04524 - Numerical Analysis

Course Unit Page

Academic Year 2021/2022

Learning outcomes

At the end of the course, students thoroughly know the theoretical and computational properties of the principal numerical methods for initial value ordinary differential equations, and of some advanced numerical methods for partial differential equations. In particular, students are able to analyze the theoretical properties of numerical methods and to critically examine the computational results.

Course contents

Review on finite arithmetic computation, on approximation theory, on bases of polynomial representation, on the solution of linear systems;

application of these concepts to the solution of ordinary differential equations: finite difference and spectral collocation methods.

Bivariate approximation of functions and solution of partial differential equations: finite difference and spectral collocation methods.

Spaces of spline functions and rational splines (univariate and bivariate case), local representation bases (B-spline), evaluation, derivation, integration, refinement and degree elevation tools, convergence properties of the spline approximation.

Interpolation and least squares approximation (discrete and continuous case) with spline functions.

Collocation and finite element methods for ordinary differential equations with boundary conditions;

Rational spline and spline shape approximation and curved domain design.

Collocation, finite elements and IsoGeometric Analysis (IGA) methods for partial differential equations with boundary conditions of a curved domain.

Introduction to the generalization of polynomial spaces and splines to Chebyshevian spaces.

The course includes a laboratory activity in which the student will be able to put into practice and experiment the proposed computational methods.

Readings/Bibliography

1.N.J.Higham, Accuracy and Stability of Numerical Algorithms, second edition, SIAM, 2002.

2.M.J.D.Powell, Approximation theory and methods, Cambridge University Press, 1981.

3.C.de Boor, A practical guide to splines, Springer Verlag, 1978.

4.L.Piegl, W.Tiller, The NURBS book, 2nd Edition, Springer, 1997.

5.J.A.Cottrel, T.J.R.Hughes, Y.Bazilevs, Isogeometric Analysis, John Wiley & Sons, Ltd, 2009.

Teaching methods

Lectures and exercises in the computer laboratory. The exercises consist in the analysis / development and use of Matlab scripts, concerning the numerical methods proposed in class. The exercises will be guided by the teacher and aimed at a better understanding of the theory as well as increasing the student's computational skills.

Assessment methods

The exam consists of an oral discussion on the topics covered in class, on the LAB exercises proposed during the course and on an in-depth study, chosen by the student, on a LAB exercise.

Teaching tools

Teacher's pantries on some topics, slides and Matlab code.

Use of the open-source software package for Octave / Matlab GeoPDEs.

Links to further information

http://www.dm.unibo.it/~casciola/html/anmat2122.html

Office hours

See the website of Giulio Casciola