Mathematics

ELISABETTA CARLINI
ELISABETTA CARLINI

General targets:

To acquire knowledge in numerical linear algebra and numerical modeling for differential problems

Specific targets:

Knowledge and understanding: At the end of the Course students will have theoretical knowledge related to methods of numerical analysis for the solution of linear systems and eigenvalue problems and for the integration of ordinary differential equations and linear partial differential equations. Also, they will have acquired techniques related to implementation of algorithms for the effective solution of the problems.

Applying knowledge and understanding: Students who have passed the exam will be able to use methodologies for the numerical solution of a linear system or of an eigenvalue problem and for the discretization of ordinary differential equations or linear partial derivatives. Also, they will be able to predict performance of such algorithms depending on the characteristics of the problem to deal with.

Making judgements: Students who have passed the exam will be able to select, among the algorithms that they will have studied during the Course, those suited to the solution of the problem to be treated, being also able to make the modifications that may be necessary to improve their performance.

Communication skills: Students will have gained the ability to communicate concepts, ideas and methodologies of numerical linear algebra and numerical modeling for differential problems.

Learning skills: The acquired knowledge will allow students who have passed the exam to face the study, at the individual level or in a Master's degree course, of more specialized aspects of numerical linear algebra and numerical modeling for differential problems, being able to understand the specific terminology and identify the most relevant topics.

At the end of the Course, students will have acquired notions related to methods for the numerical approximation of some linear partial differential equations, in particular transport equations, elliptic equations, and parabolic equations in dimensions one and two.
They will also have acquired techniques related to the implementation of algorithms for the effective solution of the problems.

The course requires familiarity with the basic instruments of mathematical analysis, and the knowledge of main theoretical aspects of Ordinary Differential and Linear Partial Differential Equations introduced in the parallel MAT/05 module, as well as the ability of writing simple Matlab programs.

Some partial differential equations of interest in the applications and the main numerical methods used to solve them. After a short recall of the main theoretical results (students are supposed to have already attended a basis course on PDE), the approximation techniques will be discussed: standard finite difference schemes and mainly the finite element method for the numerical study of linear elliptic, parabolic and hyperbolic problems in two dimensions. The course also includes computer sessions.

Hyperbolic problems
Basics of transport problems in one and two dimensions. Main finite difference schemes. Convergence analysis and study of dispersion and diffusion properties. Computer implementation of the methods.

Elliptic problems
Recall of boundary problems for second order linear equations: classical solutions, maximum principle, variational formulation in Sobolev spaces. Finite difference schemes for Poisson equation, discrete maximum principle and convergence analysis. The Galerkin method for the approximation of variational problems. Lagrange finite elements. Interpolation theory in Sobolev spaces, convergence theorems and error estimates for the finite element approximation method, computational aspects and comparison with the finite difference approach. Numerical analysis of elliptic problems with a dominant transport (or reaction) term and their resolution with finite difference or finite element techniques. Up-wind type schemes and artificial diffusion. Some notes on stabilization methods for finite element schemes in advection-diffusion problems. Computer implementation of the methods.

Parabolic problems
Recall of classical results and variational formulation for linear parabolic problems. Finite difference schemes for the heat equation, consistency error and stability estimate.Schemes base on finite elements in space and finite differences in time (theta method), stability and convergence theorems, remarks on implementation.

A. Quarteroni, Numerical Models for Differential Problems, Springer

For further information, we refer also to:
A. Quarteroni - A. Valli, Numerical Approximation of Partial Differential Equations, Springer.
L. Formaggia - F. Saleri - A. Veneziani, Applicazioni ed esercizi di modellistica numerica per problemi differenziali, Springer.
J.C. Strickwerda, Finite Difference Schemes and PDE, Wadsworth \& Brooks Cole.

The evaluation will be based on a written exam and an oral exam.
The written exam will mainly aim to verify the most theoretical knowledge.
Instead, during the oral examination practical knowledge will be verified and the programs, developed in C or Matlab, will be evaluated.

The evaluation will be based on a written exam and an oral exam.
The written exam will mainly aim to verify the most theoretical knowledge.
Instead, during the oral examination practical knowledge will be verified and the programs, developed in C or MatLab, will be evaluated.