The course is taught in English. The main goal of the course is to guide the students to acquire the expertise needed to efficiently perform the numerical simulations that, in subsequent courses or in their Master thesis, they might have to deal with. To this end, the description and analysis of the finite element method will be presented, as well as its application to classical engineering problems represented by linear partial differential equations. The necessary knowledge for setting out and solving engineering problems by means of Matlab computing programs will also be provided.

Ability to recognize and classify a partial differential equation underlying a given mathematical model. Knowledge of the basic finite element methods for the solution of engineering problems described by differential equations. Skill to select or construct efficient numerical methods for solving PDE engineering problems, using the Matlab software.

Basic notions of linear algebra, calculus, numerical methods and Matlab programming.

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. - A short description of numerical methods alternative to Finite Elements. STATIONARY PROBLEMS (40.5 hours) Lectures and problem sessions (28.5 hours): - The Finite Element Method (FEM) for second order linear elliptic problems defined on bounded intervals or on bounded plane domains (24 hours). 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 (4.5 hours) Numerical integration formulas. Conditioning of a linear system. Basic methods for the numerical solution of linear systems. Iterative methods for large sparse linear systems; in particular, preconditioned conjugate gradient and GMRes methods. Computer Lab sessions (12 hours): - Numerical solution of engineering problems using the Matlab software. FEM FOR LINEAR TIME-DEPENDENT PROBLEMS (15.5 hours) Lectures and problem sessions (9.5 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 difference 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.

Modalità di esame: Prova scritta tramite PC con l'utilizzo della piattaforma di ateneo; Prova scritta tramite l'utilizzo di vLAIB e piattaforma di ateneo;

The exam consists of two tests: the first is a 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, some 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 10 points. The Laib test lasts for about one hour. The second written test, to be held in a lecture room, consists of two exercises. In each exercise, worth up to 11 points, 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 and thirty minutes. It forbidden to use books, personal material such as notes and handouts and any electronic device. The Matlab test described above is held by using vLAIB and the Polito Exam platform integrated with proctoring tools (Respondus). Analogously, the written test described above is held by using the Polito Exam platform integrated with proctoring tools (Respondus).

Modalità di esame: Test informatizzato in laboratorio; Prova scritta (in aula); Prova scritta tramite PC con l'utilizzo della piattaforma di ateneo; Prova scritta tramite l'utilizzo di vLAIB e piattaforma di ateneo;

Both Matlab and written tests are held simultaneously, onsite and online, by following the above described procedures.