Tentative Plan of Lectures

  • Positive coefficients, monotonicity, and discrete max principle are only described in the handwritten notes and slides.
    Precise references are given in 'Curriculum' under 'Course info' in the menu.
Week Topics Literature Comments Relevant Exam problems
2 Introduction.

Elliptic boundary value problems in 1d 		BO section 2, 3.1 (until 3.1.1)

Introduction to the course.
Error analysis of a difference formula of the 1-dimensional Poisson problem in \(L^\infty\).
Truncation error, consistency, convergence.
The discrete max. principle (DMP), \(L^\infty\)-stability, schemes with positive coefficients, and monotone schemes.
Difference formulas and difference operators.
Positive definite, diagonally dominant, and tridiagonal matrices.

Handwritten notes: Week 2 (Fixed pf DMP 17.01.)
Slides: Week 2

Jupyter notebook (python code):
Numerical solution and testing of the 1d Poisson problem

Positive coeffcients, Monotonicity, and DMP are not discussed in BO.

See handwritten notes and slides. Precise references are given in 'Curriculum' under 'Course info' in the menu.

Read yourselves: Section 2 in BO
(Most of it is known from before)

Experiment with the Jupyter notebook.

3 BO section 3.

Vector, matrix and function norms. Grid functions, interpolation. Error in vector norms vs error in function norms. (Some of this not in BO - see note "Week 3" below)
Error analysis of a difference approximation of the 1-dimensional Poisson problem in \(L^2\).

Dirichlet, Neumann and mixed boundary conditions.
General linear ODEs, selfadjoint ODEs.

Handwritten notes: Week 3

June 2014 P5.
4

Parabolic PDEs, 1d+time 		Nonlinear 1d elliptic problems (BO sec. 3).

BO section 4 (and 5).

Remarks on parabolic PDEs, grids, and grid functions.
"Forward" and "Backward Euler" methods for the heat equation. Explicit/implicit methods. Consistency.

Schemes with positive coefficients, monotone methods, and the discrete max principle.
Stability in vector \(p\)-norm and function \(L^p\)-norms.

Handwritten notes: Week 4
Slides: Week 4

Jupyter notebook (python code):
Numerical solution and testing of the 1d heat equation (forward Euler in time)

Experiment with the Jupyter notebook.

5 BO sections 4 and 5, except 5.5

Convergence and error estimate for "forward" and "backward Euler" scheme for the heat equation. Interpolation and uniform convergence.

General (linear and non-linear) parabolic PDEs, semidiscretization/method of lines, Crank-Nicolson.

More on stability, consistency, convergence. Sufficient conditions for convergence, and Lax Equivalence theorem. Von Neumann stability analysis.

Handwritten notes: Week 5
Slides: Week 5

Read yourselves:
a) The \(\theta\)-method (BO sec. 4.2.5) and stability of this method (BO p. 55-56)

b) Read BO sec 5.3 on Domain of dependence.

c) Discretization of B.C. (BO sec. 4.4 - similar to 1d Poisson!)

OBS: You should be able to do simple convergence proofs.

June 2018 P2.
August 2013 P2,
June 2013 P3, P4;
May 2011 P2,
June 2010 P3.
6 Elliptic PDEs in 2d Von Neumann stability analysis.

BO section 6.

The 5-point formula for the 2d Poisson equation on square domain/grids. Consistency, stability and convergence. Error estimate. Sparsity, positive coefficients, and the discrete maximum principle.

Irregular domains: (I) Modify FDMs near boundary, (II) Fatten the boundary.

Handwritten notes: Week 6
Slides: Week 6

Jupyter notebook (python code):
Numerical solution and testing of the 2D Poisson problem.

Experiment with the Jupyter notebook.

The method of fattening the boundary is not described in BO.

See the handwritten notes and slides.

August 2014 P2,
June 2014 P2,
June 2010 P1.
7 BO section 6.

Neumann, Robin, and mixed boundary conditions. Self-adjoint problems. Variable step size in rectangular grids. Irregular grids. Method of undetermined coefficients. Beehive grid.

Sparsity, positive coefficients, and the discrete maximum principle. Error analysis and convergence.

FDMs through integration. Box-integration. Triangular grids. Monotone and conservative schemes.

Converting to 0 boundary condition.

Handwritten notes: Week 7
Slides: Week 7

The discrete max principle (DMP): A general statement and proof is given in the slides (elliptic problems, no time).

Positive coefficients, monotonicity, and DMP are not described in BO.

See handwritten notes and slides. Precise references are given in 'Curriculum' under 'Course info' in the menu.

8 Hyperbolic PDEs, 1d+time BO section 7.

Hyperbolic equations and systems. Diagonalization yields decoupled systems. Method of characteristics. Boundary conditions only at inflow. Model problems: The transport equations.

Discretisation of a BVP for the transport equation. Boundary conditions: Reachable boundary for scheme vs inflow boundary for equation. Consistency, positive coefficients, hyperbolic DMP, remarks on error analysis, and von Neumann stability. Explicit schemes, upwind scheme.

