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 (80 hours)

The course is addressed to Mathematics students, but also to 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, and it would also be useful to know how to write simple programs. As a programming language we will use Matlab, but each student is free to use other languages, such as C, C ++ or Fortran.

Final Examination: 

The exam is an oral exam, and consists of two parts, which can also be held on the same day.
In the first part, the student presents a brief report, which can also be carried out by groups of 2 or 3 people, on a project related to one of the course topics. Usually the project requires some programming, and the student must demonstrate a certain level of autonomy.
The second part of the exam is instead an oral exam on the topics covered during the course.

Voto Finale

The course on the numerical solution of PDE’s part A is dedicated to elliptic and parabolic problems. Elliptic equations arise above all in the modeling of stationary phenomena, as in the case of electrostatic or gravitational potentials, when the field sources are fixed. Another source of elliptic equations are in problems of elasticity for the study of the deformation of structures. The diffusion of heat is instead a typical parabolic problem. Convection-diffusion problems will also be considered in the course, such as the Navier Stokes equations for a viscous fluid.

The most used numerical methods for this type of models are finite element methods. In the course, we will study finite element methods for elliptic equations, considering both the theoretical point of view and implementation aspects. The aim of the course is to provide theoretical and practical tools to apply and develop programs in which finite elements are used.

At the end of the course, students should be able to solve numerically elliptic PDEs on simple domains. Above all, they should be able to use software dedicated to the solution of PDE’s in a critical and informed way.

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 P2 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.

I recommend two textbooks. The first, "Numerical solution of partial differential equations by the finite element method” by Claes Johnson, Dover," is an introductory text, and it is very clear, but not very detailed.
The second text is "Theory and Practice of Finite Element Methods" by Ern and Guermond, Springer, 2010. It is much more in-depth, up-to-date and exhaustive, but it is also more difficult.

In addition, slides and notes for lab exercises will be available.

Lectures are traditional blackboard lectures, with practical sessions at the computer lab. In the lessons, we will introduce and gradually describe the numerical methods, together with the underlying theory. In the lab sessions, we will apply the methods studied to significant test cases, using programs written on purpose for this class. The aim is to understand by direct experience the characteristics of FEMs and their properties.

Office hours are by appointment, which can be set either by email or at the end of the lessons