Lecture plan
The curriculum is taken from A. Quarteroni, Numerical Models for Differential Problems, Springer 2008.
We will also use some material from Brenner & Scott: The Mathematical Theory for Finite Element Methods, Springer 2008.
The topics will be lectured more or less in the following sequence:
- The 1-D case as an introductory example. Taken from 3.2 and 4.3
- Some functional analysis: 2.1-2.6
- Elliptic equations: Chap. 3.1-3.6
- The Galerkin method: 4.1-4.6
- Parabolic equations: 5.1-5.5
- Grid generation: 6.1-6.5.
- Solution of linear systems: 7.1-7.2: This is better covered by other courses, and will therefore be touched only briefly in this course, if at all.
- How to set up a FEM code: Chap. 8. The principles will be discussed, but we will not dig into the code described in the book.
- Diffusion-advection-reaction equations: 11.1-11.9
- Optimal control problems: Chap. 16. If time permits.
Lectures:
- 25-26.08: A one dimensional example: From Poisson equation to the variational form to the minimization problem. The Galerkin method for this problem.
- 01.09: Lax-Milgram theorem: 2.1 and 3.5. For a proof, se Brenner & Scott, 2.4-2.5, 2.7.
- 02.09: A little bit about distributions: 2.3.
- 08.09: 3.3.1 - 3.3.3. Selfstudy: 3.4.
- 09.09: 4.1 - 4.3.
- 15.09: 4.4.
- 19.09: Implementation of FEM in 1D. See the notes at It's learning.
- 22.09: Implementation of FEM in 2D. See the notes at It's learning, and chap. 4.5.1.
- 26.09: 4.5.3.
- 29.09: Spectra of the continuous and discrete Laplace, see notes at It's learning, and chap. 4.5.2.
- 30.09: 5.0-5.1.
- 06.10: 8.2 and grid-generation by Gmsh
- 27.10: 4.6-4.6.3
- 28.10: 6.1
- 03.11: 6.2-6.5. For more on grid generation, see Handbook of Grid Generation.
- 07.11: 7.2.2. CG with and without preconditioning.
- 10.11: 7.2.3. GMRES
- 11.11: 11.2-11.3 Diffusion-advection-reaction equations in 1D.
- 17.11: 11.4-11.6
- 18.11: 11.1