Lecturer: Luisa Donatella Marini  

Course name: Elements of Mathematics
Course code: 502985
Degree course: Ingegneria Civile
Disciplinary field of science: MAT/08
The course relates to:
University credits: ECTS 6
Course website: n.d.

Specific course objectives

Aim: to give the some basic instruments necessary for the theoretical and numerical analysis of Partial Differential Equaltions (Pde's) of interest for applications.

Course programme

The course is divided in two parts, strictly related to one another. In the first part a theoretical study of some model applicative problems described by Partial Differential Equations (Pde's) will be carried out. The second part is devoted to the numerical solution of the problems analysed in the first part of the course. In particular, some of the following arguments will be developed.

GENERALITIES ON DIFFERENTIAL EQUATIONS
definition of Partial Differential Equations of order m; linear, semi-linear and quasi-linear equations.

FIRST ORDER DIFFERENTIAL EQUATIONS
Transport equation; constant and variable transport coefficient; Cauchy's problem.

SECOND ORDER DIFFERENTIAL EQUATIONS
Homogeneous linear equations with constant coefficients; Characteristic lines and classification of second order Pde's; Elliptic equations: Poisson problem, weak (variational) formulation; Parabolic Pde's: the heat equation, variational formulation. Hyperbolic Pde's: wave equation, variational formulation.

FINITE DIFFERENCE AND FINITE ELEMENT METHODS
Numerical approximation of a one-dimensional elliptic model problem. Extension to two-dimensional problems.

FINITE ELEMENT FOR ADVECTION-DIFFUSION PROBLEMS
One-dimensional model problem: behaviour of the numerical solution in the advection-dominated case. Stabilization methods: artificial viscosity and up-wind Finite Elements; Petrov-galerkin schemes. Hints on artificial viscosity and streamline diffusion (SUPG) schemes for two-dimensional problems.

DISCRETIZATION OF PARABOLIC PROBLEMS
Finite Element approximation in space and theta-method in time. Hints for the case of two space dimensions.

DISCRETIZATION OF HYPERBOLIC PROBLEMS
Semidiscretization in space with Finite Elements (continuous or discontinuous). Stabilization via artificial viscosity. Space-time Finite Element approximation. Hints on nonlinear problems.

Course entry requirements

Nozioni di base del Calcolo Differenziale ed Integrale per funzioni di una e piu` variabili reali. Nozioni di base di Algebra Lineare. Nozioni di base di Calcolo Numerico.

Course structure and teaching

Lectures (hours/year in lecture theatre): 45
Practical class (hours/year in lecture theatre): 0
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

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

A. Quarteroni. Modellistica numerica per problemi differenziali. Springer Italia, 2008..

Testing and exams

Written exam: it consists of 2 questions on subjects developed in the classes. Duration: 1 hour. To pass the exam a grade of at least 18/30 must be obtained.

The maximum grade obtainable with the written exam is 26/30, and the students may accept the grade obtained.

To obtain more than 26/30 it is compulsory to undergo an oral exam. In case of failure of the oral exam the grade obtained in the written part is not granted.