## 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