundary conditions: non homogeneous Dirichlet boundary
conditions and Neumann boundary condition.
Weak form and energy estimates for parabolic equations. Semidiscretization with
finite elements.

15 Marzo  Lab 
Solution of 2D elliptic equations with PDE Toolbox of Matlab.

Terza settimana 
19 Marzo  Lez 
Time advancing scheme: θmethod. Stability. Error estimates for the
semidiscrete and the fully discrete solution.

20 Marzo  Lez 
A posteriori error estimators. Adaptive finite element schemes.

22 Marzo  Lab 
Error estimates for the finite element discretization of elliptic equations in
2D. Adaptive scheme for singularly perturbed problems or singular solutions.
