Fall 2004
Time & Place: 1:30 - 2:30pm Mondays (in HH207) & 3:30 - 5:00pm Wednesdays (in HH410)

Instructor: Dr. Bartosz Protas
Office: HH 326, Ext. 24116
Office hours: 2:30 - 3:30pm Tuesdays & Wednesdays

Outline of the Course:

The course is intended as a follow-up (but not a repetition!) of the course MATH745 taught in the Winter 2004. However, attendance of the previous course is not required as a prerequisite, hence the present course is open to new students. The focus of the course will be on advanced techniques for numerical solution of PDEs with particular emphasis on theoretical foundations of the methods as well as their implementation. 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 Approximation Theory
     a) Function-analytic background (Hilbert spaces, inner products, orthogonality and orthogonal systems)
     b) Best approximations
     c) Interpolation theory

2) Multiresolution Techniques
     a) Continuous and discrete wavelet transform
     b) Orthogonal / biorthogonal wavelets, frames
     c) Multiresolution representation of data, data compression and preconditioning of operators

3) Finite Element Method (FEM)
     a) The Galerkin method and its variants
     b) Finite element spaces
     c) Implementation issues: mesh generation, quadratures and data structures

4) Boundary Element Method (BEM)
     a) Derivation
     b) Solution of Boundary Integral Equations


Appropriate material will be made available by the instructor during the course.

Supplemental Reference:

     a) K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, Springer (TAM 39), (2001)
     b) A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer (AMS 159), (2003).
     c) S. Mallat, A Wavelet Tour of Signal Processing (Wavelet Analysis & Its Applications), Academic Press, (1999).


Numerical Analysis / Numerical Methods, Partial Differential Equations, basic programming skills in Matlab


The final grades will be based on
     a) three 20 min quizzes (3 x 15% = 45%), and
     b) a take-home project (55%).
Quiz dates:   October 4, November 1, November 29