This course is taught in English. The primary objective of the course is to equip students with the knowledge and skills required to perform efficient numerical simulations, which they may encounter in advanced coursework or their Master thesis. To achieve this, the course introduces the finite element method, covering both its theoretical foundations and practical applications to classical engineering problems governed by linear partial differential equations. Students will also gain the necessary proficiency in using Matlab to formulate and solve engineering problems through computational methods.

By the end of the course, students will be able to identify and classify partial differential equations (PDEs) underlying mathematical models of engineering problems. They will also acquire foundational knowledge of finite element methods for solving such equations. In addition, students will develop the skills to design and implement efficient numerical methods for PDE-based engineering problems using Matlab.

Basic notions of calculus and numerical linear algebra are required, along with familiarity with basic programming constructs and the Matlab software.

INTRODUCTION (4 hours) Lectures: Generalities on Partial Differential Equations (PDE) and on numerical methods. Examples of classical PDE problems, in particular with reference to the electronic engineering and nanotechnologies. Stationary problems, eigenvalue problems and evolution problems. Classification of second order equations. Boundary conditions on the physical domain, initial conditions for nonstationary problems. Strong and weak solutions. FEM FOR LINEAR STATIONARY PROBLEMS (40 hours) Lectures and problem sessions (28 hours): • The Finite Element Method (FEM) for second order linear elliptic problems defined on bounded intervals or on bounded plane domains. • Sobolev spaces and weak formulations of differential problems. Integration by parts and Green’s formulas. • Problem space domain discretization. Finite elements and finite element spaces. Definitions and examples. Some main results on approximation theory. • Application of the FEM to linear elliptic problems, first defined on bounded intervals, and then on bounded plane domains, with different types of boundary conditions (Dirichlet, Neumann, Robin, mixed-type, of periodicity). • Some basic convergence results. • Eigenvalue problems. • Construction of the FEM stiffness matrix and solution of the associated linear system. • Numerical integration formulas. Conditioning of a linear system. Basic methods for the numerical solution of linear systems. Computer Lab sessions (12 hours): • Numerical solution of engineering problems using the Matlab software. FEM FOR LINEAR TIME-DEPENDENT PROBLEMS (16 hours) Lectures and problem sessions (10 hours): • Weak formulations in the space domain of parabolic and hyperbolic problems, in particular of transport and wave propagation problems. Application of the FEM. • Basic on initial value ordinary differential equation (ODE) problems. Stiff systems. Some basic numerical methods. • Finite element schemes for the time discretization of ODE systems generated after performing the (space) FEM discretization. Computer Lab sessions (6 hours): • Numerical solution of engineering problems using the Matlab software.

The above lecture contents include also corresponding exercise sessions, for a total of 15 hours. Important aspects of the lecture topics as well as the solution of some problems will be discussed, to help the students to better understand the lectures. An additional computer room activity (18 hours) is also included. Numerical properties of the numerical methods presented in the lectures will be tested, and some engineering problems will be solved by using the Matlab software.

Lecture notes, as well as notes on the computer lab sessions, can be downloaded by the students from the Portale della Didattica. Further reading: A. Quarteroni, Numerical Models for Differential Problems, Springer Verlag, 2009.

Slides; Dispense; Esercizi; Esercizi risolti; Esercitazioni di laboratorio; Esercitazioni di laboratorio risolte; Video lezioni tratte da anni precedenti; Strumenti di auto-valutazione;

Modalità di esame: Test informatizzato in laboratorio; Prova scritta (in aula);

Exam: Computer lab-based test; Written test;

... The exam consists of two tests: the first is an computer test, the second is a written test. The first test, to be held in the informatic room under the TIL format (Test in Laboratory, widely adopted in some of the numerical courses), consists of two multiple-choice questions concerning the course topics. In this test students must show to be able to solve, by applying the finite element method and preparing short Matlab scripts, two differential problems similar to those that have been solved during the Laib course sections. This to verify the students’ knowledge of the algorithms presented during the course and their programming and computing skills. Students must use the Matlab software installed in the Laib PC, but they are not allowed to use books, notes, personal PC and any electronic device. The maximum grade is 16 points: each correct answer is worth 8 points, an unanswered question is worth 0 points, and an incorrect answer results in a 20% penalty. The Laib test lasts for about one hour. The second written test, to be held in a lecture room the same day of the Matlab test, consists of an exercise, worth up to 16 points, in which students are asked to apply the finite element method to a given problem, by performing all the required main steps, to show that they have understood how the method works and what kind of approximations it involves. The written exam lasts for about one hour. It is forbidden to use books, personal material such as notes and handouts and any electronic device. The final grade will be based on the combined scores of the two tests. A minimum score of 18 is required to pass, while full marks with all correct answers are awarded with honors (cum laude).

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.