Tentative Plan of Lectures

Week Topics Reading Comments Exercises
2 Introduction
Elliptic boundary value problems in 1d 		Slides: Introduction (pdf)
Numerical solution and testing of the 1d Poisson problem (ipynb)

Background: BO section 2.1 - 2.2.2 		The 1D Poisson problem: Difference scheme, implementation, error analysis and verification. Partly described in BO section 3, but with a different convergence proof.
The discrete maximum principle: A general result is given in BO sec. 6.9. 		Exercise 1: (pdf)
3 BO section 2.3-2.2
BO section 3 (some of it where done last week). 		Definition of consistency, convergence and stability, and how they relate. Difference operators.
Treatment of Neumann boundary conditions.
After these two weeks you should be able to: For a given BVP on an interval with boundary conditions to:
* Set up a finite difference scheme (including BC)
* Find the truncation error
* Find some bound for the global error, and prove convergence. 		Exercise 2: (pdf)
4 Parabolic PDEs BO section 4 and 5.4. Forward- and backward Euler, Crank-Nicolson.
Computational stencils.
Semi-discretization (also for nonlinear problems)
Treatment of boundary conditions.
Convergence of Forward Euler for the diffusion equation (BO 5.4).		 Exercise 3 (mandatory): (pdf)
5 BO section 5. Well posed problem
Consistency, convergence and stability.
Lax Equivalence theorem, (and how to use it), von Neumann stability. 		Exam problems:
Aug 2013, Problem 2 a) and b).
May 2013, Problem 3 and 4,
June 2018, Problem 2
May 2019, Problem 1 and 3
6 Hyperbolic PDEs BO section 5.3
BO section 7.1-7.2 		A bit more on von Neumann stability. Conditional consistency.
Domain of dependence.
Examples of hyperbolic equations. Characteristics. 		Exam problems: May 2024, Problem 2a),
May 2019,
Problem 2a), June 2018, Problem 1a).
7 BO section 7.3-7.7 Boundary conditions. Some schemes, and construction of schemes. Dissipation and dispersion. Exercise 4 (pdf)
Exam problems: May 2019, Problem 2b) and Problem 4
June 2018, Problem 1 b-c),
August 2014, Problem 4.
8 Note on conservation laws (pdf) Conservation laws
Elliptic PDEs BO section 6-6.2 The 5-point formula for the Poisson equation on a unit square. Convergence (by the maximum principle)
Implementation: (ipynb)		 Exercise 5 (pdf)
9 BO section 6.2.1-6.6, 6.9-6.10 Boundary conditions. Difference formulas on arbitrary grid. Exam problems: Aug 2014, Problem 2,
May 2019, Problem 5
10 Finite element methods CC section 1-3.2 How to set up the variational form (1d), grids and basis function, imposing boundary conditions
The space \(H^1(\Omega)\) and \(H^1_0(\Omega)\), and the Lax-Milgram theorem. Poincaré's inequality. 		Exam problems: June 2014, Problem 1
May 2019, Problem 6
11 Solution of Problem 7 Optional problems in TMA4145 (pdf) (proof of Lax-Milgram)
CC Section 3.2-4.3
Note on assembly in the 1d case (pdf)
Example of an implementation (ipynb)		 Proof of Lax-Milgram.
Galerkin orthogonality and Cea's lemma.
Assembly (how to set up the linear system) 		Exam problems: May 2017, Problem 1
May 2023, Problem 3
12 CC section 4.4-5.1 Interpolation errors and error estimates in 1d.
2d Poisson equation.
13 CC section 5.2-6 The finite element method in 2D. Implementation issues: Reference element, elemental matrices and assembly process
14 Numerical linear algebra Quarteroni et.al. ch. 4.2.1-2, 4.3.3-4, alternatively Strikwerda ch. 13.1-3 and 14.1-3. The latter have more focus on the linear algebra for discretised PDEs. Line search methods: Steepest descent and conjugate gradient methods (CG). Derivation, orthogonality properties, and convergence of CG.
Classical iterative methods: Jacobi, Gauss-Seidel, SOR. Convergence criteria.
15 Monday: No lecture
Tuesday: Old exams, or repetition. Please send me suggestions for topics.
2025-03-31, Anne Kværnø