Risorse bibliografiche
 Risorsa bibliografica obbligatoria Risorsa bibliografica facoltativa
 Scheda Riassuntiva
 Anno Accademico 2014/2015 Scuola Scuola di Ingegneria Industriale e dell'Informazione Insegnamento 095964 - NUMERICAL ANALYSIS FOR PARTIAL DIFFERENTIAL EQUATIONS Cfu 10.00 Tipo insegnamento Monodisciplinare Docenti: Titolare (Co-titolari) Perotto Simona

Corso di Studi Codice Piano di Studio preventivamente approvato Da (compreso) A (escluso) Insegnamento
Ing Ind - Inf (Mag.)(ord. 270) - MI (403) INGEGNERIA MATEMATICA*AZZZZ095964 - NUMERICAL ANALYSIS FOR PARTIAL DIFFERENTIAL EQUATIONS
Ing Ind - Inf (Mag.)(ord. 270) - MI (487) MATHEMATICAL ENGINEERING - INGEGNERIA MATEMATICA*AZZZZ095964 - NUMERICAL ANALYSIS FOR PARTIAL DIFFERENTIAL EQUATIONS

 Programma dettagliato e risultati di apprendimento attesi
 Aims :     The aim of this course is twofold: on the one hand, to extend the one-dimensional finite element analysis for partial differential equations (PDEs) to high dimensions, with an increased attention to applicative problems in Science and Engineering; on the other hand, to enrich the theoretical analysis with laboratory sessions, via FreeFem++ programming, to provide the students with a practical counterpart of what stated by the theory.Scientific talks given by expert scientists and dealing with some of the latest challenging topics in scientific modeling are also planned during the course.     Topics :   + Multidimensional elliptic problems: the 2D 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 a convection dominated problem; 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.   + The one-dimensional Galerkin spectral method: 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; extension to a two-dimensional setting; the spectral element method (SEM); the SEM-NI approach.   + One-dimensional hyperbolic problems: scalar linear equations; a priori analysis; finite element approximation in space; time discretization with the implicit and explicit Euler schemes; stability analysis; time discretization with the LF, LW, UPWIND schemes; strong stability analysis; dissipation and dispersion analysis; the Taylor-Galerkin schemes.   + Two-dimensional 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.     Labs and exercise sessions:   Most of the methods presented during the lectures will be numerically investigated during the laboratory sessions, via FreeFem++ programming. Sessions of supervised exercises are also provided in view of the written test.     Prerequisites:   Analisi Numerica delle Equazioni alle Derivate Parziali I.

 Note Sulla Modalità di valutazione
 Grading :   The exam consists of a written test in laboratory, including theory and FreeFem++ programming, and of a project, on applicative or advanced theoretical topics, not necessarily dealt with during the course.

 Bibliografia
 Quarteroni, Alfio, Numerical Models for Differential Problems, Editore: Springer, Series: MS&A, Vol. 8. Second edition., Anno edizione: 2014, ISBN: 978-88-470-5522-3

 Software utilizzato
 Nessun software richiesto

 Mix Forme Didattiche
Tipo Forma Didattica Ore didattiche
lezione
52.0
esercitazione
12.0
laboratorio informatico
18.0
laboratorio sperimentale
0.0
progetto
48.0
laboratorio di progetto
0.0

 Informazioni in lingua inglese a supporto dell'internazionalizzazione
 Insegnamento erogato in lingua Inglese 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
 schedaincarico v. 1.9.7 / 1.9.7 Area Servizi ICT 29/05/2024