Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
Academic year in which the course will be held: 
Course type: 
Compulsory subjects, characteristic of the class
Second semester
Standard lectures hours: 
Detail of lecture’s hours: 
Lesson (64 hours)

The course is of interest for Mathematics students, but also for students of other degree programs, with interests in scientific computing. Basic notions of Analysis I and II are required, as well as previous knowledge on the numerical solution of linear algebraic systems are required. Notions about Sobolev Spaces are useful, but not strictly necessary. For the lab part, each student needs a basic knowledge of a programming language: in the course we will use Matlab, but each student is free to use other languages, such as C or C++.

Final Examination: 

The exam is oral, and consists of two parts, which take place on the same day.
In the first part, the student discusses a computational project, agreed with the teacher and submitted together with the MatLab code developed. The project shall be a case-study that leverages on a topic taught during the course. The subject of the evaluation will be the appropriateness and quality of the software produced and the ability to present and discuss critically the results.
The second part of the exam is an oral examination of the material covered by the course. The evaluation will be based on knowledge of the course contents, ability to master the technical jargon of the subject, critical reasoning and ability to link the various topics.

Voto Finale

The two courses Numerical Solutions of PDEs introduce the students to the numerical techniques to approximate the solutions of partial differential equations. In particular the B course is focused on the finite element approach and its application in the context of elliptic and parabolic equations.
Elliptic equations are present in many physical models, like the equations for the electrostatic and gravitational potential, elasticity problems and deformation of structures. Heat diffusion, instead, is a typical example of parabolic problem. In the course, also convection-diffusion equations and Navier-Stokes equations for a viscous fluid will be considered.
At the end of the course the student should be able to solve numerical elliptic, parabolic and Stokes equations with finite element methods. Moreover, he/she should be able to use critically also libraries and software based on finite elements.

1. Introduction to finite elements: the elastic thread as a model problem. Abstract formulation of the Galerkin method and Lax-Milgram's lemma, convergence estimates, boundary conditions. P1 and higher order finite elements. Extension to the two-dimensional case. FEM spaces.

2. Application to several elliptic problems: convection-diffusion, the elastic beam, and the elastic membran. Stokes’ problem. Loss of coercivity and the Inf-Sup condition. Artificial viscosity and the SUPG method.

3. Parabolic problems. Convection-diffusion of heat in 1D. Semi-discretization in space. Discretization in space and time. Boundary conditions.

4. The Discontinuous Galerkin method. Stabilization techniques.

“Numerical solution of partial differential equations by the finite element method”, di Claes Johnson, Dover.

“Theory and Practice of Finite Element Methods” di Ern e Guermond, Springer.
“The Mathematical Theory of Finite Element Methods” di Brenner e Scott, Springer.
Additional material, expecially for the lab sessions, will be made available by the teacher through the e-learning web site.

Lectures (2/3 of the hours) are conducted mainly at the blackboard. Exercises to help the individual study of each pupil will be made available and discussed during the following lectures upon request.
One third of the hours will be in the computing lab to teach the students how to implement, test and employ algorithms based on finite elements (some of the tools explained in the lectures will be used for this).

Office hours are booked on demand, by email or at lecture time.