Timetable

This schedule is not final but this is the anticipated curriculum of TMA4212.

JCS = John C. Strikwerda's book on finite differences

SM = Suli and Mayers, An introduction to Numerical Analysis

N = Course note The course notes, in early form. These will likely be subject to minor revision over the course of the semester.

Timetable

Week	Date	JCS and SM	N	Subject
2	9.01, 12.01		ch. 1-3.2	Introduction to the course. Difference formulae and discretizations in 1D. Case study: ODE boundary value problems. Consistency, stability and convergence.
3	16.01, 19.01		ch. 3.3, 4.2.1-4.2.2, 4.3.1	Example: nonlinear ODE. First steps in 2d: full discretization of the heat equation with 1 spatial variable. The semi-discretization principle. (Note: Friday's lecture will be replaced by an exercise class, in the same room)
4	23.01, 26.01		ch. 4, 6.1-2	Further treatment of parabolic equations. Discretization of 2d Laplace equation on uniform grid
5	30.01, 02.02		ch. 6.1-8	Discretisation of 2d Laplace equation and other elliptic PDEs
6	06.02, 09.02		ch. 7 (minus stability analysis)	Hyperbolic equations: characteristics and discretizations
7	13.02, 15.02		ch. 5	Convergence and stability 1.
8	20.02, 23.02		ch. 5, 6.10, 7.4, 7.7	Project release and discussion. Convergence and stability 2.
9	27.02, 02.03	JCS ch. 13.1-4		Numerical linear algebra 1.
10	06.03, 09.03			Numerical linear algebra 2. Introduction to finite elements.
11	13.03, 16.03			Project sessions (exercise classes)
12-13			Excursion. No lectures
14	06.04		Excursion end (Friday lecture only).	Finite elements 1.
15	10.04, 13.04			Finite elements 2.
16	17.04, 20.04			Project presentations and hand-in.
17	24.04			Finite elements 3.

Prerequisites

Here are the theoretical course prerequisites. If some of this seems unfamiliar, new, or forgotten, do not worry too much: relevant parts will be revised during the semester, and the course note covers many topics. The first two rows of the table below are the most important and priority should be given here.

Subject	Topics	subtopics
Calculus		Taylor's theorem. Existence and uniquenes of solutions of ODEs. Solution of simple ODEs and simple (linear) PDEs. Fourier series and transform. Norms and function spaces.
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; Gershgorin'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.
Numerical ODEs		Runge-Kutta and multistep-methods, convergence of the Euler method, order conditions and stability. Note on ODEs, exercise set 8 from the course of Numerical Mathematics (TMA4215).