Welcome to MATH 745 - TOPICS IN NUMERICAL ANALYSIS

Winter 2004
Time: Monday 2:00 - 3:30pm and Thursday 1:30 - 2:30 pm
Place: HH 410


Instructor: Dr. Bartosz Protas
Office: HH 326, Ext.24116
Office hours: Tuesday 2:00 - 3:00pm and Thursday 2:30 - 3:30pm, or by appointment


Outline of the Course:

 The course will focus on advanced techniques for the numerical solution of Partial Differential Equations (PDEs) with particular emphasis on theoretical aspects, such as error and stability analysis, as well as certain implementation issues. The presented methods will be illustrated using well-known equations from mathematical physics. The specific topics that will be discussed include (optimistic variant):

1) Review of classical approximation schemes for ODEs and PDEs

2) Advanced numerical differentiation
     a) Complex step derivative
     b) Padé schemes
     c) Compact finite differences

3) Review of approximation theory
     a) Function analytic background (Hilbert spaces, inner products, orthogonality, orthogonal systems in L_2)
     b) Galerkin vs. collocation methods

4) Spectral methods for PDEs
     a) Fourier vs. Chebyshev methods
     b) Fast transforms (FFT)
     c) Application to nonlinear problems (pseudo-spectral methods, dealiasing)
     d) Domain decomposition methods (spectral elements, hp refinement)

5) Multiresolution methods for PDEs
     a) Orthogonal wavelets
     b) Discrete wavelet transform (DWT)
     c) Multiresolution representation of data

6) Finite Element Method (FEM)
     a) Finite element basis functions
     b) Approximation in curvilinear coordinate systems
     c) Implementation of boundary conditions

7) Boundary Element Method (BEM)
     a) Derivation and solution of Boundary Integral Equations


Reference:

 W. H. Press, B. P. Flanner, S. A. Teukolsky and W. T. Vetterling, ``Numerical Recipes: the Art of Scientific Computations'', Cambridge University Press, Cambridge, (1986) (also available on-line free of charge at www.nr.com). This book will serve as a point of departure. Additional material will be made available by the instructor during the course.

Prerequisites:

 Numerical Analysis / Numerical Methods, Partial Differential Equations, basic programming skills in FORTRAN, C/C++, or Matlab

Grades:

 The final grades will be based on a take-home project.