Schedule
This schedule is not final but this is the anticipated curriculum of TMA4212.
JCS = John C. Strikwerda's book on finite differences
N = course note This note is still under construction (especially chapter 7). Please let me know of any mistakes you find.
Prerequisites
| Subject | Topics | subtopics |
|---|---|---|
| Linear algebra | Basics | Vector and matrix norms; Symmetric Positive Definite matrices; Inner product spaces; Linear independence; basis of a vector space; orthonormal basis. |
| Matrix factorizations | Diagonalization and orthogonal diagonalization of matrices; LU decomposition (Gaussian elimination, pivoting); Cholesky factorization; QR factorization; SVD; Jordan Canonical form; Schur factorization. | |
| Topics of interest in numerical linear algebra | Spectral radius; Gershgoring's theorem; Condition number; Neumann series. | |
| Iterative methods | Newton method; fixed point iteration; convergence of the basic iterative methods (Jacobi, Gauss-Seidel and SOR) for linear systems. | |
| Calculus | Taylor theorem |
Schedule
| Week | Date | JCS | N | Subject |
|---|---|---|---|---|
| 2 | 10.01, 12.01 | Appendix A | ch. 1-2 | Introduction to PDEs, examples of PDEs and principles of finite difference discretizations, matrix- and vector- norms. Orthogonal diagonalization of symmetric matrices. Jordan canonical form, Schur factorization. Spectral radius and norms of matrices. Linear algebra exercises. Difference operators. Difference formulae. |
| 3 | 17.01, 19.01 | ch. 1,2,3. | Linear algebra. Boundary value problems. | |
| 4 | 24.01, 26.01 | 4.1-4.5 | Parabolic problems. | |
| 5 | 31.01, 02.02 | 5.1–5.4, 5.6. | Parabolic problems. | |
| 6 | 07.02, 09.02 | 5.8–5.9, 6 | Parabolic problems. Elliptic equations. | |
| 7 | 14.02, 16.02 | Elliptic equations. | ||
| 8 | 21.02, 23.02 | 7.1–7.4. | Advection equations and hyperbolic systems. | |
| 9 | 28.02, 01.03 | Project work. No lectures. | ||
| 10 | 06.03, 08.03 | 7.4–7.7 | Advection equations and hyperbolic systems. Dissipation and dispersion. | |
| 11 | 13.03, 15.03 | Numerical solution of linear systems. | ||
| 12 | 20.03, 22.03 | Project work. No lectures | ||
| 13 | 27.03, 29.04 | Finite element method: Rayleigh–Ritz and Galerkin principles and methods. | ||
| 14 | 03.04, 05.04 | Easter holiday. | ||
| 15 | 10.04, 12.04 | Finite elements. | ||
| 16 | 17.04, 19.04 | Finite elements. | ||
| 17 | 24.04, | Project presentation |