Lecture plan
The schedule is tentative. Details will be given as we go along.
Teaching material:
- Note on finite difference methods by Brynjulf Owren (BO)
- Note on finite element methods by Charles Curry (CC)
- J.W. Thomas: Numerical Partial Differential equations, Finite Difference Methods and Conservation Laws and Elliptic Equations (JWT)
- J.C. Strikwerda: Finite Difference Schemes and Partial Differential Equations (JCS)
| Date | Topics and reading | Relevant exam questions | |
|---|---|---|---|
| Introduction. Stationary (elliptic) PDEs | |||
| Week 2 | Introduction, classification of linear PDEs (nothing particular to read). Finite difference discretisation and error analysis of a 2-point boundary value problem. This is to some extend covered by BO 3.1, and is further discussed in Exercise 1. As background material, we have used and will to a big extend continue to use material from BO section 2, so please read it yourself. | ||
| Week 3 | Boundary conditions for BVPs and discretizations of those. More general BVPs. BO 3.1.2-3.3. Finite element methods for a 2-point boundary value problem. How to find the weak form of the problem. Sobolov space. Lax-Milgram theorem with proof and Cea's lemma. Galerkin method on a linear finite element space. The topics are covered by CC:1-2.2, 3-3.2, 4.1-4.2. | ||
| Week 4 | FEM for BVPs cont. CC –4. Poincare inequality. How to include other boundary conditions. Implementation, error estimates for the linear element method. Illustration of the assembly process. | ||
| Week 5 | Finite difference methods on elliptic problems. 5-point formula for the Poisson equation. BO 6.1-2. Poisson.ipynb Arbitrary grids, the discrete maximum principle and error analysis: BO: 6.6, 6.8-6.11 | ||
| Week 6 | A bit more on elliptic problems BO: 6.3-6.5, 6.7. Thursday: Guest lecture by Nick Trefethen. | ||
| Numerical linear algebra. | |||
| Week 7 | Classical iterative methods (Jacobi, Gauss-Seidel, SOR) and line search methods (Steepest descent, CG), both with application to linear systems of equations coming from discretisation of PDEs. JCS: 13.1-5. | ||
| Week 8 | Line search methods, CG in particular, in general and applied to discretised PDEs. JCS 14.1-3, see also cg.ipynb where discrete Poisson equation on a unit square is solved. No lecture on Thursday. | May 2017, problem 3. June 2014, problem 3. May 2013, problem 2. August 2013, problem 2c). | |
| Time dependent (parabolic and hyperbolic) PDEs | |||
| Week 9 | Finite difference methods for parabolic equations, with emphasis on the discretisation of the heat equation. BO 4. | ||
| Week 10 | Error analysis of time dependent problems. Local truncation error, consistency. Matrix and von Neumann stability. Lax equivalence theorem. BO 5. | June 2018, Problem 2. August 2013, Problem 2. June 2013, Problem 3 and 4. May 2011, Problem 2. June 2010, Problem 3. | |
| Week 11 | Hyperbolic problems. Examples of hyperbolic problems. Characteristcs. Domain of dependency and the CFL conditions. BO 7.1-2. | August 2014, problem 4. | |
| Week 12 | Some examples of methods for the transport equation, and consistency and stability analysis. BO 7.3-5. Dispersion and dissipation BO 7.7 | June 2012, problem 3. August 2013, problem 2. June 2014, problem 4. | |
| Week 13 | Systems of equations, and treatment of boundary conditions. BO 7.6 Conservation laws A short note from todays lecture. If you are interested in more, see e.g. Randall LeVeque Numerical Methods for Conservation Laws. | ||
| Week 14- | No lectures, only project work | ||