# 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 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. Existence and uniquenes of solutions of ODEs. Solution of simple ODEs and simple (linear) PDEs. Fourier series and transform. Norms and function spaces.
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).

### Timetable

Week Date JCS and SM N Subject Read on your own Some relevan exam questions
2 10.01, 13.01 ch. 1-2,3, 4.1-4.2 Introduction to the course. Difference operators and difference formulae. Classification of linear PDEs. Methods for parabolic problems. Preparation to the first two assignments: Boundary value problems. Discretization of the heat equation. Ch. 1, last part of Ch 3.1.2, beginning of Ch. 3.2, Ch. 3.3. Problem 1a, 1b, 1c, May 2009. Problem 5, June 2014.
3 17.01, 20.01 Supervision of the first two compulsory assignments (Nullrommet)
4 24.01, 27.01 4.1-4.5, 6 Forward Euler, Backward Euler and Crank-Nicholson. LTE of the theta-method. Method of lines. Methods for elliptic equations. Preparation to the third assignment: Discretization of the Laplace equation in 2D. Problem 3a) exam June 2010. Exercise 1, June 2007.
5 31.01, 03.02 6, 7.1, 7.2 Methods for elliptic equations. Methods for advection equations Exercise 1, exam June 2010. Problem 2, August 2014. Problem 2, June 2014.
6 07.02, 7.1–7.4. 5.3 Methods for advection equations and hyperbolic systems. Preparation to the fourth assignment: Hyperbolic equations. Problem 4 August 2014. Problem 3 June 2012.
7 14.02, 17.02 7.5–7.6, 5.1-5.4 5.6, 5.8 CFL condition. (Ch. 7.5 is self-study). Convergence of the Euler method, is self-study (ch 5.4). Problem 4 Exam May 2013. Ch. 5.4 note. Example page 61-62 note. Problem 3, Exam May 2013.
8 21.02, 24.02 5.9, 7.6, 7.7 Stability of parabolic problems. Matrix stability. 5.5 is not part of the curriculum. 5.7 is not part of the curriculum. Lax equivalence theorem. Von Neumann stability. Ex.1b May 2004, Ex.3c June 2007, Ex.3c June 2010, Ex.2b May 2011, Ex.1b Aug 2011, Ex.3b Aug 2014, Ex. 3c June 2006 Pb 2 August 2014.
9 28.02, 03.03 6 Elliptic equations maximum principle. Project work. Pb 4 May 2013. Pb 3 June 2013
10 07.03, 10.03 Numerical linear algebra. Pb 2c Aug 2013, Pb 2 May 2013, Pb 1c Aug 2014
11 14.03, 17.03 13 JCS p 339-349 p 354-356. 14 JCS p 373-387 p 390-391 Numerical linear algebra. Finite element method.
12 21.03, 22.03, 24.03 Project work.
13 28.03, 29.03, 31.03 Project presentation and project work. Pb 2c Aug 2013, Pb 2 May 2013, Pb 1c Aug 2014
04.04, 18.04 Easter and excursion
16 21.04 SM chaper 14 p 385-399. See material in It's learning. Finite element method: Rayleigh–Ritz and Galerkin principles and methods. Pb 1: May 2009, May 2011, May 2013, Aug 2013, June 2014, Aug. 2014.
17 25.04, 28.04 14 SM p 385-399 Finite element method. Error estimastes (from Suli and Mayers). 2D Poisson equation. Pb2 June 2010, Pb2 Aug 2011, Oppg 2 Aug 2012, Oppg 4 June 2012.