Handwritten notes: Week 8
Slides: Week 8

 June 2018 P1,
May 2016 P2,
August 2014 P4,
June 2014 P4,
June 2012 P3.
9 BO section 7.

Explicit and implicit schemes for transport equations. Consistency, positive coefficients/monotonicity, von Neumann stability, CFL conditions. High order schemes.

Discrete max principle and \(L^\infty\)-stability and convergence of monotone schemes.

Domain of dependence, dissipation and dispersion, systems of first order hyperbolic PDEs.

Handwritten notes: Week 9
Slides: Week 9

Positive coefficients/monotonicity (and convergence of such schemes) is not described in BO.

See the handwritten notes and Exercise 3.

The discrete max principle (DMP): A general statement involving the reachable boundary is given in the handwritten notes and in Exercise 3.

Read yourselves:

Von Neumann stability for Lax-Wendroff, the Leap frog method, systems of hyperbolic PDEs - BO section 7.

10 Finite element methods CC section 1, 2.1-2.2, 3.1, 5.1, 5.2

Weak formulation of PDEs, integration by parts, bilinear forms, and test functions in 1d and multi d.

Function spaces \(H^k\) and \(H^1_0\), norms and inner products.

The Galerkin method.

FEM: Subdivisions/triagulations, elements, piecewise first order polynomials \(X_h^1\), and basis (hat) functions in 1d and 2d. P1 FEM for 1d Poisson problem.

Handwritten notes: Week 10
Slides: Week 10

Jupyter notebook (python code): 1dFEM_2024.ipynb.

Start to look at the jupyter notebook on 1d FEM, but note that assembly/construction of matrices will be discussed more next week.

You could start working on Project 2.

June 2018 P3,
May 2016 P1,
May 2015 P4.
11 CC section 2.3-2.6,

Weak formulation of more general boundary value problems, lifting function, and mixed Neumann problems. Derivation of the linear system for the P1 FEM for 1d Poisson.

Implementation, element by elements computations, quadrature, and assembly process.

Handwritten notes: Week 11
Note: there is an error on the top of page 8: a(phi_{i-1},phi_{i}), and a(phi_{i},phi_{i-1}) should switch places, rather look at week 10 page 6.
Slides: Week 11

Jupyter nb: 1dFEM_2024.ipynb

Assembly process: See also jupyter notebook 1dFEM_2024.ipynb for assembly in 1d as explained in class and solution of the 1d Poisson with a P1 FEM.

Experiment with this file. Can you implement different boundary conditions and right hand sides?

12 Tuesday: CC section 3, 4.1-4.2

Continuous, coercive, and bilinear forms. Lax-Milgram and stability. The Poincare inequality. Application: Existence, uniqueness and stability for weak solutions of the Dirichlet problem for \[\partial_x(\kappa(x)\partial_x u)+c(x)u=f(x)\quad\text{in}\quad (0,1)\] Galerkin orthogonality, Cea's lemma, Energy norm, and best approximations/projections in \(V_h\).

Thursday: No lecture.

Handwritten notes: Week 12
Slides: Week 12

There is only one lecture this week.

May 2017 P1.
13 Easter vacation - No teaching
14 Tuesday: No lecture.

Thursday: CC section 4.3, 5.1-5.2.

\(H^1\) and \(L^2\) error for polynomial approximation/interpolation. \(H^1\) error estimates and convergence for \(\mathbb P_1\) FEM in 1d.

\(\mathbb P_1\) FEM for 2d Poisson. Introduction: Triangulation, elements, nodes/vertices, basis functions, linear system.

Handwritten notes: Week 14

There is only one lecture this week.

May 2015 P1,
June 2012 P4.
15




Numerical linear algebra 		CC section 5.2-5.5.

More on 2D-problems: Element wise computations, reference element, assembly.

Strikwerda chp 13.1-3 or Quarteroni et.al. chp 4.2.1-2.

On direct and iterative methods for linear systems. General iterative (fixed point) methods for linear systems. Matrix splitting and convergence. Classical iterative methods: Jacobi, Gauss-Seidel, SOR. Convergence criteria.

Handwritten notes: Week 15
Slides: Week 15

May 2017 P3,
May 2013 P2
16 Tuesday: Strikwerda chp 14.1-3 or Quarteroni et.al. chp 4.3.3-4.

Line search methods: Steepest descent and conjugate gradient methods (CG). Derivation, orthogonality properties, and convergence of CG.

Thursday: Exam problems: 2019 P1, 2023 P1.
Full error analysis for FDMs based on DMP.

Handwritten notes: Week 16
Slides: Week 16

 Read yourselves: Implementation of CG, Strickwerda 14.3.

June 2014 P3,
August 2013 P2c).
17 Tuesday: Final lecture: Exam problems, FEM.

Thursday: No lecture.

18 Tuesday 30.04.: Questions and answer session

Friday 03.05.: Questions and answer session 		14:15-16:00 in F6

13:15-15:00 in EL1

Exam Monday May 6, 9:00-13:00
2024-04-18, Espen Robstad Jakobsen