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Metodi Numerici per le Equazioni alle Derivate Parziali

academic year:	2012/2013
instructor:	Stefano Finzi Vita
degree course:	Mathematics for applications (magistrale)
type of training activity:	caratterizzante
credits:	6 (48 class hours)
scientific sector:	MAT/08 Analisi numerica
teaching language:	italiano
period:	II sem (05/03/2013 - 08/06/2013)

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: 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.

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.
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 semidiscretization approach based on finite elements in space and finite differences in time (theta method), stability and convergence theorems, remarks on implementation.
Hyperbolic problems  Numerical methods for the solution of wave equation (FD approach).
Nonlinear problems: Hamilton-Jacobi, variational inequalities.

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

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.