Mathematics

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.