L'insegnamento prevede 2.0 CFU erogati con Didattica Innovativa come segue:
Blended Learning & Flipped Classroom
Corso di Studi
Codice Piano di Studio preventivamente approvato
Ing Ind - Inf (Mag.)(ord. 270) - MI (487) MATHEMATICAL ENGINEERING - INGEGNERIA MATEMATICA
052497 - NUMERICAL ANALYSIS FOR PARTIAL DIFFERENTIAL EQUATIONS
The aim of this course is twofold: on the one hand, to systematically address multi-dimensional finite element analysis for partial differential equations (PDEs), with increased attention to applications in Science and Engineering; on the other hand, to sustain the theoretical analysis with laboratory sessions. Extensive hands-on sessions on projects of engineering relevance are planned
The complementary aim of the course is to train students: to use a widely-spread programming language as Matlab for algorithm development and numerical computations; to be able to utilize advanced simulation methods for the numerical approximations of PDE models.
Blended learning andflipped class innovative teaching methods will be exploited to allow students to acquire a good level of autonomy in learning new topics and to improve specific communication and soft skills like problem-solving and teamworking.
Risultati di apprendimento attesi
Lectures and computer labs will allow students to:
- develop a broad and deep knowledge of numerical methods for Partial Differential Equations;
- understanding and mastering the theoretical properties of numerical methods and algorithms for solving various mathematical problems arising in Science and Engineering;
- critically apply advanced numerical models to mathematical problems stemming from real-world engineering applications independently of their own field of specialization;
- motivate the choice of the adopted methods and tools;
- critical interpretation of the results obtained in the light of the theory through the implementation of the algorithms using the Matlab software;
Practical projects will allow students to:
- improve higher-order thinking skills (critical thinking, problem-solving, communicative and working group skills);
- analyze critically the obtained numerical results;
- summarize and present the results achieved during the analysis and implementation activities;
- write a scientific document.
- Multidimensional elliptic problems: the multidimensional Poisson problem; general elliptic problems; the Lax-Milgram lemma; Galerkin approximation; Galerkin orthogonality; stability and convergence analysis; the finite element method; interpolation error estimates; a priori and a posteriori error estimators; mesh adaptivity; dual problems.
- Multidimensional advection-diffusion problems: limits of the Galerkin method for convection dominated problems; stabilized finite element schemes; the generalized Galerkin formulation; the artificial diffusion and the upwind schemes; strongly consistent stabilized methods; stability and convergence analysis for the GLS scheme; stabilization based on bubble functions; the mass lumping technique for reaction dominated problems.
- Multidimensional time-dependent problems: the semidiscrete formulation; the theta-method; a priori estimates; convergence analysis for the semidiscrete form; stability analysis for the fully discrete formulation; convergence result for the fully discrete problem.
- Galerkin spectral-element methods: Legendre polynomials; Gauss-Legendre-Lobatto quadrature rules; the G-NI method; equivalence between the G-NI and a collocation scheme; the Strang Lemma; convergence of the G-NI formulation; the spectral element method (SEM); the SEM-NI approach.
- Hyperbolic problems: inflow and outflow boundaries; strong and weak imposition of the boundary conditions; stability and convergence analysis; time discretization with the implicit and the explicit Euler schemes; stability analysis; the discontinuous Galerkin (DG) method.
- Domain decomposition problems: Schwarz and Schur paradigms. Parallel preconditioners. Optimality and scalability for parallel computing
Labs and exercise sessions. Most of the methods presented during the lectures will be numerically investigated during the laboratory sessions based on employing the software Matlab.
Project. The project is an integral part of the course. The projects deal with an applicative or advanced theoretical problem, not necessarily dealt with during the course. The goal is to help students developing high-order thinking skills such as critical thinking, problem-solving attitude, communicative and working group skills. Projects will be assigned during the semester. Their evaluation will be based on the produced scientific documentation and numerical code, as well as on an oral presentation.
Some introductory level courses of Numerical Analysis, Partial Differential Equations, Functional Analysis, or equivalent, and some programming experience.
Modalità di valutazione
The exam consists of a written test and of a project
Written test. The written exam takes place in the computer room. The exam covers all the theoretical and practical arguments considered at the lectures and lab sessions. Part of the questions and problems are solved numerically with MATLAB; the exam includes the implementation and programming of numerical algorithms. The problems mainly focus on definition, application of important lemmas and theorems, and important examples. Light calculations may be needed. It is not allowed to use any form of course material. The questions will be answered without books, notes, preparations, etc.
Course project. Students are required to complete a project of their own. Each student works on an applicative or advanced theoretical problem, not necessarily dealt with during the course. The project must focus on theoretical and computational aspects. The assessment of the project will be based on subject knowledge matters, oral delivery, quality of the work, demonstration of team work and individual contributions. Project presentations last 20 minutes plus 10 minutes for questions.
Intended learning outcomes. Students are expected to
- know and understand numerical methods for differential problems, as well as evaluate their properties;
- write and interpret algorithms;
- choose the most appropriate numerical scheme for solving a given differential problem;
- solve mathematical problems with the computer via MATLAB;
- implement algorithms and use them appropriately;
- critically reason and interpret the results obtained in the light of the theory;
- model and numerically simulate Engineering problems.
Quarteroni, Alfio, Numerical Models for Differential Problems, 3rd Edition, Editore: Springer, Series: MS&A, Vol 16, Anno edizione: 2017, ISBN: 978-88-470-5522-3
Tipo Forma Didattica
Ore di attività svolte in aula
Ore di studio autonome
Laboratorio Di Progetto
Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua
Disponibilità di materiale didattico/slides in lingua inglese
Disponibilità di libri di testo/bibliografia in lingua inglese
Possibilità di sostenere l'esame in lingua inglese
Disponibilità di supporto didattico in lingua inglese