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Metodi Numerici per le Equazioni alle Derivate Parziali

academic year:	2013/2014
instructor:	Elisabetta Carlini
degree course:	Mathematics for applications (magistrale), I year
type of training activity:	caratterizzante
credits:	6 (48 class hours)
scientific sector:	MAT/08 Analisi numerica
teaching language:	italiano
period:	II sem (03/03/2014 - 13/06/2014)

Lecture meeting time and location

Presence: highly recommended

Module aims: The main goal is to introduce the students to the numerical methods for partial differential equations, with a particular emphasis on finite element schemes. The methods will be analyzed on linear model problems (stationary and evolutive ones) in two variables, with a discussion on the effective implementation of the codes. Some laboratory sessions will be devoted to the solution of problems (in Matlab, Freefem, C or Fortran languages).

Module subject: Numerical approximation of partial differential equations, which are of interest in the applications, and the main numerical methods used to solve them. Recall of the main theoretical results (students are supposed to have already attended a basis course on PDE). Recall of the standard finite difference schemes d for the numerical study of linear elliptic, parabolic and hyperbolic problems in two dimensions. Finite element methods for elliptic and parabolic problems. Computer sessions.

Hyperbolic problems. Summary of transport problems in one and two dimension. Finite difference schemes. Analysis of convergence and properties of dispersion and diffusion. 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. A semi-discretization approach based on finite elements in space and finite differences in time (theta method), stability and convergence theorems. Computer implementation of the methods.
Nonlinear problems. Hamilton-Jacobi-Bellman. Recall of essential results of existence and uniqueness for viscosity solutions. Finite difference and Semi-Lagrangian schemes. Consistency, stability and convergence analysis. Computer implementation of the methods.

Detailed module subject: Programma dettagliato svolto a lezione e Modalità d'esame

Suggested reading: A. Quarteroni, Modellistica numerica per problemi differenziali, Springer, V ediz. 2012
A. Quarteroni - A. Valli, Numerical Approximation of Partial Differential Equations, Springer, 1994
L. Formaggia - F. Saleri - A. Veneziani, Applicazioni ed esercizi di modellistica numerica per problemi differenziali, Springer, 2005
J.C. Strickwerda, Finite Difference Schemes and PDE, Wadsworth & Brooks/Cole, Pacific Gr., 1989

Issues:

Type of course: standard

Exercises:

Examination tests:

Knowledge and understanding: Successful students will be able to have a basic knowledge of the techniques for solving linear partial differential equations. They will also achieve some fundamental notions about convergence, stability, a-priori error estimates and complexity of the algorithms.

Skills and attributes: Successful students will be able to write simple codes for the solution of linear partial differential equations and to analyze their results. During the laboratory sessions on PC, the will use software tools like Matlab or Gnuplot.

Personal study: the percentage of personal study required by this course is the 65% of the total.

Examination dates on Infostud

Statistical data on examinations