Lecturer: Carlo Lovadina  

Course name: Elements of Mathematics
Course code: 502985
Degree course: Ingegneria per l'Ambiente e il Territorio, Ingegneria Civile
Disciplinary field of science: MAT/08
The course relates to:
University credits: CFU 6
Course website: n.d.

Specific course objectives

Provide some basic elements for the analytical and numerical study of partial differential equations of interest in Engineering Applications.

Course programme

The Course will deal with some of the following issues.

GENERALITIES OF PDEs
definition of PDE of order m, linear PDEs, semi-linear and quasi linear PDEs.

FIRST ORDER PDEs
Linear PDEs of constant coefficients; the Cauchy problem.

SECOND ORDER PDEs
Homogeneous linear and constant coefficient equations; characteristic curves and classification of second order PDEs. Elliptic equations: the Poisson problem; weak formulation. Parabolic equation: the heat equation; weak formulation. Hyperbolic equations: the wave equation; weak formulation.

FINITE DIFFERENCE AND FINITE ELEMENT METHODS FOR ELLIPTIC PROBLEMS
The 1D case in details. Extension to the multi-dimensional case.

FINITE ELEMENT METHODS FOR ADVECTION-DIFFUSION PROBLEMS
The 1D case: numerical solution behaviour for advection-dominant problems. Stabilization methods: artificial diffusion, streamline upwind, Petrov-Galerkin methods. Hints on the 2D case.

DISCRETIZATION OF PARABOLIC PROBLEMS
Finite element methods for spatial discretization and theta-method for time discretization.

DISCRETIZATION OF HYPERBOLIC PROBLEMS
Conforming and DG methods. Stabilized methods. Hints on non-linear hyperbolic problems.

Course entry requirements

Basic Calculus for one and several variable functions. Basic Linear Algebra. Basic Numerical Analysis.

Course structure and teaching

Lectures (hours/year in lecture theatre): 34
Practical class (hours/year in lecture theatre): 22
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

S. Salsa. Equazioni a derivate parziali. Springer Italia, 2010. .

A. Quarteroni. Numerical models for differential problems. Springer, 2009.

Testing and exams

Final oral examination.