Prima settimana 
4 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.

5 Marzo  Lez 
Finite elements for 1D elliptic equation. Piecewise linear finite element.
Construction of the linear system. Computation of the integrals by using the
reference element. Interpolation error estimates. Higher degree finite
elements.

7 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 
11 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.

12 Marzo  Lez 
Generalization to other boundary conditions: non homogeneous Dirichlet boundary
conditions and Neumann boundary condition.
Advection, diffusion and reaction equation. Weak form and finite element
discretization.

14 Marzo  Lab 
Solution of 1D elliptic equations with nonhomogeneous Dirichlet boundary
conditions and Neumann problem. Solution of an advection dominated equation in
1D.

Terza settimana 
18 Marzo  Lez 
Parabolic equation. Weak form. Semidiscretization in space and derivation of
the system of ordinary differential equations associated with the space
discretization. Time advancing scheme: θmethod.

19 Marzo  Lez 
Separation of variable for the solution of the heat equation. Eigenvalue
problem associated and representation with eigenfunctions.
Analysis of the stability of the θmethod.
Error estimates for the semidiscrete and the fully discrete solution.

21 Marzo  Lab 
Solution of partial differential equations of elliptic type using the PDE
toolbox of MATLAB. Examples.

25 Marzo  Lab 
Solution of partial differential equations of parabolic type. Behavior of the
θmethod for 1D problems. Use of the toolbox of MATLAB for solving
problem with two dimensions in space.
