## 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.
• 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