Course teached as: B018805 - COMPLEMENTI DI ANALISI NUMERICA Second Cycle Degree in MATHEMATICS Curriculum APPLICATIVO
Finite difference methods for 1D boundary value problems. Variational formulation of a boundary value problem (1D e 2D). The finite element method for stationary and parabolic problems. The finite elemnt method for Navier-Stokes. Splines and the B-spline basis. Elements of isogeometric analysis.
- W. Gautcshi (2012), Numerical Analysis, Second Edition, Birk ?auser, New York.
-A. Quarteroni (2012), Modellistica Numerica per Problemi Differenziali, quinta edizione, Springer–Verlag, Milano.
-L. Formaggia, F. Saleri, A. Veneziani, Solving numerical PDE's: problems, applications, exercises, 2012, Springer-Verlag, Milano, Italia.
-C de Boor (2001), A Practical Guide to Splines, Revised edition, Applied Mathematical Sciences 27, Springer–Verlag, New York
.- L. L. Schumaker (2007) Spline functions: basic theory, Cambridge Mathematical Library.
Knowledge: finite difference schemes for 1D boundary value problems; variational formulaton of a boundary value problem; the finite element method; splines and their application in isogeometric analysis.
Expertise: being able to implement and evaluate a numerical scheme for boundary value problems; familiarity with splines, with Matlab toolbox on splines and with their applications.
Basics of Numerical Calculus (finite arithmetic, zeros of functions, numerical methods for linear systems) and knowledge of the Matlab environment and programming language.
6 hours a week, 4 in classroom (frontal teaching) and 2 in computer lab (implementation of the introduced schemes and numerical experiments)
Type of Assessment
The exam consists in an oral interrogation and in discussing a student's work developed in Matlab.
The centered and upwind finite difference methods for 1D boundary value problems: a model problem, the diffusion-transport problem, the general second order nonlinear problem. Basics on Sobolev spaces. Variational formulation of a boundary value problem (1D and 2D). Galerkin discretization. Finite elment method for stationary and parabolic problems. Gradient and conjugate gradient methods. Mesh adaptivity. Navier-Stokes equations. The finite element method for the Stokes problem. Some hints to the non linear case. Splines and the B-spline basis. Curry-Schoenberg theorem. Witney-Schoenberg theorem. Elements of isogeometric analysis.