Lecture plan
The curriculum is taken from A. Quarteroni (Q), Numerical Models for Differential Problems, Springer 2008.
We will also use some material from Brenner & Scott (BS): The Mathematical Theory for Finite Element Methods, Springer 2008.
The topics included are:
- Introduction
- The Poisson equation:
- Weak formulation
- Finite element method
- Implementation
- Error analysis
- Finite element function spaces
- Abstract formalism
- Time dependent problems
- Grid generation
- Error estimation and adaptivity
- Diffusion-advection-reaction problems.
Schedule
Lectures: Tuesday and Thursday.
Tutorials: Wednesday.
Known changes are listed.
The topics will be described as we proceed.
| Lecture | Topics | Reading |
|---|---|---|
| 20.08 | Introduction. Weak formulation of an 1-D PDE. | BS: 0.1-0.4, Q: 3.2.1 |
| 21.08 | The linear FEM for an 1-D PDE. Weak formulation and minimization principle | Q: 2.1, 3.2.1, 4.3.1. |
| 28.08 | Definitions of Lebesgue and Sobolov spaces (1D) Lax-Milgram theorem (yet without proof). Note the assumption for Lax-Milgram! Analysis of the Galerkin method! Boundary conditions, and how they are included in the weak formulation (1D). | Q: 2.1, 3.5, 4.1-2. (4.1-2 should be known by heart!) |
| 29.08 | No tutorial | |
| 30.08 | The \(X_h^r \) spaces. Implementation in the 1D case. | Q: 4.3 - 4.3.3. A note and some corresponding slides. |
| 04.09 | More on implementation in the 1D case. Interpolation operator and error estimates. | Q: 4.3.4-4.3.5. |
| 05.09 | No tutorial session. But I will be available at my office 1348, SBII for questions regarding the course, including the exercises. | |
| 06.09 | Poisson equation in \(\mathbb{R}^d\). Definitions of the Lebesque and Sobolev spaces in \(\mathbb{R}^d\). The trace operator. | Q: 2.4, 3.3.1 and 3.3.3. |
| 11.09 | No lecture (Both Kjetil and I am out of town this week) | |
| 12.09 | No tutorial | |
| 13.09 | No lecture | |
| 18.09 | Triangulation of the domain. The abstract definition of a finite element. Lagrangian FE. The \(\mathbb{P}_r\) spaces. Barycentric coordinates. Use of reference element. | Q: 4.4 |
| 19.09 | Implementation of FEM in 2D. How to build a stiffness matrix, numerical quadratures. | Q: 4.5, see also Q: 8.1-4. |
| 20.09 | Finite element spaces | BS: chapter 3 (just an overview). |
| 25.09 | The Lax-Milgram Theorem | BS: 2.7 |
| 26.09 | Project: Part 1 | |
| 27.09 | Error estimates. | Q: 4.5.3 |
| 02.10 | Error estimate in the \(L^2 \) norm. Elliptic Regularity | Q: 4.5.4. (see also BS: 5.5) |
| 03.10 | Tutorial | |
| 04.10 | Parabolic problems | Q: 5.1 and 5.4. Started on 5.2. |
| 09.10 | A priori error estimates (just an overview) Spectra of the Laplace operators Condition number for the discrete Laplacian \( A_h\). | Q: 5.2 and 4.5.3 A note by Einar Rønquist. |
| 10.10 | Tutorial | |
| 11.10 | "Practical" error estimates | Q: 4.6-4.6.3. |
| 16.10 | Stress, strain and linear elasticity | Notes on It's learning. |
| 17.10 | Tutuorial | |
| 18.10 | Mesh generation. Moving mesh in 1D. Structured grids in 2D | Q: 6.1-6.3 |
| 23.10 | Unstructured grid: Delaunay grid, advancing front technique and regularization techniques. For more on grid generation, see Handbook of Grid Generation | Q: 6.4-6.5. |
| 24.10 | Project: Part 2 | |
| 25.10 | Grid generation with Gmsh. Solution of linear systems: CG. | Q: 7-7.2.1: Self study. Started on 7.2.2. |
| 30.10 | Conjugate gradient (CG) method. Preconditioning. GMRES. | Q: 7.2.2-7.2.3. |
| 31.10 | Tutorial | |
| 01.11 | Diffusion-transport-reaction equations | Q: 11.1-11.2, 11.6 |
| 06.11 | Tutorial (no lecture) | |
| 07.11 | Tutorial | |
| 08.11 | Tutorial (no lecture) | |
| 13.11 | Diffusion-transport-reaction equations | Q: 11.3-5 |
| 14.11 | Tutorial | |
| 15.11 | Tutorial (no lecture) | |
| 16.11 | Deadline for the project | |
| 20.11 | Project presentations. | |
| 21.11 | Project presentations. | |
| 22.11 | Summary |