Plan of Lectures
- BO = Note on finite difference methods by Brynjulf Owren
- CC = Note on finite element methods by Charles Curry
| 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, \(L^\infty\)-stability, and monotone schemes (not in BO - see note "Week 2" below). Difference formulas and difference operators. Double-corrected lecture notes: Week 2 Jupyter notebook (python code): Numerical solution and testing of the 1d Poisson problem | Read yourselves: Section 2 in BO (Most of it is known from before) Experiment with the Jupyter-file. | |
| 3 | BO section 3. Diagonally dominant and tridiagonal matrices. Vector, matrix and function norms. Grid functions, interpolation. (Some of this not in BO - see note "Week 3" below) Error analysis of a difference formula of the 1-dimensional Poisson problem in \(L^2\). Dirichlet, Neumann and mixed boundary conditions. General linear ODEs, selfadjoint ODEs. Lecture 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. Monotone methods and the discrete max principle, explicit/implicit methods. Consistency. Stability in vector \(p\)-norm and function \(L^p\)-norms. Handwritten notes: Week 4 Slides: Week 4 | ||
| 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. |
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| 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, monotonicity, and the discrete maximum principle. Irregular domains: (I) Modify FDMs near boundary, (II) Fatten the boundary. Handwritten notes: Week 6 Slides: Week 6 | 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. Methods of undetermined coefficients. Beehive grid. Sparsity, monotonicity, 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 | Monotonicity is not described in BO. See the handwritten notes and slides. The discrete max principle (DMP): A general statement and proof is given in the slides (elliptic problems, no time). Read yourselves: Converting to zero B.C., see end of lecture note. | ||
| 8 | Hyperbolic PDEs, 1d+time | BO section 7. Hyperbolic equations. Characteristics. Model problems: The wave and transport equations. Conservation laws. Discretisation of the transport equation. Consistency and von Neumann stability. Handwritten notes: Week 8 Slides: No slides. | Espen in Germany. Some technical problems in lectures with sound and zoom. | 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, 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: No slides. | Monotonicity (and convergence of monotone 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: 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 | June 2018 P3, May 2016 P1, May 2015 P4. |
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| 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 Slides: No slides. Jupyter nb: A P1 FEM for 1d Poisson | Assembly process: See handwritten lecture notes (more details added). See also jupyter notebook 1dFEM.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 | CC section 3, 4, 5.1-5.2. 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 \[\partial_x(\kappa(x)\partial_x u)+cu=f(x)\quad\text{in}\quad (0,1)\] Galerkin orthogonality, Cea's lemma, and best approximations/projections in \(V\). \(H^1\) 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 12 Slides: Week 12 | May 2017 P1. | ||
| 13 | CC section 5.2-5.5. More on 2D-problems: Element wise computations, reference element, assembly. Thursday lecture cancelled. Handwritten notes: Week 13 Slides: Week 13 | May 2015 P1, June 2012 P4. |
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| 14 | Numerical linear algebra | 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. 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. Handwritten notes: Week 14 Slides: Week 14 | Read yourselves: Implementation of CG, Strickwerda 14.3. | May 2017 P3, June 2014 P3, May 2013 P2 August 2013 P2c). |
| 15 | Easter vacation - No teaching | |||
| 16 | Tuesday 19.04.: No lecture. Thursday 21.04.: Exam problems, info about the curriculum and exam. I did most of Problem 1 from Exam 2019 in a way that is closely related to my way of presenting monotonicity, consistency, stability and convergence. In the lecture, I did not have time to give the conclusion of part (b). You can find it the notes below along with a discussion how to get error bounds and convergence rates from what we already did. These notes are highly relevant for this years exam… Handwritten notes: Week 16 Slides: No slides | |||
| 17 | Tuesday 26.04.: Final lecture: Supervision Project 3. Thursday 28.04.: No lecture. | |||
| 22 | Friday 03.06.: Questions and answer session | 14:15-16:00 in F6 (E and A) | ||
| 23 | Tuesday 07.06.: Questions and answer session | 10:15-12:00 in F6 (E in F6, A on Zoom) | ||
| Exam June 8, 9:00-13:00. | ||||