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B027566 - NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS
Main information
Teaching Language
Course Content
Suggested readings
Learning Objectives
Prerequisites
Teaching Methods
Type of Assessment
Course program
Academic Year 2021-22
Coorte 2020 - Second Cycle Degree in Mechanical Engineering
Course year
Second year - Second Semester
Belonging Department
Industrial Engineering (DIEF)
Course Type
Single education field course
Scientific Area
MAT/08 - NUMERICAL ANALYSIS
Credits
6
Teaching Hours
48
Teaching Term
28/02/2022 ⇒ 10/06/2022
Attendance required
No
Type of Evaluation
Final Grade
Course Content
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Course program
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Lectureship
Mutuality
Course teached as:
B018805 - COMPLEMENTI DI ANALISI NUMERICA
Second Cycle Degree in MATHEMATICS
Curriculum APPLICATIVO
B018805 - COMPLEMENTI DI ANALISI NUMERICA
Second Cycle Degree in MATHEMATICS
Curriculum APPLICATIVO
Teaching Language
Italian
Course Content
Finite difference methods for 1D boundary value problems. Variational formulation of a boundary value problem (1D e 2D). The finite element method for stationary and parabolic problems. The finite elemnt method for Navier-Stokes. Splines and the B-spline basis. Curves and surfaces in CAGD. Elements of isogeometric analysis.
Suggested readings (Search our library's catalogue)
- W. Gautcshi (2012), Numerical Analysis, Second Edition, Birk ̈auser, New York.
-A. Quarteroni (2012), Modellistica Numerica per Problemi Differenziali, quinta edizione, Springer–Verlag, Milano.
-L. Formaggia, F. Saleri, A. Veneziani, Solving numerical PDE's: problems, applications, exercises, 2012, Springer-Verlag, Milano, Italia.
-C de Boor (2001), A Practical Guide to Splines, Revised edition, Applied Mathematical Sciences 27, Springer–Verlag, New York
.- L. L. Schumaker (2007) Spline functions: basic theory, Cambridge Mathematical Library.
-A. Quarteroni (2012), Modellistica Numerica per Problemi Differenziali, quinta edizione, Springer–Verlag, Milano.
-L. Formaggia, F. Saleri, A. Veneziani, Solving numerical PDE's: problems, applications, exercises, 2012, Springer-Verlag, Milano, Italia.
-C de Boor (2001), A Practical Guide to Splines, Revised edition, Applied Mathematical Sciences 27, Springer–Verlag, New York
.- L. L. Schumaker (2007) Spline functions: basic theory, Cambridge Mathematical Library.
Learning Objectives
Knowledge: finite difference schemes for 1D boundary value problems; variational formulaton of a boundary value problem; the finite element method; splines and their application in CAGD and in isogeometric analysis.
Expertise: being able to implement and evaluate a numerical scheme for boundary value problems; familiarity with splines, with Matlab toolbox on splines and with their applications.
Transversal Expertise: CT1 (relations and contacts); CT4 (being able to collect data in tables and to present them graphically); CT6 (being able to do bibliographic searches); CT7 (knowing how to meet deadlines and keep commitments)
Expertise: being able to implement and evaluate a numerical scheme for boundary value problems; familiarity with splines, with Matlab toolbox on splines and with their applications.
Transversal Expertise: CT1 (relations and contacts); CT4 (being able to collect data in tables and to present them graphically); CT6 (being able to do bibliographic searches); CT7 (knowing how to meet deadlines and keep commitments)
Prerequisites
Basics of Numerical Calculus (finite arithmetic, zeros of functions, numerical methods for linear systems) and knowledge of the Matlab environment and programming language.
Teaching Methods
6 hours a week, 4 in classroom (frontal teaching) and 2 in computer lab (implementation of the introduced schemes and numerical experiments)
Type of Assessment
The exam consists in an oral interrogation on the topics presented in the course, in the presentation of a subject individually studied and in discussing a student's work collecting developed programs and the results of the related numerical experiments suitably presented.
Course program
The centered and upwind finite difference methods for 1D boundary value problems: a model problem, the diffusion-transport problem, the general second order nonlinear problem. Basics on Sobolev spaces. Variational formulation of a boundary value problem (1D and 2D). Galerkin discretization. Finite elment method for stationary and parabolic problems. The conjugate gradient method and Krylov methods. Mesh adaptivity. Navier-Stokes equations. The finite element method for the Stokes problem. Some hints to the non linear case. Splines and the B-spline basis. Curry-Schoenberg theorem. Witney-Schoenberg theorem. Elements of isogeometric analysis.