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Applied Mathematics - Introduction to Numerical Analysis & Scientific Computing

Lecturer: 
Luca Heltai, Gianluigi Rozza
Course Type: 
PhD Course
Master Course
Anno (LM): 
Second Year
Academic Year: 
2014-2015
Period: 
October-February
Duration: 
60 h
CFU (LM): 
6
Description: 

This course is also part of the Joint SISSA-ICTP HPC master 

The foundations of Numerical analysis

Frontal Lectures (20 hours - Joint with MHPC master):

  • Resolution of linear systems with direct and iterative methods
  • Polynomial approximations 
  • Numerical Integration
  • Introduction to Numerical solutions of ODEs
  • Non-linear equations and systems
  • Introduction to Finite Elements

Laboratory (20 hours - Joint with MHPC master, in parallel with the frontal lectures):

  • Introduction to python
  • Introduction to numpy
  • Vectors, Matrices and Linear Solvers 
  • Polynomial approximations
  • Numerical integration, polynomial projection
  • Introduction to Numerical solution of ODEs
  • Non-linear equations and systems
  • Introduction to Finite Elements using FENICS (python finite element toolbox)

Advanced Numerical Methods for PDEs

Frontal Lectures (20 hours, optional for MHPC):

  • The Ritz Method and the Galerkin Method
  • Finite Element Methods
  • Interpolation of Sobolev Spaces and Error Estimates
  • Finite Element Methods for Second Order Elliptic Equations

Teaching support (slides, reports, notes, links)

http://elearn.mhpc.it/moodle/course/view.php?id=3

References and Text Books:

  • A. Quarteroni, R. Sacco, and F. Saleri. Numerical mathematics, volume 37 of Texts in Applied Mathe- matics. Springer-Verlag, New York, 2000.
    [E-Book-ITA] [E-Book-ENG]
  • A. Quarteroni. Modellistica Numerica per problemi differenziali. Springer, 2008.
    [E-Book-ITA]
  • A. Quarteroni. Numerical Models for Differential Problems. Springer, 2009.
    [E-Book-ENG]
  • A. Quarteroni and A. Valli. Numerical approximation of partial differential equations. Springer Verlag, 2008.
    [E-Book-ENG]
  • S. Brenner and L. Scott. The mathematical theory of finite element methods. Springer Verlag, 2008.
    [E-Book-ENG]
  • D. Boffi, F. Brezzi, L. Demkowicz, R. Durán, R. Falk, and M. Fortin. Mixed finite elements, compatibility conditions, and applications. Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 26–July 1, 2006. Springer Verlag, 2008.
    [E-Book-ENG]
  • D. Arnold. A concise introduction to numerical analysis. Institute for Mathematics and its Applications, Minneapolis, 2001.
    [E-Book-ENG]
  • A. Quarteroni, F. Saleri, P. Gervasio. Scientific Computing with Matlab and Octave. Springer Verlag, 2006.   
    [E-Book-ENG]
  • B. Gustaffson Fundamentals of Scientific Computing, Springer, 2011
    [E-Book-ENG]
  • Tveito, A., Langtangen, H.P., Nielsen, B.F., Cai, X. Elements of Scientific Computing, Springer, 2010
    [E-Book-ENG]

Note that, when connecting from SISSA, all of the text books above are available in full text as pdf files.

Location: 
A-133

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